Optimal. Leaf size=164 \[ \frac {B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{d e (m+1)}-\frac {(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} F_1\left (\frac {m+1}{n};-p,1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c d e (m+1)} \]
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Rubi [A] time = 0.18, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {597, 365, 364, 511, 510} \[ \frac {B (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{n},-p;\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{d e (m+1)}-\frac {(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} F_1\left (\frac {m+1}{n};-p,1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c d e (m+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 510
Rule 511
Rule 597
Rubi steps
\begin {align*} \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{c+d x^n} \, dx &=\int \left (\frac {B (e x)^m \left (a+b x^n\right )^p}{d}+\frac {(-B c+A d) (e x)^m \left (a+b x^n\right )^p}{d \left (c+d x^n\right )}\right ) \, dx\\ &=\frac {B \int (e x)^m \left (a+b x^n\right )^p \, dx}{d}+\frac {(-B c+A d) \int \frac {(e x)^m \left (a+b x^n\right )^p}{c+d x^n} \, dx}{d}\\ &=\frac {\left (B \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac {b x^n}{a}\right )^p \, dx}{d}+\frac {\left ((-B c+A d) \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int \frac {(e x)^m \left (1+\frac {b x^n}{a}\right )^p}{c+d x^n} \, dx}{d}\\ &=-\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} F_1\left (\frac {1+m}{n};-p,1;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c d e (1+m)}+\frac {B (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{n},-p;\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{d e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 138, normalized size = 0.84 \[ \frac {x (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (A (m+n+1) F_1\left (\frac {m+1}{n};-p,1;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+B (m+1) x^n F_1\left (\frac {m+n+1}{n};-p,1;\frac {m+2 n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{c (m+1) (m+n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d x^{n} + c}, x\right ) \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \,x^{n}+A \right ) \left (e x \right )^{m} \left (b \,x^{n}+a \right )^{p}}{d \,x^{n}+c}\, dx \]
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 \[ \int \frac {{\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} \left (e x\right )^{m}}{d x^{n} + c}\,{d x} \]
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 \[ \int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^p}{c+d\,x^n} \,d x \]
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 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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