Optimal. Leaf size=19 \[ \left (a+b x+c x^2+d x^3\right )^{1+p} \]
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Rubi [A]
time = 0.03, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {1599, 1602}
\begin {gather*} \left (a+b x+c x^2+d x^3\right )^{p+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 1599
Rule 1602
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2+d x^3\right )^p \left (b (1+p) x+c (2+2 p) x^2+d (3+3 p) x^3\right )}{x} \, dx &=\int \left (b (1+p)+c (2+2 p) x+d (3+3 p) x^2\right ) \left (a+b x+c x^2+d x^3\right )^p \, dx\\ &=\left (a+b x+c x^2+d x^3\right )^{1+p}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 17, normalized size = 0.89 \begin {gather*} (a+x (b+x (c+d x)))^{1+p} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 20, normalized size = 1.05
method | result | size |
gosper | \(\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{1+p}\) | \(20\) |
risch | \(\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p} \left (d \,x^{3}+c \,x^{2}+b x +a \right )\) | \(34\) |
norman | \(a \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+b x \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+c \,x^{2} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+d \,x^{3} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 33, normalized size = 1.74 \begin {gather*} {\left (d x^{3} + c x^{2} + b x + a\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 = 0.39, size = 33, normalized size = 1.74 \begin {gather*} {\left (d x^{3} + c x^{2} + b x + a\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. 52 vs.
\(2 (19) = 38\).
time = 5.56, size = 52, normalized size = 2.74 \begin {gather*} \frac {{\left (d x^{3} + c x^{2} + b x + a\right )}^{p + 1} p}{p + 1} + \frac {{\left (d x^{3} + c x^{2} + b x + a\right )}^{p + 1}}{p + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.19, size = 19, normalized size = 1.00 \begin {gather*} {\left (d\,x^3+c\,x^2+b\,x+a\right )}^{p+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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