3.876 \(\int \frac{(d+e x)^{7/2}}{\left (c d^2-c e^2 x^2\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.163186, 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 \[ -\frac{2 (d+e x)^{5/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}-\frac{16 d (d+e x)^{3/2}}{3 c e \sqrt{c d^2-c e^2 x^2}}+\frac{64 d^2 \sqrt{d+e x}}{3 c e \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.4686, size = 100, normalized size = 0.84 \[ \frac{64 d^{2} \sqrt{d + e x}}{3 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{16 d \left (d + e x\right )^{\frac{3}{2}}}{3 c e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{2 \left (d + e x\right )^{\frac{5}{2}}}{3 c e \sqrt{c d^{2} - c e^{2} x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(7/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0554841, size = 55, normalized size = 0.46 \[ -\frac{2 \sqrt{d+e x} \left (-23 d^2+10 d e x+e^2 x^2\right )}{3 c e \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(7/2)/(c*d^2 - c*e^2*x^2)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(-23*d^2 + 10*d*e*x + e^2*x^2))/(3*c*e*Sqrt[c*(d^2 - e^2*x^2)]
)

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Maple [A]  time = 0.008, size = 55, normalized size = 0.5 \[{\frac{ \left ( -2\,ex+2\,d \right ) \left ( -{e}^{2}{x}^{2}-10\,dxe+23\,{d}^{2} \right ) }{3\,e} \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ( -c{e}^{2}{x}^{2}+c{d}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(7/2)/(-c*e^2*x^2+c*d^2)^(3/2),x)

[Out]

2/3*(-e*x+d)*(-e^2*x^2-10*d*e*x+23*d^2)*(e*x+d)^(3/2)/e/(-c*e^2*x^2+c*d^2)^(3/2)

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Maxima [A]  time = 0.729941, size = 58, normalized size = 0.49 \[ -\frac{2 \,{\left (\sqrt{c} e^{2} x^{2} + 10 \, \sqrt{c} d e x - 23 \, \sqrt{c} d^{2}\right )}}{3 \, \sqrt{-e x + d} c^{2} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="maxima")

[Out]

-2/3*(sqrt(c)*e^2*x^2 + 10*sqrt(c)*d*e*x - 23*sqrt(c)*d^2)/(sqrt(-e*x + d)*c^2*e
)

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Fricas [A]  time = 0.215105, size = 82, normalized size = 0.69 \[ -\frac{2 \,{\left (e^{3} x^{3} + 11 \, d e^{2} x^{2} - 13 \, d^{2} e x - 23 \, d^{3}\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} c e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="fricas")

[Out]

-2/3*(e^3*x^3 + 11*d*e^2*x^2 - 13*d^2*e*x - 23*d^3)/(sqrt(-c*e^2*x^2 + c*d^2)*sq
rt(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)**(7/2)/(-c*e**2*x**2+c*d**2)**(3/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.61128, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(7/2)/(-c*e^2*x^2 + c*d^2)^(3/2),x, algorithm="giac")

[Out]

sage0*x