3.853 \(\int \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt{d+e x}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}} \]

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(3/2))/(15*c*e*(d + e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^
2)^(3/2))/(5*c*e*Sqrt[d + e*x])

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Rubi [A]  time = 0.102985, 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 \[ -\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{5 c e \sqrt{d+e x}}-\frac{8 d \left (c d^2-c e^2 x^2\right )^{3/2}}{15 c e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-8*d*(c*d^2 - c*e^2*x^2)^(3/2))/(15*c*e*(d + e*x)^(3/2)) - (2*(c*d^2 - c*e^2*x^
2)^(3/2))/(5*c*e*Sqrt[d + e*x])

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Rubi in Sympy [A]  time = 10.1718, size = 66, normalized size = 0.85 \[ - \frac{8 d \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{15 c e \left (d + e x\right )^{\frac{3}{2}}} - \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{5 c e \sqrt{d + e x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*d*(c*d**2 - c*e**2*x**2)**(3/2)/(15*c*e*(d + e*x)**(3/2)) - 2*(c*d**2 - c*e**
2*x**2)**(3/2)/(5*c*e*sqrt(d + e*x))

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Mathematica [A]  time = 0.0387922, size = 53, normalized size = 0.68 \[ -\frac{2 \left (7 d^2-4 d e x-3 e^2 x^2\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{15 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[d + e*x]*Sqrt[c*d^2 - c*e^2*x^2],x]

[Out]

(-2*(7*d^2 - 4*d*e*x - 3*e^2*x^2)*Sqrt[c*(d^2 - e^2*x^2)])/(15*e*Sqrt[d + e*x])

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Maple [A]  time = 0.004, size = 44, normalized size = 0.6 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 3\,ex+7\,d \right ) }{15\,e}\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(-e*x+d)*(3*e*x+7*d)*(-c*e^2*x^2+c*d^2)^(1/2)/e/(e*x+d)^(1/2)

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Maxima [A]  time = 0.751899, size = 73, normalized size = 0.94 \[ \frac{2 \,{\left (3 \, \sqrt{c} e^{2} x^{2} + 4 \, \sqrt{c} d e x - 7 \, \sqrt{c} d^{2}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{15 \,{\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*sqrt(c)*e^2*x^2 + 4*sqrt(c)*d*e*x - 7*sqrt(c)*d^2)*(e*x + d)*sqrt(-e*x +
d)/(e^2*x + d*e)

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Fricas [A]  time = 0.218986, size = 101, normalized size = 1.29 \[ -\frac{2 \,{\left (3 \, c e^{4} x^{4} + 4 \, c d e^{3} x^{3} - 10 \, c d^{2} e^{2} x^{2} - 4 \, c d^{3} e x + 7 \, c d^{4}\right )}}{15 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(3*c*e^4*x^4 + 4*c*d*e^3*x^3 - 10*c*d^2*e^2*x^2 - 4*c*d^3*e*x + 7*c*d^4)/(
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. \[ \int \sqrt{- c \left (- d + e x\right ) \left (d + e x\right )} \sqrt{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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