3.1034 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=29 \[ \frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]

[Out]

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

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Rubi [A]  time = 0.0664944, antiderivative size = 29, 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 \[ \frac{c \sqrt{c d^2+2 c d e x+c e^2 x^2}}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.3049, size = 27, normalized size = 0.93 \[ \frac{c \sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0173351, size = 22, normalized size = 0.76 \[ \frac{x \left (c (d+e x)^2\right )^{3/2}}{(d+e x)^3} \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*(d + e*x)^2)^(3/2))/(d + e*x)^3

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Maple [A]  time = 0.003, size = 32, normalized size = 1.1 \[{\frac{x}{ \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^3,x)

[Out]

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

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

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.218335, size = 43, normalized size = 1.48 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}} c x}{e x + d} \]

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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.41378, size = 39, normalized size = 1.34 \[ c \left (\begin{cases} \frac{x \sqrt{c d^{2}}}{d} & \text{for}\: e = 0 \\\frac{\sqrt{c d^{2} + 2 c d e x + c e^{2} x^{2}}}{e} & \text{otherwise} \end{cases}\right ) \]

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)**3,x)

[Out]

c*Piecewise((x*sqrt(c*d**2)/d, Eq(e, 0)), (sqrt(c*d**2 + 2*c*d*e*x + c*e**2*x**2
)/e, 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)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError