Optimal. Leaf size=260 \[ \frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \, _2F_1\left (1+\frac {i n}{2},\frac {i n}{2};2+\frac {i n}{2};\frac {1}{2} (1-i a-i b x)\right )}{3 b^4 (2 i-n)} \]
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Rubi [A]
time = 0.14, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5203, 102, 152,
71} \begin {gather*} -\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} \left (-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right ) (i a+i b x+1)^{1-\frac {i n}{2}}}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{3 b^4 (-n+2 i)}+\frac {x^2 (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{4 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 152
Rule 5203
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a+b x)} x^3 \, dx &=\int x^3 (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx\\ &=\frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}+\frac {\int x (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (-2 \left (1+a^2\right )-b (6 a+n) x\right ) \, dx}{4 b^2}\\ &=\frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}-\frac {\left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx}{24 b^3}\\ &=\frac {x^2 (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{4 b^2}-\frac {(1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}} \left (6-18 a^2-10 a n-n^2+2 b (6 a+n) x\right )}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (1-i a-i b x)^{1+\frac {i n}{2}} \, _2F_1\left (1+\frac {i n}{2},\frac {i n}{2};2+\frac {i n}{2};\frac {1}{2} (1-i a-i b x)\right )}{3 b^4 (2 i-n)}\\ \end {align*}
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Mathematica [A]
time = 1.14, size = 243, normalized size = 0.93 \begin {gather*} -\frac {e^{n \text {ArcTan}(a+b x)} \left (\left (24 a^3+36 a^2 n+n \left (-8+n^2\right )+12 a \left (-2+n^2\right )\right ) (a+b x)-\left (-12+36 a^2+12 a n+n^2\right ) \left (1+a^2+2 a b x+b^2 x^2\right )+2 (12 a+n) (a+b x) \left (1+(a+b x)^2\right )-6 \left (1+(a+b x)^2\right )^2+\frac {\left (24 a^3+36 a^2 n+n \left (-8+n^2\right )+12 a \left (-2+n^2\right )\right ) \left (-e^{2 i \text {ArcTan}(a+b x)} n \, _2F_1\left (1,1-\frac {i n}{2};2-\frac {i n}{2};-e^{2 i \text {ArcTan}(a+b x)}\right )+(2 i+n) \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};-e^{2 i \text {ArcTan}(a+b x)}\right )\right )}{-2+i n}\right )}{24 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{3}\, dx\]
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^{3} e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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