Optimal. Leaf size=121 \[ \frac{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac{2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]
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Rubi [A] time = 0.0730149, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {692, 629} \[ \frac{2 d^5 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^{p+1}}{(p+2) (p+3)}+\frac{2 d^5 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{p+1}}{(p+1) (p+2) (p+3)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{p+1}}{p+3} \]
Antiderivative was successfully verified.
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Rule 692
Rule 629
Rubi steps
\begin{align*} \int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^p \, dx &=\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{1+p}}{3+p}+\frac{\left (2 \left (b^2-4 a c\right ) d^2\right ) \int (b d+2 c d x)^3 \left (a+b x+c x^2\right )^p \, dx}{3+p}\\ &=\frac{2 \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{1+p}}{(2+p) (3+p)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{1+p}}{3+p}+\frac{\left (2 \left (b^2-4 a c\right )^2 d^4\right ) \int (b d+2 c d x) \left (a+b x+c x^2\right )^p \, dx}{(2+p) (3+p)}\\ &=\frac{2 \left (b^2-4 a c\right )^2 d^5 \left (a+b x+c x^2\right )^{1+p}}{(1+p) (2+p) (3+p)}+\frac{2 \left (b^2-4 a c\right ) d^5 (b+2 c x)^2 \left (a+b x+c x^2\right )^{1+p}}{(2+p) (3+p)}+\frac{d^5 (b+2 c x)^4 \left (a+b x+c x^2\right )^{1+p}}{3+p}\\ \end{align*}
Mathematica [A] time = 0.100672, size = 145, normalized size = 1.2 \[ \frac{d^5 (a+x (b+c x))^{p+1} \left (16 c^2 \left (2 a^2-2 a c (p+1) x^2+c^2 \left (p^2+3 p+2\right ) x^4\right )-8 b^2 c \left (a (p+3)-c \left (3 p^2+10 p+7\right ) x^2\right )-32 b c^2 (p+1) x \left (a-c (p+2) x^2\right )+8 b^3 c \left (p^2+4 p+3\right ) x+b^4 \left (p^2+5 p+6\right )\right )}{(p+1) (p+2) (p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 233, normalized size = 1.9 \begin{align*}{\frac{ \left ( c{x}^{2}+bx+a \right ) ^{1+p}{d}^{5} \left ( 16\,{c}^{4}{p}^{2}{x}^{4}+32\,b{c}^{3}{p}^{2}{x}^{3}+48\,{c}^{4}p{x}^{4}+24\,{b}^{2}{c}^{2}{p}^{2}{x}^{2}+96\,b{c}^{3}p{x}^{3}+32\,{c}^{4}{x}^{4}-32\,a{c}^{3}p{x}^{2}+8\,{b}^{3}c{p}^{2}x+80\,{b}^{2}{c}^{2}p{x}^{2}+64\,b{c}^{3}{x}^{3}-32\,ab{c}^{2}px-32\,{x}^{2}a{c}^{3}+{b}^{4}{p}^{2}+32\,{b}^{3}cpx+56\,{x}^{2}{b}^{2}{c}^{2}-8\,a{b}^{2}cp-32\,ba{c}^{2}x+5\,{b}^{4}p+24\,{b}^{3}cx+32\,{a}^{2}{c}^{2}-24\,ac{b}^{2}+6\,{b}^{4} \right ) }{{p}^{3}+6\,{p}^{2}+11\,p+6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.28906, size = 398, normalized size = 3.29 \begin{align*} \frac{{\left (16 \,{\left (p^{2} + 3 \, p + 2\right )} c^{5} d^{5} x^{6} + 48 \,{\left (p^{2} + 3 \, p + 2\right )} b c^{4} d^{5} x^{5} +{\left (p^{2} + 5 \, p + 6\right )} a b^{4} d^{5} - 8 \, a^{2} b^{2} c d^{5}{\left (p + 3\right )} + 32 \, a^{3} c^{2} d^{5} + 8 \,{\left ({\left (7 \, p^{2} + 22 \, p + 15\right )} b^{2} c^{3} d^{5} + 2 \,{\left (p^{2} + p\right )} a c^{4} d^{5}\right )} x^{4} + 16 \,{\left ({\left (2 \, p^{2} + 7 \, p + 5\right )} b^{3} c^{2} d^{5} + 2 \,{\left (p^{2} + p\right )} a b c^{3} d^{5}\right )} x^{3} +{\left ({\left (9 \, p^{2} + 37 \, p + 30\right )} b^{4} c d^{5} + 8 \,{\left (3 \, p^{2} + 5 \, p\right )} a b^{2} c^{2} d^{5} - 32 \, a^{2} c^{3} d^{5} p\right )} x^{2} +{\left ({\left (p^{2} + 5 \, p + 6\right )} b^{5} d^{5} + 8 \,{\left (p^{2} + 3 \, p\right )} a b^{3} c d^{5} - 32 \, a^{2} b c^{2} d^{5} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.22443, size = 828, normalized size = 6.84 \begin{align*} \frac{{\left (a b^{4} d^{5} p^{2} +{\left (5 \, a b^{4} - 8 \, a^{2} b^{2} c\right )} d^{5} p + 16 \,{\left (c^{5} d^{5} p^{2} + 3 \, c^{5} d^{5} p + 2 \, c^{5} d^{5}\right )} x^{6} + 2 \,{\left (3 \, a b^{4} - 12 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} d^{5} + 48 \,{\left (b c^{4} d^{5} p^{2} + 3 \, b c^{4} d^{5} p + 2 \, b c^{4} d^{5}\right )} x^{5} + 8 \,{\left (15 \, b^{2} c^{3} d^{5} +{\left (7 \, b^{2} c^{3} + 2 \, a c^{4}\right )} d^{5} p^{2} + 2 \,{\left (11 \, b^{2} c^{3} + a c^{4}\right )} d^{5} p\right )} x^{4} + 16 \,{\left (5 \, b^{3} c^{2} d^{5} + 2 \,{\left (b^{3} c^{2} + a b c^{3}\right )} d^{5} p^{2} +{\left (7 \, b^{3} c^{2} + 2 \, a b c^{3}\right )} d^{5} p\right )} x^{3} +{\left (30 \, b^{4} c d^{5} + 3 \,{\left (3 \, b^{4} c + 8 \, a b^{2} c^{2}\right )} d^{5} p^{2} +{\left (37 \, b^{4} c + 40 \, a b^{2} c^{2} - 32 \, a^{2} c^{3}\right )} d^{5} p\right )} x^{2} +{\left (6 \, b^{5} d^{5} +{\left (b^{5} + 8 \, a b^{3} c\right )} d^{5} p^{2} +{\left (5 \, b^{5} + 24 \, a b^{3} c - 32 \, a^{2} b c^{2}\right )} d^{5} p\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{p}}{p^{3} + 6 \, p^{2} + 11 \, p + 6} \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 = 1.21724, size = 1184, normalized size = 9.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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