Optimal. Leaf size=30 \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
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Rubi [A] time = 0.0301744, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {372, 261} \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^4\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^4\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b (c+d x)^4\right )^{1+p}}{4 b d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0166376, size = 30, normalized size = 1. \[ \frac{\left (a+b (c+d x)^4\right )^{p+1}}{4 b d (p+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 63, normalized size = 2.1 \begin{align*}{\frac{ \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) ^{1+p}}{4\,bd \left ( 1+p \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.29188, size = 223, normalized size = 7.43 \begin{align*} \frac{{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}{\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}^{p}}{4 \,{\left (b d p + b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20282, size = 38, normalized size = 1.27 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{4} b + a\right )}^{p + 1}}{4 \, b d{\left (p + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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