Optimal. Leaf size=30 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]
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Rubi [A] time = 0.0212933, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {372, 261} \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]
Antiderivative was successfully verified.
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Rule 372
Rule 261
Rubi steps
\begin{align*} \int (c+d x) \left (a+b (c+d x)^2\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int x \left (a+b x^2\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac{\left (a+b (c+d x)^2\right )^{1+p}}{2 b d (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0089686, size = 30, normalized size = 1. \[ \frac{\left (a+b (c+d x)^2\right )^{p+1}}{2 b d (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 39, normalized size = 1.3 \begin{align*}{\frac{ \left ( b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a \right ) ^{1+p}}{2\,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 [A] time = 1.5987, size = 126, normalized size = 4.2 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\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 [B] time = 1.10946, size = 170, normalized size = 5.67 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b d^{2} x^{2} + 2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b c d x +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b c^{2} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a}{2 \,{\left (b d p + b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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