Optimal. Leaf size=140 \[ \frac {x^{m+1} (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \left (1-\frac {b x}{-a+i}\right )^{\frac {i n}{2}} \left (1+\frac {b x}{a+i}\right )^{-\frac {i n}{2}} F_1\left (m+1;-\frac {i n}{2},\frac {i n}{2};m+2;-\frac {b x}{a+i},\frac {b x}{i-a}\right )}{m+1} \]
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Rubi [A] time = 0.07, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5095, 135, 133} \[ \frac {x^{m+1} (-i a-i b x+1)^{\frac {i n}{2}} (i a+i b x+1)^{-\frac {i n}{2}} \left (1-\frac {b x}{-a+i}\right )^{\frac {i n}{2}} \left (1+\frac {b x}{a+i}\right )^{-\frac {i n}{2}} F_1\left (m+1;-\frac {i n}{2},\frac {i n}{2};m+2;-\frac {b x}{a+i},\frac {b x}{i-a}\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 5095
Rubi steps
\begin {align*} \int e^{n \tan ^{-1}(a+b x)} x^m \, dx &=\int x^m (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \, dx\\ &=\left ((1-i a-i b x)^{\frac {i n}{2}} \left (1-\frac {i b x}{1-i a}\right )^{-\frac {i n}{2}}\right ) \int x^m (1+i a+i b x)^{-\frac {i n}{2}} \left (1-\frac {i b x}{1-i a}\right )^{\frac {i n}{2}} \, dx\\ &=\left ((1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (1-\frac {i b x}{1-i a}\right )^{-\frac {i n}{2}} \left (1+\frac {i b x}{1+i a}\right )^{\frac {i n}{2}}\right ) \int x^m \left (1-\frac {i b x}{1-i a}\right )^{\frac {i n}{2}} \left (1+\frac {i b x}{1+i a}\right )^{-\frac {i n}{2}} \, dx\\ &=\frac {x^{1+m} (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (1-\frac {b x}{i-a}\right )^{\frac {i n}{2}} \left (1+\frac {b x}{i+a}\right )^{-\frac {i n}{2}} F_1\left (1+m;-\frac {i n}{2},\frac {i n}{2};2+m;-\frac {b x}{i+a},\frac {b x}{i-a}\right )}{1+m}\\ \end {align*}
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Mathematica [F] time = 0.94, size = 0, normalized size = 0.00 \[ \int e^{n \tan ^{-1}(a+b x)} x^m \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} e^{\left (n \arctan \left (b x + a\right )\right )}, x\right ) \]
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 = 0.21, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctan \left (b x +a \right )} x^{m}\, dx \]
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 \[ \int x^{m} e^{\left (n \arctan \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a+b\,x\right )} \,d x \]
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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