Integrand size = 19, antiderivative size = 605 \[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\frac {\sqrt {a+c x^4}}{2 e}-\frac {\sqrt {c} d x \sqrt {a+c x^4}}{e^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\sqrt {c d^4+a e^4} \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {a+c x^4}}\right )}{2 e^3}+\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{2 e^3}-\frac {\sqrt {c d^4+a e^4} \text {arctanh}\left (\frac {a e^2+c d^2 x^2}{\sqrt {c d^4+a e^4} \sqrt {a+c x^4}}\right )}{2 e^3}+\frac {\sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{e^2 \sqrt {a+c x^4}}-\frac {\sqrt [4]{a} c^{3/4} d^3 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{e^2 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}}-\frac {\left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {a+c x^4}} \] Output:
1/2*(c*x^4+a)^(1/2)/e-c^(1/2)*d*x*(c*x^4+a)^(1/2)/e^2/(a^(1/2)+c^(1/2)*x^2 )+1/2*(a*e^4+c*d^4)^(1/2)*arctanh((a*e^4+c*d^4)^(1/2)*x/d/e/(c*x^4+a)^(1/2 ))/e^3+1/2*c^(1/2)*d^2*arctanh(c^(1/2)*x^2/(c*x^4+a)^(1/2))/e^3-1/2*(a*e^4 +c*d^4)^(1/2)*arctanh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^4+a)^(1/2 ))/e^3+a^(1/4)*c^(1/4)*d*(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^2 /(c*x^4+a)^(1/2)-a^(1/4)*c^(3/4)*d^3*(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^2/(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)*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2 )*x^2)^2)^(1/2)*EllipticPi(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 ^4/(c^(1/2)*d^2+a^(1/2)*e^2)/(c*x^4+a)^(1/2)
Result contains complex when optimal does not.
Time = 11.41 (sec) , antiderivative size = 421, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\frac {-2 \sqrt {a} c^{3/4} d^2 e^2 \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+2 c^{3/4} d^2 \left (i \sqrt {c} d^2+\sqrt {a} e^2\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (-2 \sqrt [4]{-1} \sqrt [4]{a} \left (c d^4+a e^4\right ) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )+\sqrt [4]{c} d e \left (e^2 \left (a+c x^4\right )-2 \sqrt {-c d^4-a e^4} \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )-\sqrt {c} d^2 \sqrt {a+c x^4} \log \left (-\sqrt {c} x^2+\sqrt {a+c x^4}\right )\right )\right )}{2 \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \sqrt [4]{c} d e^4 \sqrt {a+c x^4}} \] Input:
Integrate[Sqrt[a + c*x^4]/(d + e*x),x]
Output:
(-2*Sqrt[a]*c^(3/4)*d^2*e^2*Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[( I*Sqrt[c])/Sqrt[a]]*x], -1] + 2*c^(3/4)*d^2*(I*Sqrt[c]*d^2 + Sqrt[a]*e^2)* Sqrt[1 + (c*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sqrt[a]]*(-2*(-1)^(1/4)*a^(1/4)*(c*d^4 + a*e^4)*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) - 2*Sqrt[-(c*d^4) - a*e^4]*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]] - Sqrt[c]*d^2*Sqrt[a + c*x^4]*Log[-(Sqrt[c]* x^2) + Sqrt[a + c*x^4]])))/(2*Sqrt[(I*Sqrt[c])/Sqrt[a]]*c^(1/4)*d*e^4*Sqrt [a + c*x^4])
Time = 2.02 (sec) , antiderivative size = 660, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.842, Rules used = {2267, 1524, 27, 1512, 27, 761, 1510, 1577, 493, 25, 719, 224, 219, 488, 219, 2223}
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.
| \(\displaystyle \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2267 |
| \(\displaystyle d \int \frac {\sqrt {c x^4+a}}{d^2-e^2 x^2}dx-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1524 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\int \frac {\sqrt {c} \left (\sqrt {c} d^2-\sqrt {a} e^2+\sqrt {c} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2\right )}{\sqrt {c x^4+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \int \frac {\sqrt {c} d^2-\sqrt {a} e^2+\sqrt {c} \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right ) x^2}{\sqrt {c x^4+a}}dx}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1512 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (2 \sqrt {c} d^2 \int \frac {1}{\sqrt {c x^4+a}}dx-\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+a}}dx\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (2 \sqrt {c} d^2 \int \frac {1}{\sqrt {c x^4+a}}dx-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 761 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+a}}dx}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1510 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-e \int \frac {x \sqrt {c x^4+a}}{d^2-e^2 x^2}dx\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \int \frac {\sqrt {c x^4+a}}{d^2-e^2 x^2}dx^2\) |
| \(\Big \downarrow \) 493 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (-\frac {\int -\frac {a e^2+c d^2 x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\int \frac {a e^2+c d^2 x^2}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\left (a e^4+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {c d^2 \int \frac {1}{\sqrt {c x^4+a}}dx^2}{e^2}}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\left (a e^4+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {c d^2 \int \frac {1}{1-c x^4}d\frac {x^2}{\sqrt {c x^4+a}}}{e^2}}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {\frac {\left (a e^4+c d^4\right ) \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{e^2}-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^2}}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {-\frac {\left (a e^4+c d^4\right ) \int \frac {1}{c d^4+a e^4-x^4}d\frac {-a e^2-c d^2 x^2}{\sqrt {c x^4+a}}}{e^2}-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^2}}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^2}-\frac {\sqrt {a e^4+c d^4} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
| \(\Big \downarrow \) 2223 |
| \(\displaystyle d \left (\frac {\left (a e^4+c d^4\right ) \left (\frac {\left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (\frac {\sqrt {a}}{d^2}-\frac {\sqrt {c}}{e^2}\right ) \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c} d^2+\sqrt {a} e^2\right )^2}{4 \sqrt {a} \sqrt {c} d^2 e^2},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 \sqrt [4]{a} \sqrt [4]{c} \sqrt {a+c x^4}}+\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \text {arctanh}\left (\frac {x \sqrt {a e^4+c d^4}}{d e \sqrt {a+c x^4}}\right )}{2 d e \sqrt {a e^4+c d^4}}\right )}{\sqrt {a} e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}-\frac {\sqrt {c} \left (\frac {\sqrt [4]{c} d^2 \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{\sqrt [4]{a} \sqrt {a+c x^4}}-\frac {\left (\sqrt {a} e^2+\sqrt {c} d^2\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {a+c x^4}}-\frac {x \sqrt {a+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {a}}\right )}{e^2 \left (\frac {\sqrt {c} d^2}{\sqrt {a}}+e^2\right )}\right )-\frac {1}{2} e \left (\frac {-\frac {\sqrt {c} d^2 \text {arctanh}\left (\frac {\sqrt {c} x^2}{\sqrt {a+c x^4}}\right )}{e^2}-\frac {\sqrt {a e^4+c d^4} \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{e^2}}{e^2}-\frac {\sqrt {a+c x^4}}{e^2}\right )\) |
Input:
Int[Sqrt[a + c*x^4]/(d + e*x),x]
Output:
-1/2*(e*(-(Sqrt[a + c*x^4]/e^2) + (-((Sqrt[c]*d^2*ArcTanh[(Sqrt[c]*x^2)/Sq rt[a + c*x^4]])/e^2) - (Sqrt[c*d^4 + a*e^4]*ArcTanh[(-(a*e^2) - c*d^2*x^2) /(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/e^2)/e^2)) + d*(-((Sqrt[c]*(-(((S qrt[c]*d^2 + Sqrt[a]*e^2)*(-((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]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^(1/4)*Sqrt[a + c*x^ 4])))/Sqrt[a]) + (c^(1/4)*d^2*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sq rt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(a^ (1/4)*Sqrt[a + c*x^4])))/(e^2*((Sqrt[c]*d^2)/Sqrt[a] + e^2))) + ((c*d^4 + a*e^4)*(((Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTanh[(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]*El lipticPi[(Sqrt[c]*d^2 + Sqrt[a]*e^2)^2/(4*Sqrt[a]*Sqrt[c]*d^2*e^2), 2*ArcT an[(c^(1/4)*x)/a^(1/4)], 1/2])/(4*a^(1/4)*c^(1/4)*Sqrt[a + c*x^4])))/(Sqrt [a]*e^2*((Sqrt[c]*d^2)/Sqrt[a] + e^2)))
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[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[Sqrt[(a_) + (c_.)*(x_)^4]/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d^2 + a*e^2)/(e*(e - d*q)) Int[(1 + q*x^2)/((d + e* x^2)*Sqrt[a + c*x^4]), x], x] - Simp[1/(e*(e - d*q)) Int[(c*d + a*e*q - ( c*e - a*d*q^3)*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && N eQ[c*d^2 + a*e^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]
Result contains complex when optimal does not.
Time = 1.53 (sec) , antiderivative size = 404, normalized size of antiderivative = 0.67
| method | result | size |
| default | \(\frac {\sqrt {c \,x^{4}+a}}{2 e}-\frac {c \,d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {d^{2} \sqrt {c}\, \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 e^{3}}-\frac {i \sqrt {c}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{5}}\) | \(404\) |
| elliptic | \(\frac {\sqrt {c \,x^{4}+a}}{2 e}-\frac {c \,d^{3} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{e^{4} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {d^{2} \sqrt {c}\, \ln \left (2 \sqrt {c}\, x^{2}+2 \sqrt {c \,x^{4}+a}\right )}{2 e^{3}}-\frac {i \sqrt {c}\, d \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{e^{2} \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {\left (e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{5}}\) | \(404\) |
| risch | \(\frac {\sqrt {c \,x^{4}+a}}{2 e}-\frac {-\frac {\left (e^{4} a +c \,d^{4}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {2 c \,x^{2} d^{2}}{e^{2}}+2 a}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}\, \sqrt {c \,x^{4}+a}}\right )}{2 \sqrt {a +\frac {c \,d^{4}}{e^{4}}}}+\frac {e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, -\frac {i \sqrt {a}\, e^{2}}{\sqrt {c}\, d^{2}}, \frac {\sqrt {-\frac {i \sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}}\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, d \sqrt {c \,x^{4}+a}}\right )}{e^{4}}+\frac {c d \left (\frac {d^{2} \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}+\frac {i e^{2} \sqrt {a}\, \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}-\frac {d e \ln \left (\sqrt {c}\, x^{2}+\sqrt {c \,x^{4}+a}\right )}{2 \sqrt {c}}\right )}{e^{3}}}{e}\) | \(405\) |
Input:
int((c*x^4+a)^(1/2)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/2*(c*x^4+a)^(1/2)/e-c*d^3/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)+1/2*d^2/e^3*c^(1/2)*ln(2*c^(1/2)*x^2+2*(c*x ^4+a)^(1/2))-I*c^(1/2)*d/e^2*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)*(Elli pticF(x*(I*c^(1/2)/a^(1/2))^(1/2),I)-EllipticE(x*(I*c^(1/2)/a^(1/2))^(1/2) ,I))+(a*e^4+c*d^4)/e^5*(-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)*EllipticPi(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)))
\[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{e x + d} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/(e*x+d),x, algorithm="fricas")
Output:
integral(sqrt(c*x^4 + a)/(e*x + d), x)
\[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\int \frac {\sqrt {a + c x^{4}}}{d + e x}\, dx \] Input:
integrate((c*x**4+a)**(1/2)/(e*x+d),x)
Output:
Integral(sqrt(a + c*x**4)/(d + e*x), x)
\[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{e x + d} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/(e*x+d),x, algorithm="maxima")
Output:
integrate(sqrt(c*x^4 + a)/(e*x + d), x)
\[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\int { \frac {\sqrt {c x^{4} + a}}{e x + d} \,d x } \] Input:
integrate((c*x^4+a)^(1/2)/(e*x+d),x, algorithm="giac")
Output:
integrate(sqrt(c*x^4 + a)/(e*x + d), x)
Timed out. \[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\int \frac {\sqrt {c\,x^4+a}}{d+e\,x} \,d x \] Input:
int((a + c*x^4)^(1/2)/(d + e*x),x)
Output:
int((a + c*x^4)^(1/2)/(d + e*x), x)
\[ \int \frac {\sqrt {a+c x^4}}{d+e x} \, dx=\int \frac {\sqrt {c \,x^{4}+a}}{e x +d}d x \] Input:
int((c*x^4+a)^(1/2)/(e*x+d),x)
Output:
int((c*x^4+a)^(1/2)/(e*x+d),x)