3.1094 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^p}{(d+e x)^2} \, dx\)

Optimal. Leaf size=42 \[ -\frac{c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{e (1-2 p)} \]

[Out]

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

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Rubi [A]  time = 0.0694434, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{c (d+e x) \left (c d^2+2 c d e x+c e^2 x^2\right )^{p-1}}{e (1-2 p)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.8676, size = 36, normalized size = 0.86 \[ - \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{p}}{e \left (d + e x\right ) \left (- 2 p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0291463, size = 30, normalized size = 0.71 \[ \frac{c (d+e x) \left (c (d+e x)^2\right )^{p-1}}{e (2 p-1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 41, normalized size = 1. \[{\frac{ \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{p}}{ \left ( ex+d \right ) \left ( -1+2\,p \right ) e}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.68852, size = 46, normalized size = 1.1 \[ \frac{{\left (e x + d\right )}^{2 \, p} c^{p}}{e^{2}{\left (2 \, p - 1\right )} x + d e{\left (2 \, p - 1\right )}} \]

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.245073, size = 66, normalized size = 1.57 \[ \frac{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{p}}{2 \, d e p - d e +{\left (2 \, e^{2} p - e^{2}\right )} x} \]

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,x, algorithm="fricas")

[Out]

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

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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((c*e**2*x**2+2*c*d*e*x+c*d**2)**p/(e*x+d)**2,x)

[Out]

Exception raised: TypeError

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

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,x, algorithm="giac")

[Out]

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