Optimal. Leaf size=283 \[ -\frac {c n (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{30 a^3 \sqrt {1+a^2 x^2}}+\frac {c x (1-i a x)^{\frac {1}{2} (5+i n)} (1+i a x)^{\frac {1}{2} (5-i n)} \sqrt {c+a^2 c x^2}}{6 a^2 \sqrt {1+a^2 x^2}}+\frac {2^{\frac {3}{2}-\frac {i n}{2}} c \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (5+i n)} \sqrt {c+a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-3+i n),\frac {1}{2} (5+i n);\frac {1}{2} (7+i n);\frac {1}{2} (1-i a x)\right )}{15 a^3 (5 i-n) \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.22, 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 = {5193, 5190, 92,
81, 71} \begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 81
Rule 92
Rule 5190
Rule 5193
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 = 0.19, size = 217, normalized size = 0.77 \begin {gather*} \frac {2^{-1-\frac {i n}{2}} c (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} (i+a x)^2 \sqrt {c+a^2 c x^2} \left (2^{\frac {i n}{2}} (-5 i+n) \sqrt {1+i a x} (-i+a x)^2 (-n+5 a x)-4 \sqrt {2} \left (-5+n^2\right ) (1+i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {1}{2} (5+i n),\frac {1}{2} i (3 i+n);\frac {1}{2} (7+i n);\frac {1}{2} (1-i a x)\right )\right )}{15 a^3 (-5 i+n) \sqrt {1+a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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