3.1045 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{5/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

[Out]

(c*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(3*e)

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Rubi [A]  time = 0.0657981, antiderivative size = 32, 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 \[ \frac{c \left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(c*(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(3/2))/(3*e)

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Rubi in Sympy [A]  time = 17.6832, size = 29, normalized size = 0.91 \[ \frac{c \left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e} \]

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)**(5/2)/(e*x+d)**3,x)

[Out]

c*(c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2)/(3*e)

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Mathematica [A]  time = 0.0180067, size = 21, normalized size = 0.66 \[ \frac{c \left (c (d+e x)^2\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 + 2*c*d*e*x + c*e^2*x^2)^(5/2)/(d + e*x)^3,x]

[Out]

(c*(c*(d + e*x)^2)^(3/2))/(3*e)

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Maple [A]  time = 0.004, size = 51, normalized size = 1.6 \[{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\, \left ( ex+d \right ) ^{5}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{5}{2}}}} \]

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)^(5/2)/(e*x+d)^3,x)

[Out]

1/3*x*(e^2*x^2+3*d*e*x+3*d^2)*(c*e^2*x^2+2*c*d*e*x+c*d^2)^(5/2)/(e*x+d)^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

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)^(5/2)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.226685, size = 82, normalized size = 2.56 \[ \frac{{\left (c^{2} e^{2} x^{3} + 3 \, c^{2} d e x^{2} + 3 \, c^{2} d^{2} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (e x + d\right )}} \]

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)^(5/2)/(e*x + d)^3,x, algorithm="fricas")

[Out]

1/3*(c^2*e^2*x^3 + 3*c^2*d*e*x^2 + 3*c^2*d^2*x)*sqrt(c*e^2*x^2 + 2*c*d*e*x + c*d
^2)/(e*x + d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c \left (d + e x\right )^{2}\right )^{\frac{5}{2}}}{\left (d + e x\right )^{3}}\, dx \]

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)**(5/2)/(e*x+d)**3,x)

[Out]

Integral((c*(d + e*x)**2)**(5/2)/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.317267, size = 1, normalized size = 0.03 \[ \mathit{Done} \]

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)^(5/2)/(e*x + d)^3,x, algorithm="giac")

[Out]

Done