Optimal. Leaf size=103 \[ \frac {\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac {e x}{d}\right )^{-p} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (m+1)} \]
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Rubi [A] time = 0.05, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {759, 133} \[ \frac {\left (b x+c x^2\right )^p (d+e x)^{m+1} \left (-\frac {e x}{d}\right )^{-p} \left (1-\frac {c (d+e x)}{c d-b e}\right )^{-p} F_1\left (m+1;-p,-p;m+2;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 133
Rule 759
Rubi steps
\begin {align*} \int (d+e x)^m \left (b x+c x^2\right )^p \, dx &=\frac {\left (\left (b x+c x^2\right )^p \left (1-\frac {d+e x}{d}\right )^{-p} \left (1-\frac {d+e x}{d-\frac {b e}{c}}\right )^{-p}\right ) \operatorname {Subst}\left (\int x^m \left (1-\frac {x}{d}\right )^p \left (1-\frac {c x}{c d-b e}\right )^p \, dx,x,d+e x\right )}{e}\\ &=\frac {\left (-\frac {e x}{d}\right )^{-p} (d+e x)^{1+m} \left (b x+c x^2\right )^p \left (1-\frac {c (d+e x)}{c d-b e}\right )^{-p} F_1\left (1+m;-p,-p;2+m;\frac {d+e x}{d},\frac {c (d+e x)}{c d-b e}\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 76, normalized size = 0.74 \[ \frac {x \left (\frac {b+c x}{b}\right )^{-p} (x (b+c x))^p (d+e x)^m \left (\frac {d+e x}{d}\right )^{-m} F_1\left (p+1;-p,-m;p+2;-\frac {c x}{b},-\frac {e x}{d}\right )}{p+1} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.99, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} {\left (e x + d\right )}^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \left (c \,x^{2}+b x \right )^{p} \left (e x +d \right )^{m}\, dx \]
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 \[ \int {\left (c x^{2} + b x\right )}^{p} {\left (e x + d\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (c\,x^2+b\,x\right )}^p\,{\left (d+e\,x\right )}^m \,d x \]
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 \[ \int \left (x \left (b + c x\right )\right )^{p} \left (d + e x\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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