Optimal. Leaf size=297 \[ \frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.140118, antiderivative size = 297, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {47, 63, 240, 211, 1165, 628, 1162, 617, 204} \[ \frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}+\frac{i \log \left (\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}+1\right )}{\sqrt{2} a}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 47
Rule 63
Rule 240
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{(a-i a x)^{5/4}}{(a+i a x)^{9/4}} \, dx &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\int \frac{\sqrt [4]{a-i a x}}{(a+i a x)^{5/4}} \, dx\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\int \frac{1}{(a-i a x)^{3/4} \sqrt [4]{a+i a x}} \, dx\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{2 a-x^4}} \, dx,x,\sqrt [4]{a-i a x}\right )}{a}\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac{(4 i) \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2} a}-\frac{i \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2} a}\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac{i \log \left (1+\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2} a}+\frac{i \log \left (1+\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2} a}+\frac{\left (i \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{\left (i \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}\\ &=\frac{4 i (a-i a x)^{5/4}}{5 a (a+i a x)^{5/4}}-\frac{4 i \sqrt [4]{a-i a x}}{a \sqrt [4]{a+i a x}}-\frac{i \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}+\frac{i \sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{a}-\frac{i \log \left (1+\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}-\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2} a}+\frac{i \log \left (1+\frac{\sqrt{a-i a x}}{\sqrt{a+i a x}}+\frac{\sqrt{2} \sqrt [4]{a-i a x}}{\sqrt [4]{a+i a x}}\right )}{\sqrt{2} a}\\ \end{align*}
Mathematica [C] time = 0.0240654, size = 70, normalized size = 0.24 \[ \frac{i \sqrt [4]{1+i x} (a-i a x)^{9/4} \, _2F_1\left (\frac{9}{4},\frac{9}{4};\frac{13}{4};\frac{1}{2}-\frac{i x}{2}\right )}{9 \sqrt [4]{2} a^3 \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.043, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a-iax \right ) ^{{\frac{5}{4}}} \left ( a+iax \right ) ^{-{\frac{9}{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{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.19028, size = 879, normalized size = 2.96 \begin{align*} \frac{{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) +{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} + 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (5 \, a^{2} x^{2} - 10 i \, a^{2} x - 5 \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} \log \left (-\frac{{\left (a^{2} x - i \, a^{2}\right )} \sqrt{-\frac{4 i}{a^{2}}} - 2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}}{2 \, x - 2 i}\right ) -{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (48 \, x - 32 i\right )}}{10 \, a^{2} x^{2} - 20 i \, a^{2} x - 10 \, a^{2}} \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]