Optimal. Leaf size=280 \[ -\frac {n (1-i a x)^{\frac {1}{2} (3+i n)} (1+i a x)^{\frac {1}{2} (3-i n)} \sqrt {c+a^2 c x^2}}{12 a^3 \sqrt {1+a^2 x^2}}+\frac {x (1-i a x)^{\frac {1}{2} (3+i n)} (1+i a x)^{\frac {1}{2} (3-i n)} \sqrt {c+a^2 c x^2}}{4 a^2 \sqrt {1+a^2 x^2}}+\frac {2^{-\frac {1}{2}-\frac {i n}{2}} \left (3-n^2\right ) (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {c+a^2 c x^2} \, _2F_1\left (\frac {1}{2} (-1+i n),\frac {1}{2} (3+i n);\frac {1}{2} (5+i n);\frac {1}{2} (1-i a x)\right )}{3 a^3 (3 i-n) \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.20, antiderivative size = 280, 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 {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}}+\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}} \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 \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*}
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Mathematica [A]
time = 0.15, size = 214, normalized size = 0.76 \begin {gather*} \frac {2^{-2-\frac {i n}{2}} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} (i+a x) \sqrt {c+a^2 c x^2} \left (2^{\frac {i n}{2}} (-3 i+n) \sqrt {1+i a x} (-i+a x) (-n+3 a x)-2 i \sqrt {2} \left (-3+n^2\right ) (1+i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {1}{2} (3+i n),\frac {1}{2} i (i+n);\frac {1}{2} (5+i n);\frac {1}{2} (1-i a x)\right )\right )}{3 a^3 (-3 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 \[\int {\mathrm e}^{n \arctan \left (a x \right )} x^{2} \sqrt {a^{2} c \,x^{2}+c}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} e^{n \operatorname {atan}{\left (a x \right )}}\, dx \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 )}\,\sqrt {c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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