Optimal. Leaf size=211 \[ \frac {A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} F_1\left (\frac {m+1}{n};-p,-q;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (m+1)}+\frac {B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} F_1\left (\frac {m+n+1}{n};-p,-q;\frac {m+2 n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{m+n+1} \]
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Rubi [A] time = 0.25, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {598, 511, 510} \[ \frac {A (e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} F_1\left (\frac {m+1}{n};-p,-q;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (m+1)}+\frac {B x^{n+1} (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} F_1\left (\frac {m+n+1}{n};-p,-q;\frac {m+2 n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{m+n+1} \]
Antiderivative was successfully verified.
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Rule 510
Rule 511
Rule 598
Rubi steps
\begin {align*} \int (e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right ) \left (c+d x^n\right )^q \, dx &=A \int (e x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx+\left (B x^{-m} (e x)^m\right ) \int x^{m+n} \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx\\ &=\left (A \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 \left (c+d x^n\right )^q \, dx+\left (B x^{-m} (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p}\right ) \int x^{m+n} \left (1+\frac {b x^n}{a}\right )^p \left (c+d x^n\right )^q \, dx\\ &=\left (A \left (a+b x^n\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}\right ) \int (e x)^m \left (1+\frac {b x^n}{a}\right )^p \left (1+\frac {d x^n}{c}\right )^q \, dx+\left (B x^{-m} (e x)^m \left (a+b x^n\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}\right ) \int x^{m+n} \left (1+\frac {b x^n}{a}\right )^p \left (1+\frac {d x^n}{c}\right )^q \, dx\\ &=\frac {A (e x)^{1+m} \left (a+b x^n\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}{n};-p,-q;\frac {1+m+n}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{e (1+m)}+\frac {B x^{1+n} (e x)^m \left (a+b x^n\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+n}{n};-p,-q;\frac {1+m+2 n}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+m+n}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 162, normalized size = 0.77 \[ \frac {x (e x)^m \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^q \left (\frac {d x^n}{c}+1\right )^{-q} \left (A (m+n+1) F_1\left (\frac {m+1}{n};-p,-q;\frac {m+n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+B (m+1) x^n F_1\left (\frac {m+n+1}{n};-p,-q;\frac {m+2 n+1}{n};-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{(m+1) (m+n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}, x\right ) \]
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 \[ \int {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.45, size = 0, normalized size = 0.00 \[ \int \left (B \,x^{n}+A \right ) \left (e x \right )^{m} \left (b \,x^{n}+a \right )^{p} \left (d \,x^{n}+c \right )^{q}\, dx \]
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 \[ \int {\left (B x^{n} + A\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{q} \left (e x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (a+b\,x^n\right )}^p\,{\left (c+d\,x^n\right )}^q \,d x \]
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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