Optimal. Leaf size=134 \[ \frac{2 e (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)}+\frac{(a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+1)}+\frac{e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+2)} \]
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Rubi [A] time = 0.0858001, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {770, 21, 43} \[ \frac{2 e (a+b x)^3 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)}+\frac{(a+b x)^2 (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+1)}+\frac{e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (p+2)} \]
Antiderivative was successfully verified.
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Rule 770
Rule 21
Rule 43
Rubi steps
\begin{align*} \int (a+b x) (d+e x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int (a+b x) \left (a b+b^2 x\right )^{2 p} (d+e x)^2 \, dx\\ &=\frac{\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (a b+b^2 x\right )^{1+2 p} (d+e x)^2 \, dx}{b}\\ &=\frac{\left (\left (a b+b^2 x\right )^{-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p\right ) \int \left (\frac{(b d-a e)^2 \left (a b+b^2 x\right )^{1+2 p}}{b^2}+\frac{2 e (b d-a e) \left (a b+b^2 x\right )^{2+2 p}}{b^3}+\frac{e^2 \left (a b+b^2 x\right )^{3+2 p}}{b^4}\right ) \, dx}{b}\\ &=\frac{(b d-a e)^2 (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (1+p)}+\frac{2 e (b d-a e) (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (3+2 p)}+\frac{e^2 (a+b x)^4 \left (a^2+2 a b x+b^2 x^2\right )^p}{2 b^3 (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0722754, size = 109, normalized size = 0.81 \[ \frac{\left ((a+b x)^2\right )^{p+1} \left (a^2 e^2-2 a b e (d (p+2)+e (p+1) x)+b^2 \left (d^2 \left (2 p^2+7 p+6\right )+4 d e \left (p^2+3 p+2\right ) x+e^2 \left (2 p^2+5 p+3\right ) x^2\right )\right )}{2 b^3 (p+1) (p+2) (2 p+3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 179, normalized size = 1.3 \begin{align*}{\frac{ \left ( 2\,{b}^{2}{e}^{2}{p}^{2}{x}^{2}+4\,{b}^{2}de{p}^{2}x+5\,{b}^{2}{e}^{2}p{x}^{2}-2\,ab{e}^{2}px+2\,{b}^{2}{d}^{2}{p}^{2}+12\,{b}^{2}depx+3\,{x}^{2}{b}^{2}{e}^{2}-2\,abdep-2\,xab{e}^{2}+7\,{b}^{2}{d}^{2}p+8\,x{b}^{2}de+{a}^{2}{e}^{2}-4\,abde+6\,{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) ^{2} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{2\,{b}^{3} \left ( 2\,{p}^{3}+9\,{p}^{2}+13\,p+6 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.10984, size = 545, normalized size = 4.07 \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} a d^{2}}{b{\left (2 \, p + 1\right )}} + \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} d^{2}}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b} + \frac{{\left (b^{2}{\left (2 \, p + 1\right )} x^{2} + 2 \, a b p x - a^{2}\right )}{\left (b x + a\right )}^{2 \, p} a d e}{{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} + \frac{2 \,{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} d e}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{2}} + \frac{{\left ({\left (2 \, p^{2} + 3 \, p + 1\right )} b^{3} x^{3} +{\left (2 \, p^{2} + p\right )} a b^{2} x^{2} - 2 \, a^{2} b p x + a^{3}\right )}{\left (b x + a\right )}^{2 \, p} a e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} + \frac{{\left ({\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{4} x^{4} + 2 \,{\left (2 \, p^{3} + 3 \, p^{2} + p\right )} a b^{3} x^{3} - 3 \,{\left (2 \, p^{2} + p\right )} a^{2} b^{2} x^{2} + 6 \, a^{3} b p x - 3 \, a^{4}\right )}{\left (b x + a\right )}^{2 \, p} e^{2}}{2 \,{\left (4 \, p^{4} + 20 \, p^{3} + 35 \, p^{2} + 25 \, p + 6\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.08294, size = 718, normalized size = 5.36 \begin{align*} \frac{{\left (2 \, a^{2} b^{2} d^{2} p^{2} + 6 \, a^{2} b^{2} d^{2} - 4 \, a^{3} b d e + a^{4} e^{2} +{\left (2 \, b^{4} e^{2} p^{2} + 5 \, b^{4} e^{2} p + 3 \, b^{4} e^{2}\right )} x^{4} + 4 \,{\left (2 \, b^{4} d e + a b^{3} e^{2} +{\left (b^{4} d e + a b^{3} e^{2}\right )} p^{2} +{\left (3 \, b^{4} d e + 2 \, a b^{3} e^{2}\right )} p\right )} x^{3} +{\left (6 \, b^{4} d^{2} + 12 \, a b^{3} d e + 2 \,{\left (b^{4} d^{2} + 4 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} p^{2} +{\left (7 \, b^{4} d^{2} + 22 \, a b^{3} d e + a^{2} b^{2} e^{2}\right )} p\right )} x^{2} +{\left (7 \, a^{2} b^{2} d^{2} - 2 \, a^{3} b d e\right )} p + 2 \,{\left (6 \, a b^{3} d^{2} + 2 \,{\left (a b^{3} d^{2} + a^{2} b^{2} d e\right )} p^{2} +{\left (7 \, a b^{3} d^{2} + 4 \, a^{2} b^{2} d e - a^{3} b e^{2}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (2 \, b^{3} p^{3} + 9 \, b^{3} p^{2} + 13 \, b^{3} p + 6 \, b^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1598, size = 1219, normalized size = 9.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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