Optimal. Leaf size=260 \[ -\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} \left (-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right ) (i a+i b x+1)^{1-\frac {i n}{2}}}{24 b^4}+\frac {2^{-2-\frac {i n}{2}} \left (24 a^3+36 a^2 n-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{3 b^4 (-n+2 i)}+\frac {x^2 (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{4 b^2} \]
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Rubi [A] time = 0.19, 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 = {5095, 100, 147, 69} \[ \frac {2^{-2-\frac {i n}{2}} \left (36 a^2 n+24 a^3-12 a \left (2-n^2\right )-n \left (8-n^2\right )\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{3 b^4 (-n+2 i)}-\frac {(-i a-i b x+1)^{1+\frac {i n}{2}} \left (-18 a^2+2 b x (6 a+n)-10 a n-n^2+6\right ) (i a+i b x+1)^{1-\frac {i n}{2}}}{24 b^4}+\frac {x^2 (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 100
Rule 147
Rule 5095
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 = 0.38, size = 272, normalized size = 1.05 \[ \frac {(-i (a+b x+i))^{1+\frac {i n}{2}} \left (b^2 (-n+2 i) x^2 (i a+i b x+1)^{1-\frac {i n}{2}}-2^{3-\frac {i n}{2}} (6 a+n) \, _2F_1\left (\frac {i n}{2}-2,\frac {i n}{2}+1;\frac {i n}{2}+2;-\frac {1}{2} i (a+b x+i)\right )+(1+i a) 2^{3-\frac {i n}{2}} (5 a+n-i) \, _2F_1\left (\frac {i n}{2}-1,\frac {i n}{2}+1;\frac {i n}{2}+2;-\frac {1}{2} i (a+b x+i)\right )+(a-i)^2 2^{1-\frac {i n}{2}} (4 a+n-2 i) \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;-\frac {1}{2} i (a+b x+i)\right )\right )}{4 b^4 (-n+2 i)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{3} e^{\left (n \arctan \left (b x + a\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.12, 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 \[ \int x^{3} e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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