Optimal. Leaf size=49 \[ \frac {x^2 \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+2;p+3;-\frac {c x}{b}\right )}{p+2} \]
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Rubi [A] time = 0.02, antiderivative size = 83, normalized size of antiderivative = 1.69, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {640, 624} \[ \frac {\left (b x+c x^2\right )^{p+1} \left (-\frac {c x}{b}\right )^{-p-1} \, _2F_1\left (-p,p+1;p+2;\frac {b+c x}{b}\right )}{2 c (p+1)}+\frac {\left (b x+c x^2\right )^{p+1}}{2 c (p+1)} \]
Antiderivative was successfully verified.
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Rule 624
Rule 640
Rubi steps
\begin {align*} \int x \left (b x+c x^2\right )^p \, dx &=\frac {\left (b x+c x^2\right )^{1+p}}{2 c (1+p)}-\frac {b \int \left (b x+c x^2\right )^p \, dx}{2 c}\\ &=\frac {\left (b x+c x^2\right )^{1+p}}{2 c (1+p)}+\frac {\left (-\frac {c x}{b}\right )^{-1-p} \left (b x+c x^2\right )^{1+p} \, _2F_1\left (-p,1+p;2+p;\frac {b+c x}{b}\right )}{2 c (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 47, normalized size = 0.96 \[ \frac {x^2 (x (b+c x))^p \left (\frac {c x}{b}+1\right )^{-p} \, _2F_1\left (-p,p+2;p+3;-\frac {c x}{b}\right )}{p+2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} x, x\right ) \]
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 \[ \int {\left (c x^{2} + b x\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.51, size = 0, normalized size = 0.00 \[ \int x \left (c \,x^{2}+b x \right )^{p}\, dx \]
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 \[ \int {\left (c x^{2} + b x\right )}^{p} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\left (c\,x^2+b\,x\right )}^p \,d x \]
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 \[ \int x \left (x \left (b + c x\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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