Optimal. Leaf size=283 \[ \frac{c 2^{\frac{3}{2}-\frac{i n}{2}} \left (5-n^2\right ) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} \, _2F_1\left (\frac{1}{2} (i n-3),\frac{1}{2} (i n+5);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right )}{15 a^3 (-n+5 i) \sqrt{a^2 x^2+1}}-\frac{c n \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)}}{30 a^3 \sqrt{a^2 x^2+1}}+\frac{c x \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)}}{6 a^2 \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.310511, antiderivative size = 283, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {5085, 5082, 90, 80, 69} \[ \frac{c 2^{\frac{3}{2}-\frac{i n}{2}} \left (5-n^2\right ) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} \, _2F_1\left (\frac{1}{2} (i n-3),\frac{1}{2} (i n+5);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right )}{15 a^3 (-n+5 i) \sqrt{a^2 x^2+1}}-\frac{c n \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)}}{30 a^3 \sqrt{a^2 x^2+1}}+\frac{c x \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)}}{6 a^2 \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 5085
Rule 5082
Rule 90
Rule 80
Rule 69
Rubi steps
\begin{align*} \int e^{n \tan ^{-1}(a x)} x^2 \left (c+a^2 c x^2\right )^{3/2} \, dx &=\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \int e^{n \tan ^{-1}(a x)} x^2 \left (1+a^2 x^2\right )^{3/2} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \int x^2 (1-i a x)^{\frac{3}{2}+\frac{i n}{2}} (1+i a x)^{\frac{3}{2}-\frac{i n}{2}} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{c x (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)} \sqrt{c+a^2 c x^2}}{6 a^2 \sqrt{1+a^2 x^2}}+\frac{\left (c \sqrt{c+a^2 c x^2}\right ) \int (1-i a x)^{\frac{3}{2}+\frac{i n}{2}} (1+i a x)^{\frac{3}{2}-\frac{i n}{2}} (-1-a n x) \, dx}{6 a^2 \sqrt{1+a^2 x^2}}\\ &=-\frac{c n (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)} \sqrt{c+a^2 c x^2}}{30 a^3 \sqrt{1+a^2 x^2}}+\frac{c x (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)} \sqrt{c+a^2 c x^2}}{6 a^2 \sqrt{1+a^2 x^2}}+\frac{\left (c \left (-5+n^2\right ) \sqrt{c+a^2 c x^2}\right ) \int (1-i a x)^{\frac{3}{2}+\frac{i n}{2}} (1+i a x)^{\frac{3}{2}-\frac{i n}{2}} \, dx}{30 a^2 \sqrt{1+a^2 x^2}}\\ &=-\frac{c n (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)} \sqrt{c+a^2 c x^2}}{30 a^3 \sqrt{1+a^2 x^2}}+\frac{c x (1-i a x)^{\frac{1}{2} (5+i n)} (1+i a x)^{\frac{1}{2} (5-i n)} \sqrt{c+a^2 c x^2}}{6 a^2 \sqrt{1+a^2 x^2}}+\frac{2^{\frac{3}{2}-\frac{i n}{2}} c \left (5-n^2\right ) (1-i a x)^{\frac{1}{2} (5+i n)} \sqrt{c+a^2 c x^2} \, _2F_1\left (\frac{1}{2} (-3+i n),\frac{1}{2} (5+i n);\frac{1}{2} (7+i n);\frac{1}{2} (1-i a x)\right )}{15 a^3 (5 i-n) \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.246871, size = 217, normalized size = 0.77 \[ \frac{c 2^{-1-\frac{i n}{2}} (a x+i)^2 \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left (2^{\frac{i n}{2}} (n-5 i) \sqrt{1+i a x} (a x-i)^2 (5 a x-n)-4 \sqrt{2} \left (n^2-5\right ) (1+i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{1}{2} (i n+5),\frac{1}{2} i (n+3 i);\frac{1}{2} (i n+7);\frac{1}{2} (1-i a x)\right )\right )}{15 a^3 (n-5 i) \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.286, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }}{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) ^{{\frac{3}{2}}}\, 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{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}} x^{2} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \sqrt{a^{2} c x^{2} + c} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \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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