3.869 \(\int \frac{(d+e x)^{5/2}}{\sqrt{c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=119 \[ -\frac{64 d^2 \sqrt{c d^2-c e^2 x^2}}{15 c e \sqrt{d+e x}}-\frac{16 d \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{15 c e}-\frac{2 (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{5 c e} \]

[Out]

(-64*d^2*Sqrt[c*d^2 - c*e^2*x^2])/(15*c*e*Sqrt[d + e*x]) - (16*d*Sqrt[d + e*x]*S
qrt[c*d^2 - c*e^2*x^2])/(15*c*e) - (2*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2])/(
5*c*e)

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Rubi [A]  time = 0.167346, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{64 d^2 \sqrt{c d^2-c e^2 x^2}}{15 c e \sqrt{d+e x}}-\frac{16 d \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}{15 c e}-\frac{2 (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}{5 c e} \]

Antiderivative was successfully verified.

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

[Out]

(-64*d^2*Sqrt[c*d^2 - c*e^2*x^2])/(15*c*e*Sqrt[d + e*x]) - (16*d*Sqrt[d + e*x]*S
qrt[c*d^2 - c*e^2*x^2])/(15*c*e) - (2*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2])/(
5*c*e)

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Rubi in Sympy [A]  time = 15.4749, size = 102, normalized size = 0.86 \[ - \frac{64 d^{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{15 c e \sqrt{d + e x}} - \frac{16 d \sqrt{d + e x} \sqrt{c d^{2} - c e^{2} x^{2}}}{15 c e} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \sqrt{c d^{2} - c e^{2} x^{2}}}{5 c e} \]

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)**(1/2),x)

[Out]

-64*d**2*sqrt(c*d**2 - c*e**2*x**2)/(15*c*e*sqrt(d + e*x)) - 16*d*sqrt(d + e*x)*
sqrt(c*d**2 - c*e**2*x**2)/(15*c*e) - 2*(d + e*x)**(3/2)*sqrt(c*d**2 - c*e**2*x*
*2)/(5*c*e)

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Mathematica [A]  time = 0.0499785, size = 59, normalized size = 0.5 \[ -\frac{2 (d-e x) \sqrt{d+e x} \left (43 d^2+14 d e x+3 e^2 x^2\right )}{15 e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(d - e*x)*Sqrt[d + e*x]*(43*d^2 + 14*d*e*x + 3*e^2*x^2))/(15*e*Sqrt[c*(d^2 -
e^2*x^2)])

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Maple [A]  time = 0.01, size = 55, normalized size = 0.5 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 3\,{e}^{2}{x}^{2}+14\,dxe+43\,{d}^{2} \right ) }{15\,e}\sqrt{ex+d}{\frac{1}{\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(1/2),x)

[Out]

-2/15*(-e*x+d)*(3*e^2*x^2+14*d*e*x+43*d^2)*(e*x+d)^(1/2)/e/(-c*e^2*x^2+c*d^2)^(1
/2)

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Maxima [A]  time = 0.721074, size = 78, normalized size = 0.66 \[ \frac{2 \,{\left (3 \, \sqrt{c} e^{3} x^{3} + 11 \, \sqrt{c} d e^{2} x^{2} + 29 \, \sqrt{c} d^{2} e x - 43 \, \sqrt{c} d^{3}\right )}}{15 \, \sqrt{-e x + d} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*sqrt(c)*e^3*x^3 + 11*sqrt(c)*d*e^2*x^2 + 29*sqrt(c)*d^2*e*x - 43*sqrt(c)
*d^3)/(sqrt(-e*x + d)*c*e)

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Fricas [A]  time = 0.219979, size = 95, normalized size = 0.8 \[ \frac{2 \,{\left (3 \, e^{4} x^{4} + 14 \, d e^{3} x^{3} + 40 \, d^{2} e^{2} x^{2} - 14 \, d^{3} e x - 43 \, d^{4}\right )}}{15 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*(3*e^4*x^4 + 14*d*e^3*x^3 + 40*d^2*e^2*x^2 - 14*d^3*e*x - 43*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. \[ \int \frac{\left (d + e x\right )^{\frac{5}{2}}}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(5/2)/(-c*e**2*x**2+c*d**2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((e*x + d)^(5/2)/sqrt(-c*e^2*x^2 + c*d^2), x)