3.1032 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{d+e x} \, dx\)

Optimal. Leaf size=31 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

[Out]

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

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Rubi [A]  time = 0.0679158, antiderivative size = 31, 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 \[ \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(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 = 19.1665, size = 27, normalized size = 0.87 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e} \]

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

[Out]

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

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Mathematica [A]  time = 0.0152725, size = 20, normalized size = 0.65 \[ \frac{\left (c (d+e x)^2\right )^{3/2}}{3 e} \]

Antiderivative was successfully verified.

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

[Out]

(c*(d + e*x)^2)^(3/2)/(3*e)

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Maple [A]  time = 0.004, size = 51, normalized size = 1.7 \[{\frac{x \left ({e}^{2}{x}^{2}+3\,dex+3\,{d}^{2} \right ) }{3\, \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234047, size = 74, normalized size = 2.39 \[ \frac{{\left (c e^{2} x^{3} + 3 \, c d e x^{2} + 3 \, c d^{2} x\right )} \sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}}{3 \,{\left (e x + d\right )}} \]

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

[Out]

1/3*(c*e^2*x^3 + 3*c*d*e*x^2 + 3*c*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 [A]  time = 5.03632, size = 39, normalized size = 1.26 \[ \begin{cases} \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e} & \text{for}\: e \neq 0 \\\frac{x \left (c d^{2}\right )^{\frac{3}{2}}}{d} & \text{otherwise} \end{cases} \]

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

[Out]

Piecewise(((c*d**2 + 2*c*d*e*x + c*e**2*x**2)**(3/2)/(3*e), Ne(e, 0)), (x*(c*d**
2)**(3/2)/d, True))

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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

[Out]

Exception raised: NotImplementedError