Optimal. Leaf size=127 \[ \frac{e (a+b x)^2 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (p+1)}+\frac{(a+b x) (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+1)}+\frac{e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)} \]
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Rubi [A] time = 0.0611493, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {646, 43} \[ \frac{e (a+b x)^2 (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (p+1)}+\frac{(a+b x) (b d-a e)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+1)}+\frac{e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 43
Rubi steps
\begin{align*} \int (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 \left (a b+b^2 x\right )^{2 p} (d+e x)^2 \, 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 \left (\frac{(b d-a e)^2 \left (a b+b^2 x\right )^{2 p}}{b^2}+\frac{2 e (b d-a e) \left (a b+b^2 x\right )^{1+2 p}}{b^3}+\frac{e^2 \left (a b+b^2 x\right )^{2+2 p}}{b^4}\right ) \, dx\\ &=\frac{(b d-a e)^2 (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (1+2 p)}+\frac{e (b d-a e) (a+b x)^2 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (1+p)}+\frac{e^2 (a+b x)^3 \left (a^2+2 a b x+b^2 x^2\right )^p}{b^3 (3+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0782135, size = 75, normalized size = 0.59 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p \left (\frac{e (a+b x) (b d-a e)}{p+1}+\frac{(b d-a e)^2}{2 p+1}+\frac{e^2 (a+b x)^2}{2 p+3}\right )}{b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 175, normalized size = 1.4 \begin{align*}{\frac{ \left ( 2\,{b}^{2}{e}^{2}{p}^{2}{x}^{2}+4\,{b}^{2}de{p}^{2}x+3\,{b}^{2}{e}^{2}p{x}^{2}-2\,ab{e}^{2}px+2\,{b}^{2}{d}^{2}{p}^{2}+8\,{b}^{2}depx+{e}^{2}{x}^{2}{b}^{2}-2\,abdep-ab{e}^{2}x+5\,{b}^{2}{d}^{2}p+3\,x{b}^{2}de+{a}^{2}{e}^{2}-3\,abde+3\,{b}^{2}{d}^{2} \right ) \left ( bx+a \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{{b}^{3} \left ( 4\,{p}^{3}+12\,{p}^{2}+11\,p+3 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12736, size = 212, normalized size = 1.67 \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} 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 e}{{\left (2 \, p^{2} + 3 \, p + 1\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} e^{2}}{{\left (4 \, p^{3} + 12 \, p^{2} + 11 \, p + 3\right )} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65259, size = 509, normalized size = 4.01 \begin{align*} \frac{{\left (2 \, a b^{2} d^{2} p^{2} + 3 \, a b^{2} d^{2} - 3 \, a^{2} b d e + a^{3} e^{2} +{\left (2 \, b^{3} e^{2} p^{2} + 3 \, b^{3} e^{2} p + b^{3} e^{2}\right )} x^{3} +{\left (3 \, b^{3} d e + 2 \,{\left (2 \, b^{3} d e + a b^{2} e^{2}\right )} p^{2} +{\left (8 \, b^{3} d e + a b^{2} e^{2}\right )} p\right )} x^{2} +{\left (5 \, a b^{2} d^{2} - 2 \, a^{2} b d e\right )} p +{\left (3 \, b^{3} d^{2} + 2 \,{\left (b^{3} d^{2} + 2 \, a b^{2} d e\right )} p^{2} +{\left (5 \, b^{3} d^{2} + 6 \, a b^{2} d e - 2 \, a^{2} b e^{2}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \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 = 1.14645, size = 821, normalized size = 6.46 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} p^{2} x^{3} e^{2} + 4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d p^{2} x^{2} e + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d^{2} p^{2} x + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} p^{2} x^{2} e^{2} + 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} p x^{3} e^{2} + 4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d p^{2} x e + 8 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d p x^{2} e + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d^{2} p^{2} + 5 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d^{2} p x +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} p x^{2} e^{2} +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} x^{3} e^{2} + 6 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d p x e + 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d x^{2} e + 5 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d^{2} p + 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{3} d^{2} x - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b p x e^{2} - 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b d p e + 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b^{2} d^{2} - 3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} b d e +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{3} e^{2}}{4 \, b^{3} p^{3} + 12 \, b^{3} p^{2} + 11 \, b^{3} p + 3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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