Integrand size = 19, antiderivative size = 882 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\frac {e^4 x}{\left (c d^4+a e^4\right ) \left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}}+\frac {d e \left (a e^2-c d^2 x^2\right )}{a \left (c d^4+a e^4\right ) \left (d^2-e^2 x^2\right ) \sqrt {a+c x^4}}+\frac {c x \left (d^2 \left (c d^4-5 a e^4\right )+3 e^2 \left (c d^4-a e^4\right ) x^2\right )}{2 a \left (c d^4+a e^4\right )^2 \sqrt {a+c x^4}}-\frac {3 \sqrt {c} e^2 \left (c d^4-a e^4\right ) x \sqrt {a+c x^4}}{2 a \left (c d^4+a e^4\right )^2 \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {d e^3 \left (c d^4-2 a e^4\right ) \sqrt {a+c x^4}}{a \left (c d^4+a e^4\right )^2 \left (d^2-e^2 x^2\right )}+\frac {3 c d^3 e^5 \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {a+c x^4}}\right )}{\left (c d^4+a e^4\right )^{5/2}}-\frac {3 c d^3 e^5 \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 )}{\left (c d^4+a e^4\right )^{5/2}}+\frac {3 \sqrt [4]{c} e^2 \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}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} \left (c d^4+a e^4\right )^2 \sqrt {a+c x^4}}+\frac {\sqrt [4]{c} \left (c d^4-2 \sqrt {a} \sqrt {c} d^2 e^2+3 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 {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {a+c x^4}}-\frac {3 c^{3/4} d^2 e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right )^2 \sqrt {a+c x^4}} \] Output:
e^4*x/(a*e^4+c*d^4)/(-e^2*x^2+d^2)/(c*x^4+a)^(1/2)+d*e*(-c*d^2*x^2+a*e^2)/ a/(a*e^4+c*d^4)/(-e^2*x^2+d^2)/(c*x^4+a)^(1/2)+1/2*c*x*(d^2*(-5*a*e^4+c*d^ 4)+3*e^2*(-a*e^4+c*d^4)*x^2)/a/(a*e^4+c*d^4)^2/(c*x^4+a)^(1/2)-3/2*c^(1/2) *e^2*(-a*e^4+c*d^4)*x*(c*x^4+a)^(1/2)/a/(a*e^4+c*d^4)^2/(a^(1/2)+c^(1/2)*x ^2)+d*e^3*(-2*a*e^4+c*d^4)*(c*x^4+a)^(1/2)/a/(a*e^4+c*d^4)^2/(-e^2*x^2+d^2 )+3*c*d^3*e^5*arctanh((a*e^4+c*d^4)^(1/2)*x/d/e/(c*x^4+a)^(1/2))/(a*e^4+c* d^4)^(5/2)-3*c*d^3*e^5*arctanh((c*d^2*x^2+a*e^2)/(a*e^4+c*d^4)^(1/2)/(c*x^ 4+a)^(1/2))/(a*e^4+c*d^4)^(5/2)+3/2*c^(1/4)*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)*EllipticE(sin(2*arcta n(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2/(c*x^4+a)^(1/2) +1/4*c^(1/4)*(c*d^4-2*a^(1/2)*c^(1/2)*d^2*e^2+3*a*e^4)*(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))/a^(5/4)/(c^(1/2)*d^2+a^(1/2)*e^2)/(a*e^4+c*d^4 )/(c*x^4+a)^(1/2)-3/2*c^(3/4)*d^2*e^4*(c^(1/2)*d^2-a^(1/2)*e^2)*(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*arc tan(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^(1/2)*d^2+a^(1/2)*e^2)/(a*e^4+c*d^4)^2/(c*x^ 4+a)^(1/2)
Result contains complex when optimal does not.
Time = 13.19 (sec) , antiderivative size = 578, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\frac {3 \sqrt {a} \sqrt {c} e^2 \sqrt {-c d^4-a e^4} \left (-c d^4+a e^4\right ) (d+e x) \sqrt {1+\frac {c x^4}{a}} E\left (\left .i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right )\right |-1\right )+\sqrt {c} \sqrt {-c d^4-a e^4} \left (-i c^{3/2} d^6+3 \sqrt {a} c d^4 e^2+5 i a \sqrt {c} d^2 e^4-3 a^{3/2} e^6\right ) (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+\sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (\sqrt {-c d^4-a e^4} \left (-2 a^2 e^7+c^2 d^4 x \left (d^3-d^2 e x+d e^2 x^2+3 e^3 x^3\right )+a c e^3 \left (4 d^4+d^3 e x-d^2 e^2 x^2+d e^3 x^3-3 e^4 x^4\right )\right )+12 a c d^3 e^5 (d+e x) \sqrt {a+c x^4} \arctan \left (\frac {\sqrt {c} \left (d^2-e^2 x^2\right )+e^2 \sqrt {a+c x^4}}{\sqrt {-c d^4-a e^4}}\right )-12 \sqrt [4]{-1} a^{5/4} c^{3/4} d^2 e^4 \sqrt {-c d^4-a e^4} (d+e x) \sqrt {1+\frac {c x^4}{a}} \operatorname {EllipticPi}\left (\frac {i \sqrt {a} e^2}{\sqrt {c} d^2},\arcsin \left (\frac {(-1)^{3/4} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}} \left (-c d^4-a e^4\right )^{5/2} (d+e x) \sqrt {a+c x^4}} \] Input:
Integrate[1/((d + e*x)^2*(a + c*x^4)^(3/2)),x]
Output:
(3*Sqrt[a]*Sqrt[c]*e^2*Sqrt[-(c*d^4) - a*e^4]*(-(c*d^4) + a*e^4)*(d + e*x) *Sqrt[1 + (c*x^4)/a]*EllipticE[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[c]*Sqrt[-(c*d^4) - a*e^4]*((-I)*c^(3/2)*d^6 + 3*Sqrt[a]*c*d^4*e^2 + (5*I)*a*Sqrt[c]*d^2*e^4 - 3*a^(3/2)*e^6)*(d + e*x)*Sqrt[1 + (c*x^4)/a]*E llipticF[I*ArcSinh[Sqrt[(I*Sqrt[c])/Sqrt[a]]*x], -1] + Sqrt[(I*Sqrt[c])/Sq rt[a]]*(Sqrt[-(c*d^4) - a*e^4]*(-2*a^2*e^7 + c^2*d^4*x*(d^3 - d^2*e*x + d* e^2*x^2 + 3*e^3*x^3) + a*c*e^3*(4*d^4 + d^3*e*x - d^2*e^2*x^2 + d*e^3*x^3 - 3*e^4*x^4)) + 12*a*c*d^3*e^5*(d + e*x)*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]] - 12*(-1)^(1 /4)*a^(5/4)*c^(3/4)*d^2*e^4*Sqrt[-(c*d^4) - a*e^4]*(d + e*x)*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]))/(2*a*Sqrt[(I*Sqrt[c])/Sqrt[a]]*(-(c*d^4) - a*e^4)^(5/ 2)*(d + e*x)*Sqrt[a + c*x^4])
Time = 5.06 (sec) , antiderivative size = 1336, normalized size of antiderivative = 1.51, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.632, Rules used = {2584, 2255, 27, 1577, 496, 25, 27, 679, 488, 219, 2259, 2009}
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.
| \(\displaystyle \int \frac {1}{\left (a+c x^4\right )^{3/2} (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2584 |
| \(\displaystyle \int \frac {d^2-2 d e x+e^2 x^2}{\left (a+c x^4\right )^{3/2} \left (d^2-e^2 x^2\right )^2}dx\) |
| \(\Big \downarrow \) 2255 |
| \(\displaystyle \int -\frac {2 d e x}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx+\int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-2 d e \int \frac {x}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 1577 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \int \frac {1}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx^2\) |
| \(\Big \downarrow \) 496 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \left (-\frac {\int -\frac {e^2 \left (2 a e^2-c d^2 x^2\right )}{\left (d^2-e^2 x^2\right )^2 \sqrt {c x^4+a}}dx^2}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \left (\frac {\int \frac {e^2 \left (2 a e^2-c d^2 x^2\right )}{\left (d^2-e^2 x^2\right )^2 \sqrt {c x^4+a}}dx^2}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \left (\frac {e^2 \int \frac {2 a e^2-c d^2 x^2}{\left (d^2-e^2 x^2\right )^2 \sqrt {c x^4+a}}dx^2}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \left (\frac {e^2 \left (\frac {3 a c d^2 e^2 \int \frac {1}{\left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}dx^2}{a e^4+c d^4}-\frac {\sqrt {a+c x^4} \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \left (\frac {e^2 \left (-\frac {3 a c d^2 e^2 \int \frac {1}{c d^4+a e^4-x^4}d\frac {-a e^2-c d^2 x^2}{\sqrt {c x^4+a}}}{a e^4+c d^4}-\frac {\sqrt {a+c x^4} \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \int \frac {d^2+e^2 x^2}{\left (d^2-e^2 x^2\right )^2 \left (c x^4+a\right )^{3/2}}dx-d e \left (\frac {e^2 \left (-\frac {3 a c d^2 e^2 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {\sqrt {a+c x^4} \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 2259 |
| \(\displaystyle \int \left (\frac {4 c d^4 e^4}{\left (c d^4+a e^4\right )^2 \left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}+\frac {e^4}{2 \left (c d^4+a e^4\right ) (d-e x)^2 \sqrt {c x^4+a}}+\frac {e^4}{2 \left (c d^4+a e^4\right ) (d+e x)^2 \sqrt {c x^4+a}}+\frac {c \left (\left (c d^4-3 a e^4\right ) d^2+e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{\left (c d^4+a e^4\right )^2 \left (c x^4+a\right )^{3/2}}\right )dx-d e \left (\frac {e^2 \left (-\frac {3 a c d^2 e^2 \text {arctanh}\left (\frac {-a e^2-c d^2 x^2}{\sqrt {a+c x^4} \sqrt {a e^4+c d^4}}\right )}{\left (a e^4+c d^4\right )^{3/2}}-\frac {\sqrt {a+c x^4} \left (c d^4-2 a e^4\right )}{\left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )}{a \left (a e^4+c d^4\right )}-\frac {a e^2-c d^2 x^2}{a \sqrt {a+c x^4} \left (d^2-e^2 x^2\right ) \left (a e^4+c d^4\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c x^4+a} e^7}{2 \left (c d^4+a e^4\right )^2 (d-e x)}-\frac {\sqrt {c x^4+a} e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt [4]{a} \sqrt [4]{c} \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 ) e^6}{\left (c d^4+a e^4\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt {c} x \sqrt {c x^4+a} e^6}{\left (c d^4+a e^4\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}+\frac {3 c d^3 \text {arctanh}\left (\frac {\sqrt {c d^4+a e^4} x}{d e \sqrt {c x^4+a}}\right ) e^5}{\left (c d^4+a e^4\right )^{5/2}}+\frac {\sqrt [4]{c} \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 ) e^4}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right ) \sqrt {c x^4+a}}+\frac {2 c^{5/4} d^4 \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 ) e^4}{\sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right )^2 \sqrt {c x^4+a}}-\frac {3 c^{3/4} d^2 \left (\sqrt {c} d^2-\sqrt {a} 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 ) e^4}{2 \sqrt [4]{a} \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (c d^4+a e^4\right )^2 \sqrt {c x^4+a}}+\frac {\sqrt [4]{c} \left (3 c d^4-a e^4\right ) \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 ) e^2}{2 a^{3/4} \left (c d^4+a e^4\right )^2 \sqrt {c x^4+a}}-\frac {\sqrt {c} \left (3 c d^4-a e^4\right ) x \sqrt {c x^4+a} e^2}{2 a \left (c d^4+a e^4\right )^2 \left (\sqrt {c} x^2+\sqrt {a}\right )}-d \left (\frac {e^2 \left (-\frac {3 a c d^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}{\left (c d^4+a e^4\right )^{3/2}}-\frac {\left (c d^4-2 a e^4\right ) \sqrt {c x^4+a}}{\left (c d^4+a e^4\right ) \left (d^2-e^2 x^2\right )}\right )}{a \left (c d^4+a e^4\right )}-\frac {a e^2-c d^2 x^2}{a \left (c d^4+a e^4\right ) \left (d^2-e^2 x^2\right ) \sqrt {c x^4+a}}\right ) e-\frac {\sqrt [4]{c} \left (-a e^6+3 c d^4 e^2-\frac {\sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )}{4 a^{3/4} \left (c d^4+a e^4\right )^2 \sqrt {c x^4+a}}+\frac {c x \left (\left (c d^4-3 a e^4\right ) d^2+e^2 \left (3 c d^4-a e^4\right ) x^2\right )}{2 a \left (c d^4+a e^4\right )^2 \sqrt {c x^4+a}}\) |
Input:
Int[1/((d + e*x)^2*(a + c*x^4)^(3/2)),x]
Output:
(c*x*(d^2*(c*d^4 - 3*a*e^4) + e^2*(3*c*d^4 - a*e^4)*x^2))/(2*a*(c*d^4 + a* e^4)^2*Sqrt[a + c*x^4]) + (e^7*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^4)^2*(d - e*x)) - (e^7*Sqrt[a + c*x^4])/(2*(c*d^4 + a*e^4)^2*(d + e*x)) + (Sqrt[c]*e ^6*x*Sqrt[a + c*x^4])/((c*d^4 + a*e^4)^2*(Sqrt[a] + Sqrt[c]*x^2)) - (Sqrt[ c]*e^2*(3*c*d^4 - a*e^4)*x*Sqrt[a + c*x^4])/(2*a*(c*d^4 + a*e^4)^2*(Sqrt[a ] + Sqrt[c]*x^2)) + (3*c*d^3*e^5*ArcTanh[(Sqrt[c*d^4 + a*e^4]*x)/(d*e*Sqrt [a + c*x^4])])/(c*d^4 + a*e^4)^(5/2) - d*e*(-((a*e^2 - c*d^2*x^2)/(a*(c*d^ 4 + a*e^4)*(d^2 - e^2*x^2)*Sqrt[a + c*x^4])) + (e^2*(-(((c*d^4 - 2*a*e^4)* Sqrt[a + c*x^4])/((c*d^4 + a*e^4)*(d^2 - e^2*x^2))) - (3*a*c*d^2*e^2*ArcTa nh[(-(a*e^2) - c*d^2*x^2)/(Sqrt[c*d^4 + a*e^4]*Sqrt[a + c*x^4])])/(c*d^4 + a*e^4)^(3/2)))/(a*(c*d^4 + a*e^4))) - (a^(1/4)*c^(1/4)*e^6*(Sqrt[a] + Sqr t[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*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (c^(1/4)* e^2*(3*c*d^4 - a*e^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])/(2*a^(3/4)* (c*d^4 + a*e^4)^2*Sqrt[a + c*x^4]) + (2*c^(5/4)*d^4*e^4*(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])/(a^(1/4)*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*(c*d^4 + a*e^4 )^2*Sqrt[a + c*x^4]) + (c^(1/4)*e^4*(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)], ...
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/(((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[ (-(a*d + b*c*x))*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1)*(b*c^2 + a*d^2))), x] + Simp[1/(2*a*(p + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*Simp[b*c^2*(2*p + 3) + a*d^2*(n + 2*p + 3) + b*c*d*(n + 2 *p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[p, -1] && IntQuad raticQ[a, 0, b, c, d, n, p, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
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[(Pr_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{r = Expon[Pr, x], k}, Int[Sum[Coeff[Pr, x, 2*k]*x^(2*k), {k, 0, r/2}]*(d + e*x^2)^q*(a + c*x^4)^p, x] + Int[x*Sum[Coeff[Pr, x, 2*k + 1]*x^ (2*k), {k, 0, (r - 1)/2}]*(d + e*x^2)^q*(a + c*x^4)^p, x]] /; FreeQ[{a, c, d, e, p, q}, x] && PolyQ[Pr, x] && !PolyQ[Pr, x^2]
Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e}, x] && PolyQ[Px, x] && IntegerQ[p + 1/ 2] && IntegerQ[q]
Int[((c_) + (d_.)*(x_)^(n_.))^(q_)*((a_) + (b_.)*(x_)^(nn_.))^(p_), x_Symbo l] :> Int[ExpandToSum[(c - d*x^n)^(-q), x]*((a + b*x^nn)^p/(c^2 - d^2*x^(2* n))^(-q)), x] /; FreeQ[{a, b, c, d, n, nn, p}, x] && !IntegerQ[p] && ILtQ[ q, 0] && IGtQ[Log[2, nn/n], 0]
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 642, normalized size of antiderivative = 0.73
| method | result | size |
| default | \(-\frac {e^{7} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +c \,d^{4}\right )^{2} \left (e x +d \right )}-\frac {2 c \left (\frac {e^{2} \left (e^{4} a -3 c \,d^{4}\right ) x^{3}}{4 a \left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {d e \left (e^{4} a -c \,d^{4}\right ) x^{2}}{2 a \left (e^{4} a +c \,d^{4}\right )^{2}}+\frac {d^{2} \left (3 e^{4} a -c \,d^{4}\right ) x}{4 a \left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {d^{3} e^{3}}{\left (e^{4} a +c \,d^{4}\right )^{2}}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\left (-\frac {c \,d^{2} e^{4}}{\left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {c \,d^{2} \left (3 e^{4} a -c \,d^{4}\right )}{2 a \left (e^{4} a +c \,d^{4}\right )^{2}}\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 {i \left (\frac {e^{6} c}{\left (e^{4} a +c \,d^{4}\right )^{2}}+\frac {c \,e^{2} \left (e^{4} a -3 c \,d^{4}\right )}{2 a \left (e^{4} a +c \,d^{4}\right )^{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 {6 c \,d^{3} e^{3} \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 )}{\left (e^{4} a +c \,d^{4}\right )^{2}}\) | \(642\) |
| elliptic | \(-\frac {e^{7} \sqrt {c \,x^{4}+a}}{\left (e^{4} a +c \,d^{4}\right )^{2} \left (e x +d \right )}-\frac {2 c \left (\frac {e^{2} \left (e^{4} a -3 c \,d^{4}\right ) x^{3}}{4 a \left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {d e \left (e^{4} a -c \,d^{4}\right ) x^{2}}{2 a \left (e^{4} a +c \,d^{4}\right )^{2}}+\frac {d^{2} \left (3 e^{4} a -c \,d^{4}\right ) x}{4 a \left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {d^{3} e^{3}}{\left (e^{4} a +c \,d^{4}\right )^{2}}\right )}{\sqrt {c \left (\frac {a}{c}+x^{4}\right )}}+\frac {\left (-\frac {c \,d^{2} e^{4}}{\left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {c \,d^{2} \left (3 e^{4} a -c \,d^{4}\right )}{2 a \left (e^{4} a +c \,d^{4}\right )^{2}}\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 {i \left (\frac {e^{6} c}{\left (e^{4} a +c \,d^{4}\right )^{2}}+\frac {c \,e^{2} \left (e^{4} a -3 c \,d^{4}\right )}{2 a \left (e^{4} a +c \,d^{4}\right )^{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 {6 c \,d^{3} e^{3} \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 )}{\left (e^{4} a +c \,d^{4}\right )^{2}}\) | \(642\) |
Input:
int(1/(e*x+d)^2/(c*x^4+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-e^7/(a*e^4+c*d^4)^2*(c*x^4+a)^(1/2)/(e*x+d)-2*c*(1/4*e^2*(a*e^4-3*c*d^4)/ a/(a*e^4+c*d^4)^2*x^3-1/2*d*e*(a*e^4-c*d^4)/a/(a*e^4+c*d^4)^2*x^2+1/4*d^2* (3*a*e^4-c*d^4)/a/(a*e^4+c*d^4)^2*x-d^3*e^3/(a*e^4+c*d^4)^2)/(c*(a/c+x^4)) ^(1/2)+(-c*d^2*e^4/(a*e^4+c*d^4)^2-1/2*c*d^2*(3*a*e^4-c*d^4)/a/(a*e^4+c*d^ 4)^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)*EllipticF(x*(I*c^(1/2)/a^(1/2))^(1/2 ),I)+I*(e^6*c/(a*e^4+c*d^4)^2+1/2*c*e^2*(a*e^4-3*c*d^4)/a/(a*e^4+c*d^4)^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)/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))+6*c*d^3*e^3/(a* e^4+c*d^4)^2*(-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)))
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)^2/(c*x^4+a)^(3/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a + c x^{4}\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \] Input:
integrate(1/(e*x+d)**2/(c*x**4+a)**(3/2),x)
Output:
Integral(1/((a + c*x**4)**(3/2)*(d + e*x)**2), x)
\[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x+d)^2/(c*x^4+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2), x)
\[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{4} + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{2}} \,d x } \] Input:
integrate(1/(e*x+d)^2/(c*x^4+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((c*x^4 + a)^(3/2)*(e*x + d)^2), x)
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^4+a\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \] Input:
int(1/((a + c*x^4)^(3/2)*(d + e*x)^2),x)
Output:
int(1/((a + c*x^4)^(3/2)*(d + e*x)^2), x)
\[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {c \,x^{4}+a}}{c^{2} e^{2} x^{10}+2 c^{2} d e \,x^{9}+c^{2} d^{2} x^{8}+2 a c \,e^{2} x^{6}+4 a c d e \,x^{5}+2 a c \,d^{2} x^{4}+a^{2} e^{2} x^{2}+2 a^{2} d e x +a^{2} d^{2}}d x \] Input:
int(1/(e*x+d)^2/(c*x^4+a)^(3/2),x)
Output:
int(sqrt(a + c*x**4)/(a**2*d**2 + 2*a**2*d*e*x + a**2*e**2*x**2 + 2*a*c*d* *2*x**4 + 4*a*c*d*e*x**5 + 2*a*c*e**2*x**6 + c**2*d**2*x**8 + 2*c**2*d*e*x **9 + c**2*e**2*x**10),x)