Optimal. Leaf size=322 \[ \frac {x^2 \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{3 a^2 \sqrt {a^2 c x^2+c}}+\frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (3+i n)} \, _2F_1\left (\frac {1}{2} (i n+1),\frac {1}{2} (i n+3);\frac {1}{2} (i n+5);\frac {1}{2} (1-i a x)\right )}{3 a^4 \left (4 n-i \left (3-n^2\right )\right ) \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \left (a (1+i n) n x-n^2-i n+4\right ) (1+i a x)^{\frac {1}{2} (1-i n)}}{6 a^4 (1+i n) \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.36, antiderivative size = 322, 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, 100, 146, 69} \[ \frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (3+i n)} \, _2F_1\left (\frac {1}{2} (i n+1),\frac {1}{2} (i n+3);\frac {1}{2} (i n+5);\frac {1}{2} (1-i a x)\right )}{3 a^4 \left (4 n-i \left (3-n^2\right )\right ) \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} \left (a (1+i n) n x-n^2-i n+4\right ) (1+i a x)^{\frac {1}{2} (1-i n)}}{6 a^4 (1+i n) \sqrt {a^2 c x^2+c}}+\frac {x^2 \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)}}{3 a^2 \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 146
Rule 5082
Rule 5085
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)} x^3}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{n \tan ^{-1}(a x)} x^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int x^3 (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \int x (1-i a x)^{-\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {1}{2}-\frac {i n}{2}} (-2-a n x) \, dx}{3 a^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}-\frac {(1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \left (4-i n-n^2+a (1+i n) n x\right ) \sqrt {1+a^2 x^2}}{6 a^4 (1+i n) \sqrt {c+a^2 c x^2}}+\frac {\left (n \left (5-n^2\right ) \sqrt {1+a^2 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}{6 a^3 (1+i n) \sqrt {c+a^2 c x^2}}\\ &=\frac {x^2 (1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \sqrt {1+a^2 x^2}}{3 a^2 \sqrt {c+a^2 c x^2}}-\frac {(1-i a x)^{\frac {1}{2} (1+i n)} (1+i a x)^{\frac {1}{2} (1-i n)} \left (4-i n-n^2+a (1+i n) n x\right ) \sqrt {1+a^2 x^2}}{6 a^4 (1+i n) \sqrt {c+a^2 c x^2}}+\frac {2^{-\frac {1}{2}-\frac {i n}{2}} n \left (5-n^2\right ) (1-i a x)^{\frac {1}{2} (3+i n)} \sqrt {1+a^2 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^4 \left (4 n-i \left (3-n^2\right )\right ) \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 248, normalized size = 0.77 \[ \frac {2^{-\frac {3}{2}-\frac {i n}{2}} \sqrt {a^2 x^2+1} (1-i a x)^{\frac {1}{2}+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2^{\frac {1}{2}+\frac {i n}{2}} (n-3 i) \sqrt {1+i a x} \left (n \left (2 a^2 x^2+i a x+1\right )-2 i \left (a^2 x^2-2\right )-\left (n^2 (a x+i)\right )\right )+2 n \left (n^2-5\right ) (a x+i) (1+i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+\frac {1}{2},\frac {i n}{2}+\frac {3}{2};\frac {i n}{2}+\frac {5}{2};\frac {1}{2}-\frac {i a x}{2}\right )\right )}{3 a^4 \left (n^2-4 i n-3\right ) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}, x\right ) \]
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 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{3}}{\sqrt {a^{2} c \,x^{2}+c}}\, dx \]
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 \[ \int \frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{\sqrt {a^{2} c x^{2} + c}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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 \[ \int \frac {x^{3} e^{n \operatorname {atan}{\left (a x \right )}}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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