Optimal. Leaf size=140 \[ \frac {x^{1+m} (1-i a-i b x)^{\frac {i n}{2}} (1+i a+i b x)^{-\frac {i n}{2}} \left (1-\frac {b x}{i-a}\right )^{\frac {i n}{2}} \left (1+\frac {b x}{i+a}\right )^{-\frac {i n}{2}} F_1\left (1+m;-\frac {i n}{2},\frac {i n}{2};2+m;-\frac {b x}{i+a},\frac {b x}{i-a}\right )}{1+m} \]
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Rubi [A]
time = 0.06, 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 = {5203, 140, 138}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 138
Rule 140
Rule 5203
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 = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int e^{n \text {ArcTan}(a+b x)} x^m \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.02, 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 \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^{m} 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]
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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,{\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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