Optimal. Leaf size=121 \[ -\frac {c 2^{\frac {5}{2}-\frac {i n}{2}} \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} \, _2F_1\left (\frac {1}{2} (i n-3),\frac {1}{2} (i n+5);\frac {1}{2} (i n+7);\frac {1}{2} (1-i a x)\right )}{a (-n+5 i) \sqrt {a^2 x^2+1}} \]
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Rubi [A] time = 0.10, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {5076, 5073, 69} \[ -\frac {c 2^{\frac {5}{2}-\frac {i n}{2}} \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {1}{2} (5+i n)} \, _2F_1\left (\frac {1}{2} (i n-3),\frac {1}{2} (i n+5);\frac {1}{2} (i n+7);\frac {1}{2} (1-i a x)\right )}{a (-n+5 i) \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a x)} \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)} \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 (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 {2^{\frac {5}{2}-\frac {i n}{2}} c (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 )}{a (5 i-n) \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 118, normalized size = 0.98 \[ \frac {c 2^{\frac {5}{2}-\frac {i n}{2}} \sqrt {a^2 c x^2+c} (1-i a x)^{\frac {5}{2}+\frac {i n}{2}} \, _2F_1\left (\frac {1}{2} (i n+5),\frac {1}{2} i (n+3 i);\frac {1}{2} (i n+7);\frac {1}{2} (1-i a x)\right )}{a (n-5 i) \sqrt {a^2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (n \arctan \left (a x\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (a x \right )} \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} e^{\left (n \arctan \left (a x\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} e^{n \operatorname {atan}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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