Optimal. Leaf size=49 \[ \frac {x^4 \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,4+p;5+p;-\frac {c x}{b}\right )}{4+p} \]
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Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {688, 68, 66}
\begin {gather*} \frac {x^4 \left (\frac {c x}{b}+1\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,p+4;p+5;-\frac {c x}{b}\right )}{p+4} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 68
Rule 688
Rubi steps
\begin {align*} \int x^3 \left (b x+c x^2\right )^p \, dx &=\left (x^{-p} (b+c x)^{-p} \left (b x+c x^2\right )^p\right ) \int x^{3+p} (b+c x)^p \, dx\\ &=\left (x^{-p} \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p\right ) \int x^{3+p} \left (1+\frac {c x}{b}\right )^p \, dx\\ &=\frac {x^4 \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \, _2F_1\left (-p,4+p;5+p;-\frac {c x}{b}\right )}{4+p}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 47, normalized size = 0.96 \begin {gather*} \frac {x^4 (x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \, _2F_1\left (-p,4+p;5+p;-\frac {c x}{b}\right )}{4+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int x^{3} \left (c \,x^{2}+b x \right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{3} \left (x \left (b + c x\right )\right )^{p}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\left (c\,x^2+b\,x\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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