Optimal. Leaf size=280 \[ \frac{2^{-\frac{1}{2}-\frac{i n}{2}} \left (3-n^2\right ) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left (\frac{1}{2} (i n-1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right )}{3 a^3 (-n+3 i) \sqrt{a^2 x^2+1}}-\frac{n \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)}}{12 a^3 \sqrt{a^2 x^2+1}}+\frac{x \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)}}{4 a^2 \sqrt{a^2 x^2+1}} \]
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Rubi [A] time = 0.288978, antiderivative size = 280, 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} \[ \frac{2^{-\frac{1}{2}-\frac{i n}{2}} \left (3-n^2\right ) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} \, _2F_1\left (\frac{1}{2} (i n-1),\frac{1}{2} (i n+3);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right )}{3 a^3 (-n+3 i) \sqrt{a^2 x^2+1}}-\frac{n \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)}}{12 a^3 \sqrt{a^2 x^2+1}}+\frac{x \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)}}{4 a^2 \sqrt{a^2 x^2+1}} \]
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 \sqrt{c+a^2 c x^2} \, dx &=\frac{\sqrt{c+a^2 c x^2} \int e^{n \tan ^{-1}(a x)} x^2 \sqrt{1+a^2 x^2} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{\sqrt{c+a^2 c x^2} \int x^2 (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{\frac{1}{2}-\frac{i n}{2}} \, dx}{\sqrt{1+a^2 x^2}}\\ &=\frac{x (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)} \sqrt{c+a^2 c x^2}}{4 a^2 \sqrt{1+a^2 x^2}}+\frac{\sqrt{c+a^2 c x^2} \int (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{\frac{1}{2}-\frac{i n}{2}} (-1-a n x) \, dx}{4 a^2 \sqrt{1+a^2 x^2}}\\ &=-\frac{n (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)} \sqrt{c+a^2 c x^2}}{12 a^3 \sqrt{1+a^2 x^2}}+\frac{x (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)} \sqrt{c+a^2 c x^2}}{4 a^2 \sqrt{1+a^2 x^2}}+\frac{\left (\left (-3+n^2\right ) \sqrt{c+a^2 c x^2}\right ) \int (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{\frac{1}{2}-\frac{i n}{2}} \, dx}{12 a^2 \sqrt{1+a^2 x^2}}\\ &=-\frac{n (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)} \sqrt{c+a^2 c x^2}}{12 a^3 \sqrt{1+a^2 x^2}}+\frac{x (1-i a x)^{\frac{1}{2} (3+i n)} (1+i a x)^{\frac{1}{2} (3-i n)} \sqrt{c+a^2 c x^2}}{4 a^2 \sqrt{1+a^2 x^2}}+\frac{2^{-\frac{1}{2}-\frac{i n}{2}} \left (3-n^2\right ) (1-i a x)^{\frac{1}{2} (3+i n)} \sqrt{c+a^2 c x^2} \, _2F_1\left (\frac{1}{2} (-1+i n),\frac{1}{2} (3+i n);\frac{1}{2} (5+i n);\frac{1}{2} (1-i a x)\right )}{3 a^3 (3 i-n) \sqrt{1+a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.19168, size = 214, normalized size = 0.76 \[ \frac{2^{-2-\frac{i n}{2}} (a x+i) \sqrt{a^2 c x^2+c} (1-i a x)^{\frac{1}{2}+\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left (2^{\frac{i n}{2}} (n-3 i) \sqrt{1+i a x} (a x-i) (3 a x-n)-2 i \sqrt{2} \left (n^2-3\right ) (1+i a x)^{\frac{i n}{2}} \, _2F_1\left (\frac{1}{2} (i n+3),\frac{1}{2} i (n+i);\frac{1}{2} (i n+5);\frac{1}{2} (1-i a x)\right )\right )}{3 a^3 (n-3 i) \sqrt{a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.302, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{n\arctan \left ( ax \right ) }}{x}^{2}\sqrt{{a}^{2}c{x}^{2}+c}\, 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 \sqrt{a^{2} c x^{2} + c} 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 (\sqrt{a^{2} c x^{2} + c} x^{2} 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{c \left (a^{2} x^{2} + 1\right )} e^{n \operatorname{atan}{\left (a x \right )}}\, dx \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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