Optimal. Leaf size=62 \[ \frac{\left (a+b (c+d x)^2\right )^{p+2}}{2 b^2 d (p+2)}-\frac{a \left (a+b (c+d x)^2\right )^{p+1}}{2 b^2 d (p+1)} \]
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Rubi [A] time = 0.0590412, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {372, 266, 43} \[ \frac{\left (a+b (c+d x)^2\right )^{p+2}}{2 b^2 d (p+2)}-\frac{a \left (a+b (c+d x)^2\right )^{p+1}}{2 b^2 d (p+1)} \]
Antiderivative was successfully verified.
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Rule 372
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (c+d x)^3 \left (a+b (c+d x)^2\right )^p \, dx &=\frac{\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right )^p \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int x (a+b x)^p \, dx,x,(c+d x)^2\right )}{2 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^p}{b}+\frac{(a+b x)^{1+p}}{b}\right ) \, dx,x,(c+d x)^2\right )}{2 d}\\ &=-\frac{a \left (a+b (c+d x)^2\right )^{1+p}}{2 b^2 d (1+p)}+\frac{\left (a+b (c+d x)^2\right )^{2+p}}{2 b^2 d (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0416965, size = 51, normalized size = 0.82 \[ \frac{\left (a+b (c+d x)^2\right )^{p+1} \left (b (p+1) (c+d x)^2-a\right )}{2 b^2 d (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 91, normalized size = 1.5 \begin{align*} -{\frac{ \left ( b{d}^{2}{x}^{2}+2\,bcdx+b{c}^{2}+a \right ) ^{1+p} \left ( -b{d}^{2}p{x}^{2}-2\,bcdpx-b{d}^{2}{x}^{2}-b{c}^{2}p-2\,bcdx-b{c}^{2}+a \right ) }{2\,{b}^{2}d \left ({p}^{2}+3\,p+2 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.39967, size = 189, normalized size = 3.05 \begin{align*} \frac{{\left (b^{2} d^{4}{\left (p + 1\right )} x^{4} + 4 \, b^{2} c d^{3}{\left (p + 1\right )} x^{3} + b^{2} c^{4}{\left (p + 1\right )} + a b c^{2} p +{\left (6 \, b^{2} c^{2} d^{2}{\left (p + 1\right )} + a b d^{2} p\right )} x^{2} - a^{2} + 2 \,{\left (2 \, b^{2} c^{3} d{\left (p + 1\right )} + a b c d p\right )} x\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (p^{2} + 3 \, p + 2\right )} b^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.6184, size = 369, normalized size = 5.95 \begin{align*} \frac{{\left (b^{2} c^{4} +{\left (b^{2} d^{4} p + b^{2} d^{4}\right )} x^{4} + 4 \,{\left (b^{2} c d^{3} p + b^{2} c d^{3}\right )} x^{3} +{\left (6 \, b^{2} c^{2} d^{2} +{\left (6 \, b^{2} c^{2} + a b\right )} d^{2} p\right )} x^{2} - a^{2} +{\left (b^{2} c^{4} + a b c^{2}\right )} p + 2 \,{\left (2 \, b^{2} c^{3} d +{\left (2 \, b^{2} c^{3} + a b c\right )} d p\right )} x\right )}{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p}}{2 \,{\left (b^{2} d p^{2} + 3 \, b^{2} d p + 2 \, b^{2} 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 = 1.12388, size = 662, normalized size = 10.68 \begin{align*} \frac{{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} d^{4} p x^{4} + 4 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c d^{3} p x^{3} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} d^{4} x^{4} + 6 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c^{2} d^{2} p x^{2} + 4 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c d^{3} x^{3} + 4 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c^{3} d p x + 6 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c^{2} d^{2} x^{2} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c^{4} p + 4 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c^{3} d x +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a b d^{2} p x^{2} +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} b^{2} c^{4} + 2 \,{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a b c d p x +{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a b c^{2} p -{\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a\right )}^{p} a^{2}}{2 \,{\left (b^{2} d p^{2} + 3 \, b^{2} d p + 2 \, b^{2} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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