Integrand size = 19, antiderivative size = 800 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\frac {\left (c d^4+a e^4\right ) \sqrt {a+c x^4}}{2 e^5}-\frac {c d^3 x \sqrt {a+c x^4}}{3 e^4}+\frac {c d^2 x^2 \sqrt {a+c x^4}}{4 e^3}-\frac {c d x^3 \sqrt {a+c x^4}}{5 e^2}-\frac {\sqrt {c} d \left (5 c d^4+7 a e^4\right ) x \sqrt {a+c x^4}}{5 e^6 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\left (a+c x^4\right )^{3/2}}{6 e}+\frac {\left (c d^4+a e^4\right )^{3/2} \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 e^7}+\frac {\sqrt {c} d^2 \left (2 c d^4+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{4 e^7}-\frac {\left (c d^4+a e^4\right )^{3/2} \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{2 e^7}+\frac {\sqrt [4]{a} \sqrt [4]{c} d \left (5 c d^4+7 a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{5 e^6 \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} \sqrt [4]{c} d \left (15 c^{3/2} d^6+5 \sqrt {a} c d^4 e^2+23 a \sqrt {c} d^2 e^4+3 a^{3/2} e^6\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{15 e^6 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e^8 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}} \] Output:
1/2*(a*e^4+c*d^4)*(c*x^4+a)^(1/2)/e^5-1/3*c*d^3*x*(c*x^4+a)^(1/2)/e^4+1/4* c*d^2*x^2*(c*x^4+a)^(1/2)/e^3-1/5*c*d*x^3*(c*x^4+a)^(1/2)/e^2-1/5*c^(1/2)* d*(7*a*e^4+5*c*d^4)*x*(c*x^4+a)^(1/2)/e^6/(a^(1/2)+c^(1/2)*x^2)+1/6*(c*x^4 +a)^(3/2)/e+1/2*(a*e^4+c*d^4)^(3/2)*arctanh((a*e^4+c*d^4)^(1/2)*x/d/e/(c*x ^4+a)^(1/2))/e^7+1/4*c^(1/2)*d^2*(3*a*e^4+2*c*d^4)*arctanh(c^(1/2)*x^2/(c* x^4+a)^(1/2))/e^7-1/2*(a*e^4+c*d^4)^(3/2)*arctanh((c*d^2*x^2+a*e^2)/(a*e^4 +c*d^4)^(1/2)/(c*x^4+a)^(1/2))/e^7+1/5*a^(1/4)*c^(1/4)*d*(7*a*e^4+5*c*d^4) *(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE (sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/e^6/(c*x^4+a)^(1/2)-1/15*a^ (1/4)*c^(1/4)*d*(15*c^(3/2)*d^6+5*a^(1/2)*c*d^4*e^2+23*a*c^(1/2)*d^2*e^4+3 *a^(3/2)*e^6)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1 /2)*InverseJacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*2^(1/2))/e^6/(c^(1/2)* d^2+a^(1/2)*e^2)/(c*x^4+a)^(1/2)-1/4*(c^(1/2)*d^2-a^(1/2)*e^2)*(a*e^4+c*d^ 4)^2*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Ellip ticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/4*(c^(1/2)*d^2+a^(1/2)*e^2)^2/a^( 1/2)/c^(1/2)/d^2/e^2,1/2*2^(1/2))/a^(1/4)/c^(1/4)/d/e^8/(c^(1/2)*d^2+a^(1/ 2)*e^2)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 12.56 (sec) , antiderivative size = 530, normalized size of antiderivative = 0.66 \[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\frac {-12 \sqrt {a} c^{3/4} d^2 e^2 \left (5 c d^4+7 a e^4\right ) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+4 c^{3/4} d^2 \left (15 i c^{3/2} d^6+15 \sqrt {a} c d^4 e^2+25 i a \sqrt {c} d^2 e^4+21 a^{3/2} e^6\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (-60 \sqrt [4]{-1} \sqrt [4]{a} \left (c d^4+a e^4\right )^2 \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )+\sqrt [4]{c} d e \left (e^2 \left (a+c x^4\right ) \left (40 a e^4+c \left (30 d^4-20 d^3 e x+15 d^2 e^2 x^2-12 d e^3 x^3+10 e^4 x^4\right )\right )+60 \left (-c d^4-a e^4\right )^{3/2} \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )-15 \sqrt {c} d^2 \left (2 c d^4+3 a e^4\right ) \sqrt {a+c x^4} \log \left (-\sqrt {c} x^2+\sqrt {a+c x^4}\right )\right )\right )}{60 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt [4]{c} d e^8 \sqrt {a+c x^4}} \] Input:
Integrate[(a + c*x^4)^(3/2)/(d + e*x),x]
Output:
(-12*Sqrt[a]*c^(3/4)*d^2*e^2*(5*c*d^4 + 7*a*e^4)*Sqrt[1 + (c*x^4)/a]*Ellip ticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + 4*c^(3/4)*d^2*((15*I)*c ^(3/2)*d^6 + 15*Sqrt[a]*c*d^4*e^2 + (25*I)*a*Sqrt[c]*d^2*e^4 + 21*a^(3/2)* e^6)*Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(-60*(-1)^(1/4)*a^(1/4)*(c*d^4 + a*e^4)^2 *Sqrt[1 + (c*x^4)/a]*EllipticPi[(I*Sqrt[a]*e^2)/(Sqrt[c]*d^2), ArcSin[((-1 )^(3/4)*c^(1/4)*x)/a^(1/4)], -1] + c^(1/4)*d*e*(e^2*(a + c*x^4)*(40*a*e^4 + c*(30*d^4 - 20*d^3*e*x + 15*d^2*e^2*x^2 - 12*d*e^3*x^3 + 10*e^4*x^4)) + 60*(-(c*d^4) - a*e^4)^(3/2)*Sqrt[a + c*x^4]*ArcTan[(Sqrt[c]*(d^2 - e^2*x^2 ) + e^2*Sqrt[a + c*x^4])/Sqrt[-(c*d^4) - a*e^4]] - 15*Sqrt[c]*d^2*(2*c*d^4 + 3*a*e^4)*Sqrt[a + c*x^4]*Log[-(Sqrt[c]*x^2) + Sqrt[a + c*x^4]])))/(60*S qrt[(I*Sqrt[c])/Sqrt[a]]*c^(1/4)*d*e^8*Sqrt[a + c*x^4])
Time = 3.58 (sec) , antiderivative size = 876, normalized size of antiderivative = 1.10, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.105, Rules used = {2267, 1531, 27, 1577, 493, 25, 682, 27, 719, 224, 219, 488, 219, 2223, 2427, 2427, 27, 1512, 27, 761, 1510}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2267 |
| \(\displaystyle d \int \frac {\left (c x^4+a\right )^{3/2}}{d^2-e^2 x^2}dx-e \int \frac {x \left (c x^4+a\right )^{3/2}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1531 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right )^2 \int \frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {c} x^2+\sqrt {a}\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-e \int \frac {x \left (c x^4+a\right )^{3/2}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-e \int \frac {x \left (c x^4+a\right )^{3/2}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \int \frac {\left (c x^4+a\right )^{3/2}}{d^2-e^2 x^2}dx^2\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (-\frac {\int -\frac {\left (a e^2+c d^2 x^2\right ) \sqrt {c x^4+a}}{d^2-e^2 x^2}dx^2}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\int \frac {\left (a e^2+c d^2 x^2\right ) \sqrt {c x^4+a}}{d^2-e^2 x^2}dx^2}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\int \frac {c \left (a \left (c d^4+2 a e^4\right ) e^2+c d^2 \left (2 c d^4+3 a e^4\right ) x^2\right )}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{2 c e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\int \frac {a \left (c d^4+2 a e^4\right ) e^2+c d^2 \left (2 c d^4+3 a e^4\right ) x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {2 \left (a e^4+c d^4\right )^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {c d^2 \left (3 a e^4+2 c d^4\right ) \int \frac {1}{\sqrt {c x^4+a}}dx^2}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {2 \left (a e^4+c d^4\right )^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {c d^2 \left (3 a e^4+2 c d^4\right ) \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+a}}}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\frac {2 \left (a e^4+c d^4\right )^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {2 \left (a e^4+c d^4\right )^2 \int \frac {1}{c d^4+a e^4-x^4}d\frac {-a e^2-c d^2 x^2}{\sqrt {c x^4+a}}}{e^2}-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\int \frac {c^2 e^4 \left (c d^4-a e^4\right ) x^6+c^2 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c \left (c d^4-a e^4\right ) \left (c d^4+2 a e^4\right ) x^2+\sqrt {a} \sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {\int \frac {5 c^3 d^2 e^2 \left (c d^4-a e^4\right ) x^4+c^2 \left (c d^4-a e^4\right ) \left (5 c d^4+7 a e^4\right ) x^2+5 \sqrt {a} c^{3/2} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c^{3/2} d^6+2 a \sqrt {c} e^4 d^2-a^{3/2} e^6\right )}{\sqrt {c x^4+a}}dx}{5 c}+\frac {1}{5} c e^4 x^3 \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 2427 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {\frac {\int \frac {c^{5/2} \left (3 \sqrt {c} \left (c d^4-a e^4\right ) \left (5 c d^4+7 a e^4\right ) x^2+5 \sqrt {a} \left (3 c^2 d^8-4 \sqrt {a} c^{3/2} e^2 d^6+6 a c e^4 d^4-8 a^{3/2} \sqrt {c} e^6 d^2+3 a^2 e^8\right )\right )}{\sqrt {c x^4+a}}dx}{3 c}+\frac {5}{3} c^2 d^2 e^2 x \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{5 c}+\frac {1}{5} c e^4 x^3 \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {\frac {1}{3} c^{3/2} \int \frac {3 \sqrt {c} \left (c d^4-a e^4\right ) \left (5 c d^4+7 a e^4\right ) x^2+5 \sqrt {a} \left (3 c^2 d^8-4 \sqrt {a} c^{3/2} e^2 d^6+6 a c e^4 d^4-8 a^{3/2} \sqrt {c} e^6 d^2+3 a^2 e^8\right )}{\sqrt {c x^4+a}}dx+\frac {5}{3} c^2 d^2 e^2 x \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{5 c}+\frac {1}{5} c e^4 x^3 \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {\frac {1}{3} c^{3/2} \left (2 \sqrt {a} \left (-20 a^{3/2} \sqrt {c} d^2 e^6-3 a^2 e^8-10 \sqrt {a} c^{3/2} d^6 e^2+18 a c d^4 e^4+15 c^2 d^8\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-3 \sqrt {a} \left (c d^4-a e^4\right ) \left (7 a e^4+5 c d^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx\right )+\frac {5}{3} c^2 d^2 e^2 x \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{5 c}+\frac {1}{5} c e^4 x^3 \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {\frac {1}{3} c^{3/2} \left (2 \sqrt {a} \left (-20 a^{3/2} \sqrt {c} d^2 e^6-3 a^2 e^8-10 \sqrt {a} c^{3/2} d^6 e^2+18 a c d^4 e^4+15 c^2 d^8\right ) \int \frac {1}{\sqrt {c x^4+a}}dx-3 \left (c d^4-a e^4\right ) \left (7 a e^4+5 c d^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx\right )+\frac {5}{3} c^2 d^2 e^2 x \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{5 c}+\frac {1}{5} c e^4 x^3 \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (a e^4+c d^4\right )^2 \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {\frac {1}{3} c^{3/2} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (-20 a^{3/2} \sqrt {c} d^2 e^6-3 a^2 e^8-10 \sqrt {a} c^{3/2} d^6 e^2+18 a c d^4 e^4+15 c^2 d^8\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-3 \left (c d^4-a e^4\right ) \left (7 a e^4+5 c d^4\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx\right )+\frac {5}{3} c^2 d^2 e^2 x \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{5 c}+\frac {1}{5} c e^4 x^3 \sqrt {a+c x^4} \left (c d^4-a e^4\right )}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right ) \left (3 a e^4+2 c d^4\right )}{e^2}-\frac {2 \left (a e^4+c d^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{2 e^4}-\frac {\sqrt {a+c x^4} \left (2 \left (a e^4+c d^4\right )+c d^2 e^2 x^2\right )}{2 e^4}}{e^2}-\frac {\left (a+c x^4\right )^{3/2}}{3 e^2}\right )\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle d \left (\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right )^2 \left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {c x^4+a}}\right )}{2 d e \sqrt {c d^4+a e^4}}+\frac {\left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {c x^4+a}}\right )}{e^6 \left (c d^4-a e^4\right )}-\frac {\frac {1}{5} c \left (c d^4-a e^4\right ) x^3 \sqrt {c x^4+a} e^4+\frac {\frac {5}{3} c^2 d^2 \left (c d^4-a e^4\right ) x \sqrt {c x^4+a} e^2+\frac {1}{3} c^{3/2} \left (\frac {\sqrt [4]{a} \left (15 c^2 d^8-10 \sqrt {a} c^{3/2} e^2 d^6+18 a c e^4 d^4-20 a^{3/2} \sqrt {c} e^6 d^2-3 a^2 e^8\right ) \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {c x^4+a}}-3 \left (c d^4-a e^4\right ) \left (5 c d^4+7 a e^4\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {c} x^2+\sqrt {a}\right ) \sqrt {\frac {c x^4+a}{\left (\sqrt {c} x^2+\sqrt {a}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {c x^4+a}}-\frac {x \sqrt {c x^4+a}}{\sqrt {c} x^2+\sqrt {a}}\right )\right )}{5 c}}{e^6 \left (c d^4-a e^4\right )}\right )-\frac {1}{2} e \left (\frac {\frac {-\frac {\sqrt {c} \left (2 c d^4+3 a e^4\right ) \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {c x^4+a}}\right ) d^2}{e^2}-\frac {2 \left (c d^4+a e^4\right )^{3/2} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {c x^4+a}}\right )}{e^2}}{2 e^4}-\frac {\left (c d^2 e^2 x^2+2 \left (c d^4+a e^4\right )\right ) \sqrt {c x^4+a}}{2 e^4}}{e^2}-\frac {\left (c x^4+a\right )^{3/2}}{3 e^2}\right )\) |
Input:
Int[(a + c*x^4)^(3/2)/(d + e*x),x]
Output:
-1/2*(e*(-1/3*(a + c*x^4)^(3/2)/e^2 + (-1/2*((2*(c*d^4 + a*e^4) + c*d^2*e^ 2*x^2)*Sqrt[a + c*x^4])/e^4 + (-((Sqrt[c]*d^2*(2*c*d^4 + 3*a*e^4)*ArcTanh[ (Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/e^2) - (2*(c*d^4 + a*e^4)^(3/2)*ArcTanh[(- (a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/e^2)/(2*e^4)) /e^2)) + d*(-(((c*e^4*(c*d^4 - a*e^4)*x^3*Sqrt[a + c*x^4])/5 + ((5*c^2*d^2 *e^2*(c*d^4 - a*e^4)*x*Sqrt[a + c*x^4])/3 + (c^(3/2)*(-3*(c*d^4 - a*e^4)*( 5*c*d^4 + 7*a*e^4)*(-((x*Sqrt[a + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1 /4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*El lipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^4])) + (a^(1/4)*(15*c^2*d^8 - 10*Sqrt[a]*c^(3/2)*d^6*e^2 + 18*a*c*d^4*e^4 - 20*a^ (3/2)*Sqrt[c]*d^2*e^6 - 3*a^2*e^8)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4 )/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2] )/(c^(1/4)*Sqrt[a + c*x^4])))/3)/(5*c))/(e^6*(c*d^4 - a*e^4))) + ((Sqrt[c] *d^2 - Sqrt[a]*e^2)*(c*d^4 + a*e^4)^2*(((Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan h[(Sqrt[c*d^4 + a*e^4]*x)/(d*e*Sqrt[a + c*x^4])])/(2*d*e*Sqrt[c*d^4 + a*e^ 4]) + ((Sqrt[a]/d^2 - Sqrt[c]/e^2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4 )/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*S qrt[a]*Sqrt[c]*d^2*e^2), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c ^(1/4)*Sqrt[a + c*x^4])))/(e^6*(c*d^4 - a*e^4)))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*((a + b*x^2)^p/(d*(n + 2*p + 1))), x] + Simp[2*(p/(d*(n + 2*p + 1))) Int[(c + d*x)^n*(a + b*x^2)^(p - 1)*(a*d - b*c*x), x], x] /; FreeQ[{a, b, c, d, n}, x] && GtQ[p, 0] && NeQ[n + 2*p + 1, 0] && ( !Rationa lQ[n] || LtQ[n, 1]) && !ILtQ[n + 2*p, 0] && IntQuadraticQ[a, 0, b, c, d, n , p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* (1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e }, x] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + c*x^4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, c , d, e}, x] && PosQ[c/a]
Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[-(c *d^2 + a*e^2)^(p + 1/2)/(e^(2*p)*(c*d^2 - a*e^2)) Int[(a*d*Rt[c/a, 2] + a *e + (c*d + a*e*Rt[c/a, 2])*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] + Si mp[1/(e^(2*p)*(c*d^2 - a*e^2)) Int[(1/Sqrt[a + c*x^4])*ExpandToSum[(e^(2* p)*(c*d^2 - a*e^2)*(a + c*x^4)^(p + 1/2) + (c*d^2 + a*e^2)^(p + 1/2)*(a*d*R t[c/a, 2] + a*e + (c*d + a*e*Rt[c/a, 2])*x^2))/(d + e*x^2), x], x], x] /; F reeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p - 1/2, 0] && NeQ[c *d^2 - a*e^2, 0] && PosQ[c/a]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, c, d, e, p, q}, x]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(ArcTanh[Rt[(-c)* (d/e) - a*(e/d), 2]*(x/Sqrt[a + c*x^4])]/(2*d*e*Rt[(-c)*(d/e) - a*(e/d), 2] )), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^ 2)]/(4*d*e*q*Sqrt[a + c*x^4]))*EllipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*Arc Tan[q*x], 1/2], x]] /; FreeQ[{a, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0 ] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[c*(d/e) + a*(e /d)]
Int[((a_) + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[d Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x], x] - Simp[e Int[x*((a + c*x^4)^p/( d^2 - e^2*x^2)), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{q = Expon[Pq, x ]}, With[{Pqq = Coeff[Pq, x, q]}, Simp[Pqq*x^(q - n + 1)*((a + b*x^n)^(p + 1)/(b*(q + n*p + 1))), x] + Simp[1/(b*(q + n*p + 1)) Int[ExpandToSum[b*(q + n*p + 1)*(Pq - Pqq*x^q) - a*Pqq*(q - n + 1)*x^(q - n), x]*(a + b*x^n)^p, x], x]] /; NeQ[q + n*p + 1, 0] && q - n >= 0 && (IntegerQ[2*p] || IntegerQ [p + (q + 1)/(2*n)])] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n, 0]
Result contains complex when optimal does not.
Time = 2.19 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.72
| method | result | size |
| risch | \(\frac {\left (10 x^{4} c \,e^{4}-12 c d \,x^{3} e^{3}+15 x^{2} c \,d^{2} e^{2}-20 c \,d^{3} x e +40 e^{4} a +30 c \,d^{4}\right ) \sqrt {c \,x^{4}+a}}{60 e^{5}}-\frac {-\frac {30 \left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{4}}+\frac {c d \left (\frac {6 i e^{2} \left (7 e^{4} a +5 c \,d^{4}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {15 d e \left (3 e^{4} a +2 c \,d^{4}\right ) \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {30 c \,d^{6} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {50 a \,d^{2} e^{4} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )}{e^{3}}}{30 e^{5}}\) | \(576\) |
| default | \(\frac {c \,x^{4} \sqrt {c \,x^{4}+a}}{6 e}-\frac {c d \,x^{3} \sqrt {c \,x^{4}+a}}{5 e^{2}}+\frac {c \,d^{2} x^{2} \sqrt {c \,x^{4}+a}}{4 e^{3}}-\frac {c \,d^{3} x \sqrt {c \,x^{4}+a}}{3 e^{4}}+\frac {\left (\frac {c \left (2 e^{4} a +c \,d^{4}\right )}{e^{5}}-\frac {2 c a}{3 e}\right ) \sqrt {c \,x^{4}+a}}{2 c}+\frac {\left (-\frac {c \,d^{3} \left (2 e^{4} a +c \,d^{4}\right )}{e^{8}}+\frac {c \,d^{3} a}{3 e^{4}}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (\frac {d^{2} c \left (2 e^{4} a +c \,d^{4}\right )}{e^{7}}-\frac {d^{2} c a}{2 e^{3}}\right ) \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {i \left (-\frac {c d \left (2 e^{4} a +c \,d^{4}\right )}{e^{6}}+\frac {3 c d a}{5 e^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\frac {\left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{9}}\) | \(593\) |
| elliptic | \(\frac {c \,x^{4} \sqrt {c \,x^{4}+a}}{6 e}-\frac {c d \,x^{3} \sqrt {c \,x^{4}+a}}{5 e^{2}}+\frac {c \,d^{2} x^{2} \sqrt {c \,x^{4}+a}}{4 e^{3}}-\frac {c \,d^{3} x \sqrt {c \,x^{4}+a}}{3 e^{4}}+\frac {\left (\frac {c \left (2 e^{4} a +c \,d^{4}\right )}{e^{5}}-\frac {2 c a}{3 e}\right ) \sqrt {c \,x^{4}+a}}{2 c}+\frac {\left (-\frac {c \,d^{3} \left (2 e^{4} a +c \,d^{4}\right )}{e^{8}}+\frac {c \,d^{3} a}{3 e^{4}}\right ) \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (\frac {d^{2} c \left (2 e^{4} a +c \,d^{4}\right )}{e^{7}}-\frac {d^{2} c a}{2 e^{3}}\right ) \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}+\frac {i \left (-\frac {c d \left (2 e^{4} a +c \,d^{4}\right )}{e^{6}}+\frac {3 c d a}{5 e^{2}}\right ) \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}+\frac {\left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{9}}\) | \(593\) |
Input:
int((c*x^4+a)^(3/2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/60*(10*c*e^4*x^4-12*c*d*e^3*x^3+15*c*d^2*e^2*x^2-20*c*d^3*e*x+40*a*e^4+3 0*c*d^4)*(c*x^4+a)^(1/2)/e^5-1/30/e^5*(-30*(a^2*e^8+2*a*c*d^4*e^4+c^2*d^8) /e^4*(-1/2/(a+c*d^4/e^4)^(1/2)*arctanh(1/2*(2*c*x^2*d^2/e^2+2*a)/(a+c*d^4/ e^4)^(1/2)/(c*x^4+a)^(1/2))+1/(I*c^(1/2)/a^(1/2))^(1/2)/d*e*(1-I*c^(1/2)*x ^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)*Elliptic Pi(x*(I*c^(1/2)/a^(1/2))^(1/2),-I/c^(1/2)*a^(1/2)/d^2*e^2,(-I/a^(1/2)*c^(1 /2))^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)))+c*d/e^3*(6*I*e^2*(7*a*e^4+5*c*d^4)* a^(1/2)/(I*c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^( 1/2)*x^2/a^(1/2))^(1/2)/(c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(I*c^(1/2)/a^ (1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2),I))-15/2*d*e*(3*a*e^ 4+2*c*d^4)*ln(c^(1/2)*x^2+(c*x^4+a)^(1/2))/c^(1/2)+30*c*d^6/(I*c^(1/2)/a^( 1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a^(1/2))^(1/2 )/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)+50*a*d^2*e^4/(I *c^(1/2)/a^(1/2))^(1/2)*(1-I*c^(1/2)*x^2/a^(1/2))^(1/2)*(1+I*c^(1/2)*x^2/a ^(1/2))^(1/2)/(c*x^4+a)^(1/2)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)))
Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\text {Timed out} \] Input:
integrate((c*x^4+a)^(3/2)/(e*x+d),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (a + c x^{4}\right )^{\frac {3}{2}}}{d + e x}\, dx \] Input:
integrate((c*x**4+a)**(3/2)/(e*x+d),x)
Output:
Integral((a + c*x**4)**(3/2)/(d + e*x), x)
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{e x + d} \,d x } \] Input:
integrate((c*x^4+a)^(3/2)/(e*x+d),x, algorithm="maxima")
Output:
integrate((c*x^4 + a)^(3/2)/(e*x + d), x)
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\int { \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{e x + d} \,d x } \] Input:
integrate((c*x^4+a)^(3/2)/(e*x+d),x, algorithm="giac")
Output:
integrate((c*x^4 + a)^(3/2)/(e*x + d), x)
Timed out. \[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^4+a\right )}^{3/2}}{d+e\,x} \,d x \] Input:
int((a + c*x^4)^(3/2)/(d + e*x),x)
Output:
int((a + c*x^4)^(3/2)/(d + e*x), x)
\[ \int \frac {\left (a+c x^4\right )^{3/2}}{d+e x} \, dx=\int \frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{e x +d}d x \] Input:
int((c*x^4+a)^(3/2)/(e*x+d),x)
Output:
int((c*x^4+a)^(3/2)/(e*x+d),x)