Optimal. Leaf size=288 \[ \frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^3}+\frac{(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac{3 c d (d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) (p+4) \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.432876, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051 \[ \frac{6 c^3 d^3 (d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{6 c^2 d^2 (d+e x)^{-2 p-3} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^3}+\frac{(d+e x)^{-2 p-5} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+4) \left (c d^2-a e^2\right )}+\frac{3 c d (d+e x)^{-2 (p+2)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+3) (p+4) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 126.583, size = 264, normalized size = 0.92 \[ \frac{6 c^{3} d^{3} \left (d + e x\right )^{- 2 p - 2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 1\right ) \left (p + 2\right ) \left (p + 3\right ) \left (p + 4\right ) \left (a e^{2} - c d^{2}\right )^{4}} - \frac{6 c^{2} d^{2} \left (d + e x\right )^{- 2 p - 3} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 2\right ) \left (p + 3\right ) \left (p + 4\right ) \left (a e^{2} - c d^{2}\right )^{3}} + \frac{3 c d \left (d + e x\right )^{- 2 p - 4} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 3\right ) \left (p + 4\right ) \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\left (d + e x\right )^{- 2 p - 5} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 4\right ) \left (a e^{2} - c d^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(-5-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
[Out]
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Mathematica [A] time = 0.649324, size = 216, normalized size = 0.75 \[ \frac{(d+e x)^{-2 p} ((d+e x) (a e+c d x))^p \left (\frac{6 c^4 d^4}{(p+1) (p+2) (p+3) (p+4) \left (c d^2-a e^2\right )^4}+\frac{6 c^3 d^3 p}{(p+1) \left (p^3+9 p^2+26 p+24\right ) (d+e x) \left (c d^2-a e^2\right )^3}+\frac{3 c^2 d^2 p}{(p+2) \left (p^2+7 p+12\right ) (d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{c d p}{(p+3) (p+4) (d+e x)^3 \left (c d^2-a e^2\right )}-\frac{1}{(p+4) (d+e x)^4}\right )}{e} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^(-5 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]
[Out]
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Maple [B] time = 0.016, size = 745, normalized size = 2.6 \[ -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-4-2\,p} \left ({a}^{3}{e}^{6}{p}^{3}-3\,{a}^{2}c{d}^{2}{e}^{4}{p}^{3}-3\,{a}^{2}cd{e}^{5}{p}^{2}x+3\,a{c}^{2}{d}^{4}{e}^{2}{p}^{3}+6\,a{c}^{2}{d}^{3}{e}^{3}{p}^{2}x+6\,a{c}^{2}{d}^{2}{e}^{4}p{x}^{2}-{c}^{3}{d}^{6}{p}^{3}-3\,{c}^{3}{d}^{5}e{p}^{2}x-6\,{c}^{3}{d}^{4}{e}^{2}p{x}^{2}-6\,{x}^{3}{c}^{3}{d}^{3}{e}^{3}+6\,{a}^{3}{e}^{6}{p}^{2}-21\,{a}^{2}c{d}^{2}{e}^{4}{p}^{2}-9\,{a}^{2}cd{e}^{5}px+24\,a{c}^{2}{d}^{4}{e}^{2}{p}^{2}+30\,a{c}^{2}{d}^{3}{e}^{3}px+6\,{x}^{2}a{c}^{2}{d}^{2}{e}^{4}-9\,{c}^{3}{d}^{6}{p}^{2}-21\,{c}^{3}{d}^{5}epx-24\,{x}^{2}{c}^{3}{d}^{4}{e}^{2}+11\,{a}^{3}{e}^{6}p-42\,{a}^{2}c{d}^{2}{e}^{4}p-6\,x{a}^{2}cd{e}^{5}+57\,a{c}^{2}{d}^{4}{e}^{2}p+24\,xa{c}^{2}{d}^{3}{e}^{3}-26\,{c}^{3}{d}^{6}p-36\,{c}^{3}{d}^{5}ex+6\,{a}^{3}{e}^{6}-24\,{a}^{2}c{d}^{2}{e}^{4}+36\,{c}^{2}{d}^{4}a{e}^{2}-24\,{c}^{3}{d}^{6} \right ) \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{p}}{{a}^{4}{e}^{8}{p}^{4}-4\,{a}^{3}c{d}^{2}{e}^{6}{p}^{4}+6\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{4}-4\,a{c}^{3}{d}^{6}{e}^{2}{p}^{4}+{c}^{4}{d}^{8}{p}^{4}+10\,{a}^{4}{e}^{8}{p}^{3}-40\,{a}^{3}c{d}^{2}{e}^{6}{p}^{3}+60\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{3}-40\,a{c}^{3}{d}^{6}{e}^{2}{p}^{3}+10\,{c}^{4}{d}^{8}{p}^{3}+35\,{a}^{4}{e}^{8}{p}^{2}-140\,{a}^{3}c{d}^{2}{e}^{6}{p}^{2}+210\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}{p}^{2}-140\,a{c}^{3}{d}^{6}{e}^{2}{p}^{2}+35\,{c}^{4}{d}^{8}{p}^{2}+50\,{a}^{4}{e}^{8}p-200\,{a}^{3}c{d}^{2}{e}^{6}p+300\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}p-200\,a{c}^{3}{d}^{6}{e}^{2}p+50\,{c}^{4}{d}^{8}p+24\,{a}^{4}{e}^{8}-96\,{a}^{3}c{d}^{2}{e}^{6}+144\,{a}^{2}{c}^{2}{d}^{4}{e}^{4}-96\,a{c}^{3}{d}^{6}{e}^{2}+24\,{c}^{4}{d}^{8}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(-5-2*p)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]
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 - 5),x, algorithm="maxima")
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Fricas [A] time = 0.244581, size = 1419, normalized size = 4.93 \[ \text{result too large to display} \]
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 - 5),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)**(-5-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 5}\,{d x} \]
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 - 5),x, algorithm="giac")
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