3.875 \(\int \frac{(d+e x)^{9/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}} \]

[Out]

(256*d^3*Sqrt[d + e*x])/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (64*d^2*(d + e*x)^(3/2
))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (8*d*(d + e*x)^(5/2))/(5*c*e*Sqrt[c*d^2 - c
*e^2*x^2]) - (2*(d + e*x)^(7/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2])

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Rubi [A]  time = 0.231663, 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 \[ -\frac{2 (d+e x)^{7/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{8 d (d+e x)^{5/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}-\frac{64 d^2 (d+e x)^{3/2}}{5 c e \sqrt{c d^2-c e^2 x^2}}+\frac{256 d^3 \sqrt{d+e x}}{5 c e \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(256*d^3*Sqrt[d + e*x])/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (64*d^2*(d + e*x)^(3/2
))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2]) - (8*d*(d + e*x)^(5/2))/(5*c*e*Sqrt[c*d^2 - c
*e^2*x^2]) - (2*(d + e*x)^(7/2))/(5*c*e*Sqrt[c*d^2 - c*e^2*x^2])

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Rubi in Sympy [A]  time = 21.5651, size = 136, normalized size = 0.85 \[ \frac{256 d^{3} \sqrt{d + e x}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{64 d^{2} \left (d + e x\right )^{\frac{3}{2}}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{8 d \left (d + e x\right )^{\frac{5}{2}}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{7}{2}}}{5 c e \sqrt{c d^{2} - c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(9/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

256*d**3*sqrt(d + e*x)/(5*c*e*sqrt(c*d**2 - c*e**2*x**2)) - 64*d**2*(d + e*x)**(
3/2)/(5*c*e*sqrt(c*d**2 - c*e**2*x**2)) - 8*d*(d + e*x)**(5/2)/(5*c*e*sqrt(c*d**
2 - c*e**2*x**2)) - 2*(d + e*x)**(7/2)/(5*c*e*sqrt(c*d**2 - c*e**2*x**2))

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Mathematica [A]  time = 0.0675603, size = 66, normalized size = 0.41 \[ -\frac{2 \sqrt{d+e x} \left (-91 d^3+43 d^2 e x+7 d e^2 x^2+e^3 x^3\right )}{5 c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-91*d^3 + 43*d^2*e*x + 7*d*e^2*x^2 + e^3*x^3))/(5*c*e*Sqrt[c*
(d^2 - e^2*x^2)])

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Maple [A]  time = 0.009, size = 66, normalized size = 0.4 \[{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{3}{x}^{3}-7\,d{e}^{2}{x}^{2}-43\,{d}^{2}xe+91\,{d}^{3} \right ) }{5\,e} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2/5*(-e*x+d)*(-e^3*x^3-7*d*e^2*x^2-43*d^2*e*x+91*d^3)*(e*x+d)^(3/2)/e/(-c*e^2*x^
2+c*d^2)^(3/2)

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Maxima [A]  time = 0.730802, size = 61, normalized size = 0.38 \[ -\frac{2 \,{\left (e^{3} x^{3} + 7 \, d e^{2} x^{2} + 43 \, d^{2} e x - 91 \, d^{3}\right )}}{5 \, \sqrt{-e x + d} c^{\frac{3}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/5*(e^3*x^3 + 7*d*e^2*x^2 + 43*d^2*e*x - 91*d^3)/(sqrt(-e*x + d)*c^(3/2)*e)

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Fricas [A]  time = 0.2111, size = 97, normalized size = 0.61 \[ -\frac{2 \,{\left (e^{4} x^{4} + 8 \, d e^{3} x^{3} + 50 \, d^{2} e^{2} x^{2} - 48 \, d^{3} e x - 91 \, d^{4}\right )}}{5 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/5*(e^4*x^4 + 8*d*e^3*x^3 + 50*d^2*e^2*x^2 - 48*d^3*e*x - 91*d^4)/(sqrt(-c*e^2
*x^2 + c*d^2)*sqrt(e*x + d)*c*e)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(9/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.612206, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x