3.1097 \(\int (d+e x)^{-1+2 p} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx\)

Optimal. Leaf size=43 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e} \]

[Out]

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

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Rubi [A]  time = 0.0484752, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e} \]

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]

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

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Rubi in Sympy [A]  time = 19.753, size = 39, normalized size = 0.91 \[ \frac{\left (d + e x\right )^{2 p} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{- p} \log{\left (d + e x \right )}}{e} \]

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 = 0.0133801, size = 32, normalized size = 0.74 \[ \frac{(d+e x)^{2 p} \log (d+e x) \left (c (d+e x)^2\right )^{-p}}{e} \]

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]

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

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Maple [A]  time = 0.065, size = 74, normalized size = 1.7 \[{\frac{1}{{{\rm e}^{p\ln \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) }}} \left ( x\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1+2\,p \right ) \ln \left ( ex+d \right ) }}+{\frac{d\ln \left ( ex+d \right ){{\rm e}^{ \left ( -1+2\,p \right ) \ln \left ( ex+d \right ) }}}{e}} \right ) } \]

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))+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 = 0.686852, size = 20, normalized size = 0.47 \[ \frac{c^{-p} \log \left (e x + d\right )}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c^(-p)*log(e*x + d)/e

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Fricas [A]  time = 0.231118, size = 20, normalized size = 0.47 \[ \frac{\log \left (e x + d\right )}{c^{p} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(e*x + d)/(c^p*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. \[ \int \frac{{\left (e x + d\right )}^{2 \, p - 1}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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