Optimal. Leaf size=113 \[ -\frac{10 a \left (x^2+1\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \tan ^{-1}(x),2\right )}{3 (a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}+\frac{10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}{3 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0242478, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {47, 50, 42, 233, 231} \[ -\frac{10 a \left (x^2+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a+i a x)^{3/4} (a-i a x)^{3/4}}+\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}+\frac{10 i \sqrt [4]{a+i a x} \sqrt [4]{a-i a x}}{3 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 50
Rule 42
Rule 233
Rule 231
Rubi steps
\begin{align*} \int \frac{(a-i a x)^{5/4}}{(a+i a x)^{7/4}} \, dx &=\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}-\frac{5}{3} \int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{3/4}} \, dx\\ &=\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}+\frac{10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{3 a}-\frac{1}{3} (5 a) \int \frac{1}{(a-i a x)^{3/4} (a+i a x)^{3/4}} \, dx\\ &=\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}+\frac{10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{3 a}-\frac{\left (5 a \left (a^2+a^2 x^2\right )^{3/4}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}+\frac{10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{3 a}-\frac{\left (5 a \left (1+x^2\right )^{3/4}\right ) \int \frac{1}{\left (1+x^2\right )^{3/4}} \, dx}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ &=\frac{4 i (a-i a x)^{5/4}}{3 a (a+i a x)^{3/4}}+\frac{10 i \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}{3 a}-\frac{10 a \left (1+x^2\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{3 (a-i a x)^{3/4} (a+i a x)^{3/4}}\\ \end{align*}
Mathematica [C] time = 0.0252346, size = 70, normalized size = 0.62 \[ \frac{i \sqrt [4]{2} (1+i x)^{3/4} (a-i a x)^{9/4} \, _2F_1\left (\frac{7}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2}-\frac{i x}{2}\right )}{9 a^2 (a+i a x)^{3/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{{\frac{5}{4}}} \left ( a+iax \right ) ^{-{\frac{7}{4}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-i \, a x + a\right )}^{\frac{5}{4}}}{{\left (i \, a x + a\right )}^{\frac{7}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3 \,{\left (a x - i \, a\right )}{\rm integral}\left (-\frac{5 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{3 \,{\left (a x^{2} + a\right )}}, x\right ) - 2 \,{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (-3 i \, x - 7\right )}}{3 \,{\left (a x - i \, a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]