Optimal. Leaf size=64 \[ \frac {a^x}{b^2 (1+b x)}+\frac {a^{-1/b} \text {Ei}\left (\frac {(1+b x) \log (a)}{b}\right )}{b^2}-\frac {a^{-1/b} \text {Ei}\left (\frac {(1+b x) \log (a)}{b}\right ) \log (a)}{b^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2230, 2208,
2209} \begin {gather*} -\frac {a^{-1/b} \log (a) \text {Ei}\left (\frac {(b x+1) \log (a)}{b}\right )}{b^3}+\frac {a^{-1/b} \text {Ei}\left (\frac {(b x+1) \log (a)}{b}\right )}{b^2}+\frac {a^x}{b^2 (b x+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2208
Rule 2209
Rule 2230
Rubi steps
\begin {align*} \int \frac {a^x x}{(1+b x)^2} \, dx &=\int \left (-\frac {a^x}{b (1+b x)^2}+\frac {a^x}{b (1+b x)}\right ) \, dx\\ &=-\frac {\int \frac {a^x}{(1+b x)^2} \, dx}{b}+\frac {\int \frac {a^x}{1+b x} \, dx}{b}\\ &=\frac {a^x}{b^2 (1+b x)}+\frac {a^{-1/b} \text {Ei}\left (\frac {(1+b x) \log (a)}{b}\right )}{b^2}-\frac {\log (a) \int \frac {a^x}{1+b x} \, dx}{b^2}\\ &=\frac {a^x}{b^2 (1+b x)}+\frac {a^{-1/b} \text {Ei}\left (\frac {(1+b x) \log (a)}{b}\right )}{b^2}-\frac {a^{-1/b} \text {Ei}\left (\frac {(1+b x) \log (a)}{b}\right ) \log (a)}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 43, normalized size = 0.67 \begin {gather*} \frac {\frac {a^x b}{1+b x}+a^{-1/b} \text {Ei}\left (\frac {(1+b x) \log (a)}{b}\right ) (b-\log (a))}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded while calling a Python object} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.04, size = 79, normalized size = 1.23
method | result | size |
risch | \(-\frac {a^{-\frac {1}{b}} \expIntegral \left (1, -x \ln \left (a \right )-\frac {\ln \left (a \right )}{b}\right )}{b^{2}}+\frac {\ln \left (a \right ) a^{x}}{b^{3} \left (x \ln \left (a \right )+\frac {\ln \left (a \right )}{b}\right )}+\frac {\ln \left (a \right ) a^{-\frac {1}{b}} \expIntegral \left (1, -x \ln \left (a \right )-\frac {\ln \left (a \right )}{b}\right )}{b^{3}}\) | \(79\) |
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 [A]
time = 0.31, size = 54, normalized size = 0.84 \begin {gather*} \frac {a^{x} b + \frac {{\left (b^{2} x - {\left (b x + 1\right )} \log \left (a\right ) + b\right )} {\rm Ei}\left (\frac {{\left (b x + 1\right )} \log \left (a\right )}{b}\right )}{a^{\left (\frac {1}{b}\right )}}}{b^{4} x + b^{3}} \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 \frac {a^{x} x}{\left (b x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 = -0.02 \begin {gather*} \int \frac {a^x\,x}{{\left (b\,x+1\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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