Optimal. Leaf size=154 \[ \frac{14 x}{65 a^6 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0399608, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 42, 199, 197, 196} \[ \frac{14 x}{65 a^6 \left (x^2+1\right ) \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}+\frac{42 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 42
Rule 199
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{17/4} (a+i a x)^{9/4}} \, dx &=-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}+\frac{9 \int \frac{1}{(a-i a x)^{13/4} (a+i a x)^{9/4}} \, dx}{13 a}\\ &=-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{7 \int \frac{1}{(a-i a x)^{9/4} (a+i a x)^{9/4}} \, dx}{13 a^2}\\ &=-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{\left (7 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{9/4}} \, dx}{13 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{\left (21 \sqrt [4]{a^2+a^2 x^2}\right ) \int \frac{1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{65 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{\left (21 \sqrt [4]{1+x^2}\right ) \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{2 i}{13 a^2 (a-i a x)^{13/4} (a+i a x)^{5/4}}-\frac{2 i}{13 a^3 (a-i a x)^{9/4} (a+i a x)^{5/4}}+\frac{14 x}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x} \left (1+x^2\right )}+\frac{42 \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{65 a^6 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0336205, size = 70, normalized size = 0.45 \[ -\frac{i \sqrt [4]{1+i x} \, _2F_1\left (-\frac{13}{4},\frac{9}{4};-\frac{9}{4};\frac{1}{2}-\frac{i x}{2}\right )}{13 \sqrt [4]{2} a^3 (a-i a x)^{13/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.086, size = 130, normalized size = 0.8 \begin{align*}{\frac{84\,i{x}^{4}+42\,{x}^{5}+112\,i{x}^{2}-46\,x+14\,{x}^{3}+20\,i}{ \left ( 65\,x-65\,i \right ) \left ( x+i \right ) ^{3}{a}^{6}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{21\,x}{65\,{a}^{6}}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{3}{2}};\,-{x}^{2})}\sqrt [4]{-{a}^{2} \left ( -1+ix \right ) \left ( 1+ix \right ) }{\frac{1}{\sqrt [4]{{a}^{2}}}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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{{\left (42 \, x^{5} + 84 i \, x^{4} + 14 \, x^{3} + 112 i \, x^{2} - 46 \, x + 20 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (65 \, a^{8} x^{6} + 130 i \, a^{8} x^{5} + 65 \, a^{8} x^{4} + 260 i \, a^{8} x^{3} - 65 \, a^{8} x^{2} + 130 i \, a^{8} x - 65 \, a^{8}\right )}{\rm integral}\left (-\frac{21 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{65 \,{\left (a^{8} x^{2} + a^{8}\right )}}, x\right )}{65 \, a^{8} x^{6} + 130 i \, a^{8} x^{5} + 65 \, a^{8} x^{4} + 260 i \, a^{8} x^{3} - 65 \, a^{8} x^{2} + 130 i \, a^{8} x - 65 \, a^{8}} \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]