Optimal. Leaf size=206 \[ \frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0931066, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {658, 650} \[ \frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 658
Rule 650
Rubi steps
\begin{align*} \int (d+e x)^{-4-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx &=\frac{(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}+\frac{(2 c d) \int (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{\left (c d^2-a e^2\right ) (3+p)}\\ &=\frac{2 c d (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}+\frac{(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}+\frac{\left (2 c^2 d^2\right ) \int (d+e x)^{-2-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}\\ &=\frac{2 c d (d+e x)^{-3-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^2 (2+p) (3+p)}+\frac{2 c^2 d^2 (d+e x)^{-2 (1+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right )^3 (1+p) (2+p) (3+p)}+\frac{(d+e x)^{-2 (2+p)} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p}}{\left (c d^2-a e^2\right ) (3+p)}\\ \end{align*}
Mathematica [A] time = 0.0906075, size = 131, normalized size = 0.64 \[ \frac{(d+e x)^{-2 (p+2)} ((d+e x) (a e+c d x))^{p+1} \left (a^2 e^4 \left (p^2+3 p+2\right )-2 a c d e^2 (p+1) (d (p+3)+e x)+c^2 d^2 \left (d^2 \left (p^2+5 p+6\right )+2 d e (p+3) x+2 e^2 x^2\right )\right )}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 381, normalized size = 1.9 \begin{align*} -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-3-2\,p} \left ({a}^{2}{e}^{4}{p}^{2}-2\,ac{d}^{2}{e}^{2}{p}^{2}-2\,acd{e}^{3}px+{c}^{2}{d}^{4}{p}^{2}+2\,{c}^{2}{d}^{3}epx+2\,{c}^{2}{d}^{2}{e}^{2}{x}^{2}+3\,{a}^{2}{e}^{4}p-8\,ac{d}^{2}{e}^{2}p-2\,acd{e}^{3}x+5\,{c}^{2}{d}^{4}p+6\,{c}^{2}{d}^{3}ex+2\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}+6\,{c}^{2}{d}^{4} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{p}}{{a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}-{c}^{3}{d}^{6}{p}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-18\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}+18\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}-6\,{c}^{3}{d}^{6}{p}^{2}+11\,{a}^{3}{e}^{6}p-33\,{a}^{2}c{d}^{2}{e}^{4}p+33\,a{c}^{2}{d}^{4}{e}^{2}p-11\,{c}^{3}{d}^{6}p+6\,{a}^{3}{e}^{6}-18\,{a}^{2}c{d}^{2}{e}^{4}+18\,a{c}^{2}{d}^{4}{e}^{2}-6\,{c}^{3}{d}^{6}}} \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 (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 \, p - 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.40401, size = 1145, normalized size = 5.56 \begin{align*} \frac{{\left (2 \, c^{3} d^{3} e^{3} x^{4} + 6 \, a c^{2} d^{5} e - 6 \, a^{2} c d^{3} e^{3} + 2 \, a^{3} d e^{5} + 2 \,{\left (4 \, c^{3} d^{4} e^{2} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} p\right )} x^{3} +{\left (a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} p^{2} +{\left (12 \, c^{3} d^{5} e +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p^{2} +{\left (7 \, c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p\right )} x^{2} +{\left (5 \, a c^{2} d^{5} e - 8 \, a^{2} c d^{3} e^{3} + 3 \, a^{3} d e^{5}\right )} p +{\left (6 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} +{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} p^{2} +{\left (5 \, c^{3} d^{6} - a c^{2} d^{4} e^{2} - 7 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6}\right )} p\right )} x\right )}{\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 \, p - 4}}{6 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 6 \, a^{3} e^{6} +{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{3} + 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{2} + 11 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p} \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 (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 \, p - 4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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