Optimal. Leaf size=55 \[ \frac {x (d x)^m \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac {c x}{b}\right )}{m+p+1} \]
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Rubi [A] time = 0.02, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {674, 66, 64} \[ \frac {x (d x)^m \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,m+p+1;m+p+2;-\frac {c x}{b}\right )}{m+p+1} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 674
Rubi steps
\begin {align*} \int (d x)^m \left (b x+c x^2\right )^p \, dx &=\left (x^{-m-p} (d x)^m (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{m+p} (b+c x)^p \, dx\\ &=\left (x^{-m-p} (d x)^m \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{m+p} \left (1+\frac {c x}{b}\right )^p \, dx\\ &=\frac {x (d x)^m \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,1+m+p;2+m+p;-\frac {c x}{b}\right )}{1+m+p}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.96 \[ \frac {x (d x)^m (x (b+c x))^p \left (\frac {c x}{b}+1\right )^{-p} \, _2F_1\left (-p,m+p+1;m+p+2;-\frac {c x}{b}\right )}{m+p+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + b x\right )}^{p} \left (d x\right )^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + b x\right )}^{p} \left (d x\right )^{m}\,{d x} \]
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 \[ \int \left (d x \right )^{m} \left (c \,x^{2}+b x \right )^{p}\, dx \]
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 \[ \int {\left (c x^{2} + b x\right )}^{p} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (c\,x^2+b\,x\right )}^p\,{\left (d\,x\right )}^m \,d x \]
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 \[ \int \left (d x\right )^{m} \left (x \left (b + c x\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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