3.2090 \(\int (d+e x)^{-2-2 p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx\)

Optimal. Leaf size=60 \[ \frac{(d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \]

[Out]

(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p)/((c*d^2 - a*e^2)*(1 + p)*(d + e*
x)^(2*(1 + p)))

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Rubi [A]  time = 0.0581803, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026 \[ \frac{(d+e x)^{-2 (p+1)} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-2 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(1 + p)/((c*d^2 - a*e^2)*(1 + p)*(d + e*
x)^(2*(1 + p)))

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Rubi in Sympy [A]  time = 20.2881, size = 54, normalized size = 0.9 \[ - \frac{\left (d + e x\right )^{- 2 p - 2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{p + 1}}{\left (p + 1\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-2-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)

[Out]

-(d + e*x)**(-2*p - 2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(p + 1)/((p +
1)*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.119598, size = 49, normalized size = 0.82 \[ \frac{(d+e x)^{-2 (p+1)} ((d+e x) (a e+c d x))^{p+1}}{(p+1) \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(-2 - 2*p)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p,x]

[Out]

((a*e + c*d*x)*(d + e*x))^(1 + p)/((c*d^2 - a*e^2)*(1 + p)*(d + e*x)^(2*(1 + p))
)

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Maple [A]  time = 0.006, size = 75, normalized size = 1.3 \[ -{\frac{ \left ( cdx+ae \right ) \left ( ex+d \right ) ^{-1-2\,p} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed \right ) ^{p}}{a{e}^{2}p-c{d}^{2}p+a{e}^{2}-c{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-2-2*p)*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^p,x)

[Out]

-(c*d*x+a*e)*(e*x+d)^(-1-2*p)/(a*e^2*p-c*d^2*p+a*e^2-c*d^2)*(c*d*e*x^2+a*e^2*x+c
*d^2*x+a*d*e)^p

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 2}\,{d x} \]

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 - 2),x, algorithm="maxima")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 2), x)

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Fricas [A]  time = 0.228687, size = 124, normalized size = 2.07 \[ \frac{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 2}}{c d^{2} - a e^{2} +{\left (c d^{2} - a e^{2}\right )} p} \]

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 - 2),x, algorithm="fricas")

[Out]

(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^
p*(e*x + d)^(-2*p - 2)/(c*d^2 - a*e^2 + (c*d^2 - a*e^2)*p)

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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)**(-2-2*p)*(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**p,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 2}\,{d x} \]

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 - 2),x, algorithm="giac")

[Out]

integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^p*(e*x + d)^(-2*p - 2), x)