Optimal. Leaf size=220 \[ \frac {2^{-\frac {i n}{2}} \left (-6 a^2-6 a n-n^2+2\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{3 b^3 (-n+2 i)}-\frac {(4 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{3 b^2} \]
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Rubi [A] time = 0.14, antiderivative size = 220, 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, 90, 80, 69} \[ \frac {2^{-\frac {i n}{2}} \left (-6 a^2-6 a n-n^2+2\right ) (-i a-i b x+1)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;\frac {1}{2} (-i a-i b x+1)\right )}{3 b^3 (-n+2 i)}-\frac {(4 a+n) (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (-i a-i b x+1)^{1+\frac {i n}{2}} (i a+i b x+1)^{1-\frac {i n}{2}}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 69
Rule 80
Rule 90
Rule 5095
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a+b x)} x^2 \, dx &=\int x^2 (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx\\ &=\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}+\frac {\int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (-1-a^2-b (4 a+n) x\right ) \, dx}{3 b^2}\\ &=-\frac {(4 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}-\frac {\left (2-6 a^2-6 a n-n^2\right ) \int (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx}{6 b^2}\\ &=-\frac {(4 a+n) (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{6 b^3}+\frac {x (1-i a-i b x)^{1+\frac {i n}{2}} (1+i a+i b x)^{1-\frac {i n}{2}}}{3 b^2}+\frac {2^{-\frac {i n}{2}} \left (2-6 a^2-6 a n-n^2\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^3 (2 i-n)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 160, normalized size = 0.73 \[ \frac {(-i (a+b x+i))^{1+\frac {i n}{2}} \left (\frac {2^{1-\frac {i n}{2}} \left (6 a^2+6 a n+n^2-2\right ) \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2};\frac {i n}{2}+2;-\frac {1}{2} i (a+b x+i)\right )}{n-2 i}-\left ((4 a+n) (i a+i b x+1)^{1-\frac {i n}{2}}\right )+2 b x (i a+i b x+1)^{1-\frac {i n}{2}}\right )}{6 b^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{2} e^{\left (n \arctan \left (b x + a\right )\right )}, x\right ) \]
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 = 0.07, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{2}\, dx \]
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 \[ \int x^{2} e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \]
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 \[ \int x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} e^{n \operatorname {atan}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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