Optimal. Leaf size=100 \[ -\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]
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Rubi [A] time = 0.0182027, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {45, 37} \[ -\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}} \]
Antiderivative was successfully verified.
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Rule 45
Rule 37
Rubi steps
\begin{align*} \int \frac{1}{(a-i a x)^{15/4} \sqrt [4]{a+i a x}} \, dx &=-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}+\frac{4 \int \frac{1}{(a-i a x)^{11/4} \sqrt [4]{a+i a x}} \, dx}{11 a}\\ &=-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}+\frac{8 \int \frac{1}{(a-i a x)^{7/4} \sqrt [4]{a+i a x}} \, dx}{77 a^2}\\ &=-\frac{2 i (a+i a x)^{3/4}}{11 a^2 (a-i a x)^{11/4}}-\frac{8 i (a+i a x)^{3/4}}{77 a^3 (a-i a x)^{7/4}}-\frac{16 i (a+i a x)^{3/4}}{231 a^4 (a-i a x)^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0245436, size = 52, normalized size = 0.52 \[ \frac{2 \left (-8 i x^2+28 x+41 i\right ) (a+i a x)^{3/4}}{231 a^4 (x+i)^2 (a-i a x)^{3/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 50, normalized size = 0.5 \begin{align*}{\frac{40\,i{x}^{2}+16\,{x}^{3}-26\,x+82\,i}{231\,{a}^{3} \left ( x+i \right ) ^{2}} \left ( -a \left ( -1+ix \right ) \right ) ^{-{\frac{3}{4}}}{\frac{1}{\sqrt [4]{a \left ( 1+ix \right ) }}}} \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 \frac{1}{{\left (i \, a x + a\right )}^{\frac{1}{4}}{\left (-i \, a x + a\right )}^{\frac{15}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55214, size = 157, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (i \, a x + a\right )}^{\frac{3}{4}}{\left (-i \, a x + a\right )}^{\frac{1}{4}}{\left (8 \, x^{2} + 28 i \, x - 41\right )}}{231 \, a^{5} x^{3} + 693 i \, a^{5} x^{2} - 693 \, a^{5} x - 231 i \, a^{5}} \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(-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.
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