Optimal. Leaf size=373 \[ -\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{475 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}} \]
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Rubi [A] time = 0.255284, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.812, Rules used = {5062, 97, 153, 147, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (-452 a x+521 i) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}+\frac{475 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \log \left (\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}+1\right )}{128 \sqrt{2} a^4}-\frac{475 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}} \]
Antiderivative was successfully verified.
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Rule 5062
Rule 97
Rule 153
Rule 147
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int e^{\frac{5}{2} i \tan ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1+i a x)^{5/4}}{(1-i a x)^{5/4}} \, dx\\ &=-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}+\frac{(4 i) \int \frac{x^2 \sqrt [4]{1+i a x} \left (3+\frac{17 i a x}{4}\right )}{\sqrt [4]{1-i a x}} \, dx}{a}\\ &=-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}+\frac{i \int \frac{x \sqrt [4]{1+i a x} \left (-\frac{17 i a}{2}+\frac{113 a^2 x}{8}\right )}{\sqrt [4]{1-i a x}} \, dx}{a^3}\\ &=-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac{(475 i) \int \frac{\sqrt [4]{1+i a x}}{\sqrt [4]{1-i a x}} \, dx}{64 a^3}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac{(475 i) \int \frac{1}{\sqrt [4]{1-i a x} (1+i a x)^{3/4}} \, dx}{128 a^3}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-i a x}\right )}{32 a^4}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{32 a^4}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac{475 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 a^4}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt{2} a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt{2} a^4}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}+\frac{475 \log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt{2} a^4}-\frac{475 \log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt{2} a^4}+\frac{475 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}-\frac{475 \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}\\ &=\frac{475 (1-i a x)^{3/4} \sqrt [4]{1+i a x}}{64 a^4}-\frac{4 i x^3 (1+i a x)^{5/4}}{a \sqrt [4]{1-i a x}}-\frac{17 x^2 (1-i a x)^{3/4} (1+i a x)^{5/4}}{4 a^2}-\frac{i (521 i-452 a x) (1-i a x)^{3/4} (1+i a x)^{5/4}}{96 a^4}-\frac{475 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{475 \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{64 \sqrt{2} a^4}+\frac{475 \log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}-\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt{2} a^4}-\frac{475 \log \left (1+\frac{\sqrt{1-i a x}}{\sqrt{1+i a x}}+\frac{\sqrt{2} \sqrt [4]{1-i a x}}{\sqrt [4]{1+i a x}}\right )}{128 \sqrt{2} a^4}\\ \end{align*}
Mathematica [C] time = 0.0427348, size = 96, normalized size = 0.26 \[ \frac{380 \sqrt [4]{2} (1-i a x) \text{Hypergeometric2F1}\left (-\frac{5}{4},\frac{3}{4},\frac{7}{4},\frac{1}{2} (1-i a x)\right )-\sqrt [4]{1+i a x} (a x-i)^2 \left (6 a^2 x^2-5 i a x+59\right )}{24 a^4 \sqrt [4]{1-i a x}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.189, size = 0, normalized size = 0. \begin{align*} \int \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) ^{{\frac{5}{2}}}{x}^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7259, size = 759, normalized size = 2.03 \begin{align*} -\frac{96 \, a^{4} \sqrt{\frac{225625 i}{4096 \, a^{8}}} \log \left (\frac{64}{475} \, a^{4} \sqrt{\frac{225625 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt{\frac{225625 i}{4096 \, a^{8}}} \log \left (-\frac{64}{475} \, a^{4} \sqrt{\frac{225625 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) + 96 \, a^{4} \sqrt{-\frac{225625 i}{4096 \, a^{8}}} \log \left (\frac{64}{475} \, a^{4} \sqrt{-\frac{225625 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) - 96 \, a^{4} \sqrt{-\frac{225625 i}{4096 \, a^{8}}} \log \left (-\frac{64}{475} \, a^{4} \sqrt{-\frac{225625 i}{4096 \, a^{8}}} + \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}\right ) +{\left (48 \, a^{4} x^{4} - 136 i \, a^{3} x^{3} - 226 \, a^{2} x^{2} + 521 i \, a x - 2467\right )} \sqrt{\frac{i \, \sqrt{a^{2} x^{2} + 1}}{a x + i}}}{192 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \left (\frac{i \, a x + 1}{\sqrt{a^{2} x^{2} + 1}}\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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