Optimal. Leaf size=283 \[ \frac {c x \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)}}{6 a^2 \sqrt {a^2 x^2+1}}+\frac {c 2^{\frac {3}{2}-\frac {i n}{2}} \left (5-n^2\right ) \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} \, _2F_1\left (\frac {1}{2} (i n-3),\frac {1}{2} (i n+5);\frac {1}{2} (i n+7);\frac {1}{2} (1-i a x)\right )}{15 a^3 (-n+5 i) \sqrt {a^2 x^2+1}}-\frac {c n \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)}}{30 a^3 \sqrt {a^2 x^2+1}} \]
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Rubi [A] time = 0.31, antiderivative size = 283, normalized size of antiderivative = 1.00, 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 69
Rule 80
Rule 90
Rule 5082
Rule 5085
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*}
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Mathematica [A] time = 0.27, 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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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \sqrt {a^{2} c x^{2} + c} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (a x \right )} x^{2} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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