3.1088 \(\int (d+e x)^p \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p} \, dx\)

Optimal. Leaf size=44 \[ \frac{(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]

[Out]

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

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Rubi [A]  time = 0.0532202, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{(d+e x)^{p+1} \left (c d^2+2 c d e x+c e^2 x^2\right )^{-p}}{e (1-p)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.3838, size = 36, normalized size = 0.82 \[ \frac{\left (d + e x\right )^{p + 1} \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{- p}}{e \left (- p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0278744, size = 31, normalized size = 0.7 \[ \frac{(d+e x)^{p+1} \left (c (d+e x)^2\right )^{-p}}{e-e p} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 44, normalized size = 1. \[ -{\frac{ \left ( ex+d \right ) ^{1+p}}{e \left ( -1+p \right ) \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.689457, size = 39, normalized size = 0.89 \[ -\frac{{\left (e x + d\right )}{\left (e x + d\right )}^{-p} c^{-p}}{e{\left (p - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.236138, size = 41, normalized size = 0.93 \[ -\frac{e x + d}{{\left (e p - e\right )}{\left (e x + d\right )}^{p} c^{p}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**p/((c*e**2*x**2+2*c*d*e*x+c*d**2)**p),x)

[Out]

Exception raised: TypeError

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GIAC/XCAS [A]  time = 0.226931, size = 72, normalized size = 1.64 \[ -\frac{x e^{\left (-p{\rm ln}\left (x e + d\right ) - p{\rm ln}\left (c\right ) + 1\right )} + d e^{\left (-p{\rm ln}\left (x e + d\right ) - p{\rm ln}\left (c\right )\right )}}{p e - e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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