Optimal. Leaf size=113 \[ \frac {x (e x)^m \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \left (a x^j+b x^{j+n}\right )^p F_1\left (\frac {1+m+j p}{n};-p,-q;\frac {1+m+n+j p}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+m+j p} \]
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Rubi [A]
time = 0.15, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2067, 525, 524}
\begin {gather*} \frac {x (e x)^m \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \left (a x^j+b x^{j+n}\right )^p F_1\left (\frac {m+j p+1}{n};-p,-q;\frac {m+n+j p+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{j p+m+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 524
Rule 525
Rule 2067
Rubi steps
\begin {align*} \int (e x)^m \left (c+d x^n\right )^q \left (a x^j+b x^{j+n}\right )^p \, dx &=\left (x^{-m-j p} (e x)^m \left (a+b x^n\right )^{-p} \left (a x^j+b x^{j+n}\right )^p\right ) \int x^{m+j p} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx\\ &=\left (x^{-m-j p} (e x)^m \left (1+\frac {b x^n}{a}\right )^{-p} \left (a x^j+b x^{j+n}\right )^p\right ) \int x^{m+j p} \left (1+\frac {b x^n}{a}\right )^p \left (c+d x^n\right )^q \, dx\\ &=\left (x^{-m-j p} (e x)^m \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \left (a x^j+b x^{j+n}\right )^p\right ) \int x^{m+j p} \left (1+\frac {b x^n}{a}\right )^p \left (1+\frac {d x^n}{c}\right )^q \, dx\\ &=\frac {x (e x)^m \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \left (a x^j+b x^{j+n}\right )^p F_1\left (\frac {1+m+j p}{n};-p,-q;\frac {1+m+n+j p}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+m+j p}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 111, normalized size = 0.98 \begin {gather*} \frac {x (e x)^m \left (x^j \left (a+b x^n\right )\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} F_1\left (\frac {1+m+j p}{n};-p,-q;\frac {1+m+n+j p}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+m+j p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.55, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (c +d \,x^{n}\right )^{q} \left (a \,x^{j}+b \,x^{j +n}\right )^{p}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (a\,x^j+b\,x^{j+n}\right )}^p\,{\left (e\,x\right )}^m\,{\left (c+d\,x^n\right )}^q \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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