3.1096 \(\int (d+e x)^{-1-2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^p \, dx\)

Optimal. Leaf size=41 \[ \frac{(d+e x)^{-2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e} \]

[Out]

((c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p*Log[d + e*x])/(e*(d + e*x)^(2*p))

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Rubi [A]  time = 0.0499682, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{(d+e x)^{-2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^p}{e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(-1 - 2*p)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]

[Out]

((c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p*Log[d + e*x])/(e*(d + e*x)^(2*p))

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Rubi in Sympy [A]  time = 19.2604, size = 39, normalized size = 0.95 \[ \frac{\left (d + e x\right )^{- 2 p} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p} \log{\left (d + e x \right )}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(-1-2*p)*(c*e**2*x**2+2*c*d*e*x+c*d**2)**p,x)

[Out]

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

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Mathematica [A]  time = 0.0172093, size = 30, normalized size = 0.73 \[ \frac{(d+e x)^{-2 p} \log (d+e x) \left (c (d+e x)^2\right )^p}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(-1 - 2*p)*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^p,x]

[Out]

((c*(d + e*x)^2)^p*Log[d + e*x])/(e*(d + e*x)^(2*p))

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Maple [B]  time = 0.063, size = 95, normalized size = 2.3 \[ x\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1-2\,p \right ) \ln \left ( ex+d \right ) }}{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}+{\frac{d\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1-2\,p \right ) \ln \left ( ex+d \right ) }}{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}}{e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(-1-2*p)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^p,x)

[Out]

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

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Maxima [A]  time = 0.688156, size = 18, normalized size = 0.44 \[ \frac{c^{p} \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p*(e*x + d)^(-2*p - 1),x, algorithm="maxima")

[Out]

c^p*log(e*x + d)/e

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Fricas [A]  time = 0.221603, size = 18, normalized size = 0.44 \[ \frac{c^{p} \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p*(e*x + d)^(-2*p - 1),x, algorithm="fricas")

[Out]

c^p*log(e*x + d)/e

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}{\left (e x + d\right )}^{-2 \, p - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p*(e*x + d)^(-2*p - 1),x, algorithm="giac")

[Out]

integrate((c*e^2*x^2 + 2*c*d*e*x + c*d^2)^p*(e*x + d)^(-2*p - 1), x)