Optimal. Leaf size=76 \[ \frac{(a+b x) (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (2 p+1)}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b^2 (p+1)} \]
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Rubi [A] time = 0.0238942, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {640, 609} \[ \frac{(a+b x) (b d-a e) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (2 p+1)}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{p+1}}{2 b^2 (p+1)} \]
Antiderivative was successfully verified.
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Rule 640
Rule 609
Rubi steps
\begin{align*} \int (d+e x) \left (a^2+2 a b x+b^2 x^2\right )^p \, dx &=\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 b^2 (1+p)}+\frac{\left (2 b^2 d-2 a b e\right ) \int \left (a^2+2 a b x+b^2 x^2\right )^p \, dx}{2 b^2}\\ &=\frac{(b d-a e) (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p}{b^2 (1+2 p)}+\frac{e \left (a^2+2 a b x+b^2 x^2\right )^{1+p}}{2 b^2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0340394, size = 54, normalized size = 0.71 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p (-a e+2 b d (p+1)+b e (2 p+1) x)}{2 b^2 (p+1) (2 p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 65, normalized size = 0.9 \begin{align*} -{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p} \left ( -2\,bepx-2\,bdp-bxe+ae-2\,bd \right ) \left ( bx+a \right ) }{2\,{b}^{2} \left ( 2\,{p}^{2}+3\,p+1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08004, size = 105, normalized size = 1.38 \begin{align*} \frac{{\left (b x + a\right )}{\left (b x + a\right )}^{2 \, p} d}{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} e}{2 \,{\left (2 \, p^{2} + 3 \, p + 1\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68205, size = 204, normalized size = 2.68 \begin{align*} \frac{{\left (2 \, a b d p + 2 \, a b d - a^{2} e +{\left (2 \, b^{2} e p + b^{2} e\right )} x^{2} + 2 \,{\left (b^{2} d +{\left (b^{2} d + a b e\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{2 \,{\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\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 = 1.15629, size = 308, normalized size = 4.05 \begin{align*} \frac{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} p x^{2} e + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} d p x + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b p x e +{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} x^{2} e + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b d p + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} b^{2} d x + 2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a b d -{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p} a^{2} e}{2 \,{\left (2 \, b^{2} p^{2} + 3 \, b^{2} p + b^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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