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

Optimal. Leaf size=74 \[ \frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}} \]

[Out]

(8*d*Sqrt[d + e*x])/(c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (2*(d + e*x)^(3/2))/(c*e*Sqr
t[c*d^2 - c*e^2*x^2])

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Rubi [A]  time = 0.103128, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{8 d \sqrt{d+e x}}{c e \sqrt{c d^2-c e^2 x^2}}-\frac{2 (d+e x)^{3/2}}{c e \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(8*d*Sqrt[d + e*x])/(c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (2*(d + e*x)^(3/2))/(c*e*Sqr
t[c*d^2 - c*e^2*x^2])

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Rubi in Sympy [A]  time = 10.0446, size = 61, normalized size = 0.82 \[ \frac{8 d \sqrt{d + e x}}{c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{3}{2}}}{c e \sqrt{c d^{2} - c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*d*sqrt(d + e*x)/(c*e*sqrt(c*d**2 - c*e**2*x**2)) - 2*(d + e*x)**(3/2)/(c*e*sqr
t(c*d**2 - c*e**2*x**2))

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Mathematica [A]  time = 0.0422803, size = 43, normalized size = 0.58 \[ \frac{2 (3 d-e x) \sqrt{d+e x}}{c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 44, normalized size = 0.6 \[ 2\,{\frac{ \left ( -ex+d \right ) \left ( -ex+3\,d \right ) \left ( ex+d \right ) ^{3/2}}{e \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{3/2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.723812, size = 31, normalized size = 0.42 \[ -\frac{2 \,{\left (e x - 3 \, d\right )}}{\sqrt{-e x + d} c^{\frac{3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.225875, size = 68, normalized size = 0.92 \[ -\frac{2 \,{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )}}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.617146, size = 4, normalized size = 0.05 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x