Optimal. Leaf size=148 \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0384858, antiderivative size = 148, 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, 46, 42, 197, 196} \[ \frac{2 \sqrt [4]{x^2+1} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 46
Rule 42
Rule 197
Rule 196
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{17/4} \sqrt [4]{a+i a x}} \, dx &=-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}+\frac{5 \int \frac{1}{(a-i a x)^{13/4} \sqrt [4]{a+i a x}} \, dx}{13 a}\\ &=-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}+\frac{5 \int \frac{1}{(a-i a x)^{9/4} \sqrt [4]{a+i a x}} \, dx}{39 a^2}\\ &=-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}+\frac{\int \frac{1}{(a-i a x)^{5/4} (a+i a x)^{5/4}} \, dx}{39 a^2}\\ &=-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}+\frac{\sqrt [4]{a^2+a^2 x^2} \int \frac{1}{\left (a^2+a^2 x^2\right )^{5/4}} \, dx}{39 a^2 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}+\frac{\sqrt [4]{1+x^2} \int \frac{1}{\left (1+x^2\right )^{5/4}} \, dx}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ &=-\frac{4 i}{39 a^3 (a-i a x)^{5/4} \sqrt [4]{a+i a x}}-\frac{2 i (a+i a x)^{3/4}}{13 a^2 (a-i a x)^{13/4}}-\frac{10 i (a+i a x)^{3/4}}{117 a^3 (a-i a x)^{9/4}}+\frac{2 \sqrt [4]{1+x^2} E\left (\left .\frac{1}{2} \tan ^{-1}(x)\right |2\right )}{39 a^4 \sqrt [4]{a-i a x} \sqrt [4]{a+i a x}}\\ \end{align*}
Mathematica [C] time = 0.0281626, size = 70, normalized size = 0.47 \[ -\frac{2 i 2^{3/4} \sqrt [4]{1+i x} \, _2F_1\left (-\frac{13}{4},\frac{1}{4};-\frac{9}{4};\frac{1}{2}-\frac{i x}{2}\right )}{13 a (a-i a x)^{13/4} \sqrt [4]{a+i a x}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.059, size = 114, normalized size = 0.8 \begin{align*}{\frac{18\,i{x}^{3}+6\,{x}^{4}-40-16\,{x}^{2}}{117\, \left ( x+i \right ) ^{3}{a}^{4}}{\frac{1}{\sqrt [4]{-a \left ( -1+ix \right ) }}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}}-{\frac{x}{39\,{a}^{4}}{\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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{17}{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{{\left (6 \, x^{3} + 24 i \, x^{2} - 40 \, x - 40 i\right )}{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}} +{\left (117 \, a^{6} x^{4} + 468 i \, a^{6} x^{3} - 702 \, a^{6} x^{2} - 468 i \, a^{6} x + 117 \, a^{6}\right )}{\rm integral}\left (-\frac{{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{3}{4}}}{39 \,{\left (a^{6} x^{2} + a^{6}\right )}}, x\right )}{117 \, a^{6} x^{4} + 468 i \, a^{6} x^{3} - 702 \, a^{6} x^{2} - 468 i \, a^{6} x + 117 \, a^{6}} \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]