Optimal. Leaf size=92 \[ \frac{(a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (1,2 p;p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-p) (d+e x) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0700642, antiderivative size = 120, normalized size of antiderivative = 1.3, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.086, Rules used = {677, 70, 69} \[ \frac{c d (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (2-p,p+1;p+2;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 677
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p}{(d+e x)^2} \, dx &=\frac{\left ((a e+c d x)^{-p} \left (1+\frac{e x}{d}\right )^{-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int (a e+c d x)^p \left (1+\frac{e x}{d}\right )^{-2+p} \, dx}{d^2}\\ &=\frac{\left (c^2 (a e+c d x)^{-p} \left (\frac{c d \left (1+\frac{e x}{d}\right )}{c d-\frac{a e^2}{d}}\right )^{-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p\right ) \int (a e+c d x)^p \left (\frac{c d^2}{c d^2-a e^2}+\frac{c d e x}{c d^2-a e^2}\right )^{-2+p} \, dx}{\left (c d-\frac{a e^2}{d}\right )^2}\\ &=\frac{c d (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, _2F_1\left (2-p,1+p;2+p;-\frac{e (a e+c d x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.0376085, size = 108, normalized size = 1.17 \[ \frac{c d (a e+c d x) \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (2-p,p+1;p+2;\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{(p+1) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.165, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\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(-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 \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\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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