Optimal. Leaf size=72 \[ \frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)^2} \]
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Rubi [A] time = 0.0304151, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {646, 68} \[ \frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (2,2 p+1;2 (p+1);-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Rule 646
Rule 68
Rubi steps
\begin{align*} \int \frac{\left (a^2+2 a b x+b^2 x^2\right )^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 \frac{\left (a b+b^2 x\right )^{2 p}}{(d+e x)^2} \, dx\\ &=\frac{b (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^p \, _2F_1\left (2,1+2 p;2 (1+p);-\frac{e (a+b x)}{b d-a e}\right )}{(b d-a e)^2 (1+2 p)}\\ \end{align*}
Mathematica [A] time = 0.0160252, size = 63, normalized size = 0.88 \[ \frac{b (a+b x) \left ((a+b x)^2\right )^p \, _2F_1\left (2,2 p+1;2 p+2;-\frac{e (a+b x)}{b d-a e}\right )}{(2 p+1) (b d-a e)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.188, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}}{ \left ( ex+d \right ) ^{2}}}\, dx \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 \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{e^{2} x^{2} + 2 \, d e x + d^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\left (a + b x\right )^{2}\right )^{p}}{\left (d + e x\right )^{2}}\, dx \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 \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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