Optimal. Leaf size=115 \[ \frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{2 (p+1) (2 p+1) (b d-a e)^2}+\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
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Rubi [A] time = 0.0482377, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {646, 45, 37} \[ \frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 p-1}}{2 (p+1) (2 p+1) (b d-a e)^2}+\frac{(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p (d+e x)^{-2 (p+1)}}{2 (p+1) (b d-a e)} \]
Antiderivative was successfully verified.
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Rule 646
Rule 45
Rule 37
Rubi steps
\begin{align*} \int (d+e x)^{-3-2 p} \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)^{-3-2 p} \, dx\\ &=\frac{(a+b x) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e) (1+p)}+\frac{\left (b \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 (1+p)} \, dx}{2 (b d-a e) (1+p)}\\ &=\frac{b (a+b x) (d+e x)^{-1-2 p} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e)^2 (1+p) (1+2 p)}+\frac{(a+b x) (d+e x)^{-2 (1+p)} \left (a^2+2 a b x+b^2 x^2\right )^p}{2 (b d-a e) (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0399814, size = 72, normalized size = 0.63 \[ \frac{(a+b x) \left ((a+b x)^2\right )^p (d+e x)^{-2 (p+1)} (-a e (2 p+1)+2 b d (p+1)+b e x)}{2 (p+1) (2 p+1) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 139, normalized size = 1.2 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{-2-2\,p} \left ( bx+a \right ) \left ( 2\,aep-2\,bdp-bxe+ae-2\,bd \right ) \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{4\,{a}^{2}{e}^{2}{p}^{2}-8\,abde{p}^{2}+4\,{b}^{2}{d}^{2}{p}^{2}+6\,{a}^{2}{e}^{2}p-12\,abdep+6\,{b}^{2}{d}^{2}p+2\,{a}^{2}{e}^{2}-4\,abde+2\,{b}^{2}{d}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92459, size = 451, normalized size = 3.92 \begin{align*} \frac{{\left (b^{2} e^{2} x^{3} + 2 \, a b d^{2} - a^{2} d e +{\left (3 \, b^{2} d e + 2 \,{\left (b^{2} d e - a b e^{2}\right )} p\right )} x^{2} + 2 \,{\left (a b d^{2} - a^{2} d e\right )} p +{\left (2 \, b^{2} d^{2} + 2 \, a b d e - a^{2} e^{2} + 2 \,{\left (b^{2} d^{2} - a^{2} e^{2}\right )} p\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}}{2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2} + 2 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p^{2} + 3 \,{\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} p\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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