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

Optimal. Leaf size=78 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}} \]

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^
2)^(5/2))/(7*c*e*(d + e*x)^(3/2))

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Rubi [A]  time = 0.10536, antiderivative size = 78, 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 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{5/2}}{7 c e (d+e x)^{3/2}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{5/2}}{35 c e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(5/2))/(35*c*e*(d + e*x)^(5/2)) - (2*(c*d^2 - c*e^2*x^
2)^(5/2))/(7*c*e*(d + e*x)^(3/2))

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Rubi in Sympy [A]  time = 10.2135, size = 66, normalized size = 0.85 \[ - \frac{8 d \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{35 c e \left (d + e x\right )^{\frac{5}{2}}} - \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{5}{2}}}{7 c e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

-8*d*(c*d**2 - c*e**2*x**2)**(5/2)/(35*c*e*(d + e*x)**(5/2)) - 2*(c*d**2 - c*e**
2*x**2)**(5/2)/(7*c*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.0465399, size = 51, normalized size = 0.65 \[ -\frac{2 c (d-e x)^2 (9 d+5 e x) \sqrt{c \left (d^2-e^2 x^2\right )}}{35 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*c*(d - e*x)^2*(9*d + 5*e*x)*Sqrt[c*(d^2 - e^2*x^2)])/(35*e*Sqrt[d + e*x])

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Maple [A]  time = 0.007, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 5\,ex+9\,d \right ) }{35\,e} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(1/2),x)

[Out]

-2/35*(-e*x+d)*(5*e*x+9*d)*(-c*e^2*x^2+c*d^2)^(3/2)/e/(e*x+d)^(3/2)

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Maxima [A]  time = 0.73343, size = 74, normalized size = 0.95 \[ -\frac{2 \,{\left (5 \, c^{\frac{3}{2}} e^{3} x^{3} - c^{\frac{3}{2}} d e^{2} x^{2} - 13 \, c^{\frac{3}{2}} d^{2} e x + 9 \, c^{\frac{3}{2}} d^{3}\right )} \sqrt{-e x + d}}{35 \, e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/35*(5*c^(3/2)*e^3*x^3 - c^(3/2)*d*e^2*x^2 - 13*c^(3/2)*d^2*e*x + 9*c^(3/2)*d^
3)*sqrt(-e*x + d)/e

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Fricas [A]  time = 0.216099, size = 134, normalized size = 1.72 \[ \frac{2 \,{\left (5 \, c^{2} e^{5} x^{5} - c^{2} d e^{4} x^{4} - 18 \, c^{2} d^{2} e^{3} x^{3} + 10 \, c^{2} d^{3} e^{2} x^{2} + 13 \, c^{2} d^{4} e x - 9 \, c^{2} d^{5}\right )}}{35 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/35*(5*c^2*e^5*x^5 - c^2*d*e^4*x^4 - 18*c^2*d^2*e^3*x^3 + 10*c^2*d^3*e^2*x^2 +
13*c^2*d^4*e*x - 9*c^2*d^5)/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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