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

Optimal. Leaf size=78 \[ -\frac{8 d \sqrt{c d^2-c e^2 x^2}}{3 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e} \]

[Out]

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

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Rubi [A]  time = 0.106187, 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 \[ -\frac{8 d \sqrt{c d^2-c e^2 x^2}}{3 c e \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{3 c e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 10.4312, size = 66, normalized size = 0.85 \[ - \frac{8 d \sqrt{c d^{2} - c e^{2} x^{2}}}{3 c e \sqrt{d + e x}} - \frac{2 \sqrt{d + e x} \sqrt{c d^{2} - c e^{2} x^{2}}}{3 c e} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0425046, size = 47, normalized size = 0.6 \[ -\frac{2 (d-e x) \sqrt{d+e x} (5 d+e x)}{3 e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 43, normalized size = 0.6 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( ex+5\,d \right ) }{3\,e}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}}} \]

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)^(1/2),x)

[Out]

-2/3*(-e*x+d)*(e*x+5*d)*(e*x+d)^(1/2)/e/(-c*e^2*x^2+c*d^2)^(1/2)

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Maxima [A]  time = 0.724304, size = 46, normalized size = 0.59 \[ \frac{2 \,{\left (e^{2} x^{2} + 4 \, d e x - 5 \, d^{2}\right )}}{3 \, \sqrt{-e x + d} \sqrt{c} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e^2*x^2 + 4*d*e*x - 5*d^2)/(sqrt(-e*x + d)*sqrt(c)*e)

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Fricas [A]  time = 0.223471, size = 78, normalized size = 1. \[ \frac{2 \,{\left (e^{3} x^{3} + 5 \, d e^{2} x^{2} - d^{2} e x - 5 \, d^{3}\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*(e^3*x^3 + 5*d*e^2*x^2 - d^2*e*x - 5*d^3)/(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. \[ \int \frac{\left (d + e x\right )^{\frac{3}{2}}}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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