3.1033 \(\int \frac{\left (c d^2+2 c d e x+c e^2 x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=37 \[ \frac{c (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]

[Out]

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

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Rubi [A]  time = 0.0689954, antiderivative size = 37, 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 \[ \frac{c (d+e x) \sqrt{c d^2+2 c d e x+c e^2 x^2}}{2 e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 18.5876, size = 32, normalized size = 0.86 \[ \frac{\left (c d^{2} + 2 c d e x + c e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )} \]

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

[Out]

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

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Mathematica [A]  time = 0.00404043, size = 33, normalized size = 0.89 \[ \frac{c^2 x (d+e x) (2 d+e x)}{2 \sqrt{c (d+e x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 40, normalized size = 1.1 \[{\frac{x \left ( ex+2\,d \right ) }{2\, \left ( ex+d \right ) ^{3}} \left ( c{e}^{2}{x}^{2}+2\,cdex+c{d}^{2} \right ) ^{{\frac{3}{2}}}} \]

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)^2,x)

[Out]

1/2*x*(e*x+2*d)*(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. \[ \text{Exception raised: RuntimeError} \]

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)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.226246, size = 58, normalized size = 1.57 \[ \frac{\sqrt{c e^{2} x^{2} + 2 \, c d e x + c d^{2}}{\left (c e x^{2} + 2 \, c d x\right )}}{2 \,{\left (e x + d\right )}} \]

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)^2,x, algorithm="fricas")

[Out]

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

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

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

[Out]

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

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

[Out]

Exception raised: TypeError