Optimal. Leaf size=93 \[ \frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
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Rubi [A] time = 0.0683955, antiderivative size = 93, normalized size of antiderivative = 1., 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} \[ \frac{\left (-\frac{e (a e+c d x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \, _2F_1\left (-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
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} \, 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 )^{-1+p} \, dx}{d}\\ &=\frac{\left (\left (\frac{e (a e+c d x)}{d \left (-c d+\frac{a e^2}{d}\right )}\right )^{-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 \left (1+\frac{e x}{d}\right )^{-1+p} \left (-\frac{a e^2}{c d^2-a e^2}-\frac{c e x}{c d-\frac{a e^2}{d}}\right )^p \, dx}{d}\\ &=\frac{\left (-\frac{e (a e+c d 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 (-p,p;1+p;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p}\\ \end{align*}
Mathematica [A] time = 0.0267257, size = 81, normalized size = 0.87 \[ \frac{\left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p,p;p+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e p} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.174, 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}}{ex+d}}\, 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}}{e x + d}\,{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 x + d}, 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 (d + e x\right ) \left (a e + c d x\right )\right )^{p}}{d + e x}\, 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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}}{e x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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