Optimal. Leaf size=23 \[ x^2 \left (a+b x+c x^2+d x^3\right )^{1+p} \]
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Rubi [A]
time = 0.04, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, Rules used = {1602}
\begin {gather*} x^2 \left (a+b x+c x^2+d x^3\right )^{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 1602
Rubi steps
\begin {align*} \int x \left (a+b x+c x^2+d x^3\right )^p \left (2 a+b (3+p) x+c (4+2 p) x^2+d (5+3 p) x^3\right ) \, dx &=x^2 \left (a+b x+c x^2+d x^3\right )^{1+p}\\ \end {align*}
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Mathematica [A]
time = 0.96, size = 21, normalized size = 0.91 \begin {gather*} x^2 (a+x (b+x (c+d x)))^{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 24, normalized size = 1.04
method | result | size |
gosper | \(x^{2} \left (d \,x^{3}+c \,x^{2}+b x +a \right )^{1+p}\) | \(24\) |
risch | \(\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p} x^{2} \left (d \,x^{3}+c \,x^{2}+b x +a \right )\) | \(37\) |
norman | \(a \,x^{2} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+b \,x^{3} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+c \,x^{4} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+d \,x^{5} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 39, normalized size = 1.70 \begin {gather*} {\left (d x^{5} + c x^{4} + b x^{3} + a x^{2}\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 39, normalized size = 1.70 \begin {gather*} {\left (d x^{5} + c x^{4} + b x^{3} + a x^{2}\right )} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 89 vs.
\(2 (23) = 46\).
time = 5.19, size = 89, normalized size = 3.87 \begin {gather*} {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} d x^{5} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} c x^{4} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} b x^{3} + {\left (d x^{3} + c x^{2} + b x + a\right )}^{p} a x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.32, size = 39, normalized size = 1.70 \begin {gather*} {\left (d\,x^3+c\,x^2+b\,x+a\right )}^p\,\left (d\,x^5+c\,x^4+b\,x^3+a\,x^2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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