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

Optimal. Leaf size=160 \[ -\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e \sqrt{d+e x}}-\frac{8 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{3/2}}{315 c e (d+e x)^{3/2}} \]

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(3/2))/(315*c*e*(d + e*x)^(3/2)) - (64*d^2*(c*d^2
- c*e^2*x^2)^(3/2))/(105*c*e*Sqrt[d + e*x]) - (8*d*Sqrt[d + e*x]*(c*d^2 - c*e^2*
x^2)^(3/2))/(21*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(3/2))/(9*c*e)

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Rubi [A]  time = 0.235265, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ -\frac{64 d^2 \left (c d^2-c e^2 x^2\right )^{3/2}}{105 c e \sqrt{d+e x}}-\frac{8 d \sqrt{d+e x} \left (c d^2-c e^2 x^2\right )^{3/2}}{21 c e}-\frac{2 (d+e x)^{3/2} \left (c d^2-c e^2 x^2\right )^{3/2}}{9 c e}-\frac{256 d^3 \left (c d^2-c e^2 x^2\right )^{3/2}}{315 c e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-256*d^3*(c*d^2 - c*e^2*x^2)^(3/2))/(315*c*e*(d + e*x)^(3/2)) - (64*d^2*(c*d^2
- c*e^2*x^2)^(3/2))/(105*c*e*Sqrt[d + e*x]) - (8*d*Sqrt[d + e*x]*(c*d^2 - c*e^2*
x^2)^(3/2))/(21*c*e) - (2*(d + e*x)^(3/2)*(c*d^2 - c*e^2*x^2)^(3/2))/(9*c*e)

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Rubi in Sympy [A]  time = 21.6635, size = 138, normalized size = 0.86 \[ - \frac{256 d^{3} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{315 c e \left (d + e x\right )^{\frac{3}{2}}} - \frac{64 d^{2} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{105 c e \sqrt{d + e x}} - \frac{8 d \sqrt{d + e x} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{21 c e} - \frac{2 \left (d + e x\right )^{\frac{3}{2}} \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{9 c e} \]

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]

-256*d**3*(c*d**2 - c*e**2*x**2)**(3/2)/(315*c*e*(d + e*x)**(3/2)) - 64*d**2*(c*
d**2 - c*e**2*x**2)**(3/2)/(105*c*e*sqrt(d + e*x)) - 8*d*sqrt(d + e*x)*(c*d**2 -
c*e**2*x**2)**(3/2)/(21*c*e) - 2*(d + e*x)**(3/2)*(c*d**2 - c*e**2*x**2)**(3/2)
/(9*c*e)

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Mathematica [A]  time = 0.0562649, size = 75, normalized size = 0.47 \[ -\frac{2 \left (319 d^4+2 d^3 e x-156 d^2 e^2 x^2-130 d e^3 x^3-35 e^4 x^4\right ) \sqrt{c \left (d^2-e^2 x^2\right )}}{315 e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c*(d^2 - e^2*x^2)]*(319*d^4 + 2*d^3*e*x - 156*d^2*e^2*x^2 - 130*d*e^3*x
^3 - 35*e^4*x^4))/(315*e*Sqrt[d + e*x])

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Maple [A]  time = 0.009, size = 66, normalized size = 0.4 \[ -{\frac{ \left ( -2\,ex+2\,d \right ) \left ( 35\,{e}^{3}{x}^{3}+165\,d{e}^{2}{x}^{2}+321\,{d}^{2}xe+319\,{d}^{3} \right ) }{315\,e}\sqrt{-c{e}^{2}{x}^{2}+c{d}^{2}}{\frac{1}{\sqrt{ex+d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(-e*x+d)*(35*e^3*x^3+165*d*e^2*x^2+321*d^2*e*x+319*d^3)*(-c*e^2*x^2+c*d^2
)^(1/2)/e/(e*x+d)^(1/2)

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Maxima [A]  time = 0.731075, size = 111, normalized size = 0.69 \[ \frac{2 \,{\left (35 \, \sqrt{c} e^{4} x^{4} + 130 \, \sqrt{c} d e^{3} x^{3} + 156 \, \sqrt{c} d^{2} e^{2} x^{2} - 2 \, \sqrt{c} d^{3} e x - 319 \, \sqrt{c} d^{4}\right )}{\left (e x + d\right )} \sqrt{-e x + d}}{315 \,{\left (e^{2} x + d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/315*(35*sqrt(c)*e^4*x^4 + 130*sqrt(c)*d*e^3*x^3 + 156*sqrt(c)*d^2*e^2*x^2 - 2*
sqrt(c)*d^3*e*x - 319*sqrt(c)*d^4)*(e*x + d)*sqrt(-e*x + d)/(e^2*x + d*e)

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Fricas [A]  time = 0.215531, size = 134, normalized size = 0.84 \[ -\frac{2 \,{\left (35 \, c e^{6} x^{6} + 130 \, c d e^{5} x^{5} + 121 \, c d^{2} e^{4} x^{4} - 132 \, c d^{3} e^{3} x^{3} - 475 \, c d^{4} e^{2} x^{2} + 2 \, c d^{5} e x + 319 \, c d^{6}\right )}}{315 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/315*(35*c*e^6*x^6 + 130*c*d*e^5*x^5 + 121*c*d^2*e^4*x^4 - 132*c*d^3*e^3*x^3 -
475*c*d^4*e^2*x^2 + 2*c*d^5*e*x + 319*c*d^6)/(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. \[ \int \sqrt{- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac{5}{2}}\, dx \]

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)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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