Optimal. Leaf size=65 \[ \frac{(-3 a x+i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (1-i a x)^3 (1+i a x)^6 \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.208288, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {5085, 5082, 81} \[ \frac{(-3 a x+i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (1-i a x)^3 (1+i a x)^6 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5085
Rule 5082
Rule 81
Rubi steps
\begin{align*} \int \frac{e^{-3 i \tan ^{-1}(a x)} x^2}{\left (c+a^2 c x^2\right )^{11/2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{-3 i \tan ^{-1}(a x)} x^2}{\left (1+a^2 x^2\right )^{11/2}} \, dx}{c^5 \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int \frac{x^2}{(1-i a x)^4 (1+i a x)^7} \, dx}{c^5 \sqrt{c+a^2 c x^2}}\\ &=\frac{(i-3 a x) \sqrt{1+a^2 x^2}}{24 a^3 c^5 (1-i a x)^3 (1+i a x)^6 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.100985, size = 65, normalized size = 1. \[ -\frac{i (3 a x-i) \sqrt{a^2 x^2+1}}{24 a^3 c^5 (a x-i)^6 (a x+i)^3 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 58, normalized size = 0.9 \begin{align*} -{\frac{ \left ( -ax+i \right ) \left ( ax+i \right ) \left ( i-3\,ax \right ) }{24\,{a}^{3} \left ( 1+iax \right ) ^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08692, size = 126, normalized size = 1.94 \begin{align*} \frac{3 \, a x - i}{24 i \, a^{12} c^{\frac{11}{2}} x^{9} + 72 \, a^{11} c^{\frac{11}{2}} x^{8} + 192 \, a^{9} c^{\frac{11}{2}} x^{6} - 144 i \, a^{8} c^{\frac{11}{2}} x^{5} + 144 \, a^{7} c^{\frac{11}{2}} x^{4} - 192 i \, a^{6} c^{\frac{11}{2}} x^{3} - 72 i \, a^{4} c^{\frac{11}{2}} x - 24 \, a^{3} c^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.41703, size = 437, normalized size = 6.72 \begin{align*} \frac{{\left (-i \, a^{6} x^{9} - 3 \, a^{5} x^{8} - 8 \, a^{3} x^{6} + 6 i \, a^{2} x^{5} - 6 \, a x^{4} + 8 i \, x^{3}\right )} \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1}}{24 \, a^{11} c^{6} x^{11} - 72 i \, a^{10} c^{6} x^{10} + 24 \, a^{9} c^{6} x^{9} - 264 i \, a^{8} c^{6} x^{8} - 144 \, a^{7} c^{6} x^{7} - 336 i \, a^{6} c^{6} x^{6} - 336 \, a^{5} c^{6} x^{5} - 144 i \, a^{4} c^{6} x^{4} - 264 \, a^{3} c^{6} x^{3} + 24 i \, a^{2} c^{6} x^{2} - 72 \, a c^{6} x + 24 i \, c^{6}} \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 \frac{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{11}{2}}{\left (i \, a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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