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 = 0.257502, 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 \[ \frac{2 c^2 d^2 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) \left (c d^2-a e^2\right )^3}+\frac{2 c d (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) \left (c d^2-a e^2\right )^2}+\frac{(d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
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Rubi in Sympy [A] time = 78.1269, size = 189, normalized size = 0.92 \[ - \frac{2 c^{2} d^{2} \left (d + e x\right )^{- 2 p - 2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right ) \left (p + 3\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{2 c d \left (d + e x\right )^{- 2 p - 3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 2\right ) \left (p + 3\right ) \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\left (d + e x\right )^{- 2 p - 4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 3\right ) \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(-4-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
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Mathematica [C] time = 0.149826, size = 101, normalized size = 0.49 \[ -\frac{(d+e x)^{-2 p-3} \left (\frac{e (a e+c d x)}{a e^2-c d^2}\right )^{-p} ((d+e x) (a e+c d x))^p \, _2F_1\left (-p-3,-p;-p-2;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{e (p+3)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(-4 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
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Maple [A] time = 0.009, size = 381, normalized size = 1.9 \[ -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-3-2\,p} \left ({a}^{2}{e}^{4}{p}^{2}-2\,ac{d}^{2}{e}^{2}{p}^{2}-2\,acd{e}^{3}px+{c}^{2}{d}^{4}{p}^{2}+2\,{c}^{2}{d}^{3}epx+2\,{x}^{2}{c}^{2}{d}^{2}{e}^{2}+3\,{a}^{2}{e}^{4}p-8\,ac{d}^{2}{e}^{2}p-2\,xacd{e}^{3}+5\,{c}^{2}{d}^{4}p+6\,x{c}^{2}{d}^{3}e+2\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}+6\,{c}^{2}{d}^{4} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{p}}{{a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}-{c}^{3}{d}^{6}{p}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-18\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}+18\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}-6\,{c}^{3}{d}^{6}{p}^{2}+11\,{a}^{3}{e}^{6}p-33\,{a}^{2}c{d}^{2}{e}^{4}p+33\,a{c}^{2}{d}^{4}{e}^{2}p-11\,{c}^{3}{d}^{6}p+6\,{a}^{3}{e}^{6}-18\,{a}^{2}c{d}^{2}{e}^{4}+18\,{c}^{2}{d}^{4}a{e}^{2}-6\,{c}^{3}{d}^{6}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(-4-2*p)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4),x, algorithm="maxima")
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Fricas [A] time = 0.239958, size = 783, normalized size = 3.8 \[ \frac{{\left (2 \, c^{3} d^{3} e^{3} x^{4} + 6 \, a c^{2} d^{5} e - 6 \, a^{2} c d^{3} e^{3} + 2 \, a^{3} d e^{5} + 2 \,{\left (4 \, c^{3} d^{4} e^{2} +{\left (c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} p\right )} x^{3} +{\left (a c^{2} d^{5} e - 2 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} p^{2} +{\left (12 \, c^{3} d^{5} e +{\left (c^{3} d^{5} e - 2 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p^{2} +{\left (7 \, c^{3} d^{5} e - 8 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} p\right )} x^{2} +{\left (5 \, a c^{2} d^{5} e - 8 \, a^{2} c d^{3} e^{3} + 3 \, a^{3} d e^{5}\right )} p +{\left (6 \, c^{3} d^{6} + 6 \, a c^{2} d^{4} e^{2} - 6 \, a^{2} c d^{2} e^{4} + 2 \, a^{3} e^{6} +{\left (c^{3} d^{6} - a c^{2} d^{4} e^{2} - a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} p^{2} +{\left (5 \, c^{3} d^{6} - a c^{2} d^{4} e^{2} - 7 \, a^{2} c d^{2} e^{4} + 3 \, a^{3} e^{6}\right )} p\right )} x\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}}{6 \, c^{3} d^{6} - 18 \, a c^{2} d^{4} e^{2} + 18 \, a^{2} c d^{2} e^{4} - 6 \, a^{3} e^{6} +{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{3} + 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p^{2} + 11 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} p} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(-4-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 4),x, algorithm="giac")
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