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

Optimal. Leaf size=36 \[ \frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0496786, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034 \[ \frac{2 \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.13237, size = 29, normalized size = 0.81 \[ \frac{2 \sqrt{d + e x}}{c e \sqrt{c d^{2} - c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0368579, size = 35, normalized size = 0.97 \[ \frac{2 \sqrt{d+e x}}{c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 36, normalized size = 1. \[ 2\,{\frac{ \left ( -ex+d \right ) \left ( ex+d \right ) ^{3/2}}{e \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.722348, size = 22, normalized size = 0.61 \[ \frac{2}{\sqrt{-e x + d} c^{\frac{3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

2/(sqrt(-e*x + d)*c^(3/2)*e)

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Fricas [A]  time = 0.225013, size = 65, normalized size = 1.81 \[ -\frac{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{c^{2} e^{3} x^{2} - c^{2} d^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.610746, size = 4, normalized size = 0.11 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x