3.250 \(\int \frac{x^3 \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^6}+\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^4}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2} \]

[Out]

(2*(4*b + 5*a*c^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^6) + ((4*b + 5*a*c^2)*x^2
*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^4) + (b*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(
5*c^2)

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Rubi [A]  time = 0.26647, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^6}+\frac{x^2 \sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2+4 b\right )}{15 c^4}+\frac{b x^4 \sqrt{c x-1} \sqrt{c x+1}}{5 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^3*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

(2*(4*b + 5*a*c^2)*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^6) + ((4*b + 5*a*c^2)*x^2
*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(15*c^4) + (b*x^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(
5*c^2)

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Rubi in Sympy [A]  time = 13.3902, size = 95, normalized size = 0.92 \[ \frac{b x^{4} \sqrt{c x - 1} \sqrt{c x + 1}}{5 c^{2}} + \frac{x^{2} \left (5 a c^{2} + 4 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{15 c^{4}} + \frac{2 \left (5 a c^{2} + 4 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{15 c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

b*x**4*sqrt(c*x - 1)*sqrt(c*x + 1)/(5*c**2) + x**2*(5*a*c**2 + 4*b)*sqrt(c*x - 1
)*sqrt(c*x + 1)/(15*c**4) + 2*(5*a*c**2 + 4*b)*sqrt(c*x - 1)*sqrt(c*x + 1)/(15*c
**6)

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Mathematica [A]  time = 0.0631832, size = 61, normalized size = 0.59 \[ \frac{\sqrt{c x-1} \sqrt{c x+1} \left (5 a c^2 \left (c^2 x^2+2\right )+b \left (3 c^4 x^4+4 c^2 x^2+8\right )\right )}{15 c^6} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^3*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(5*a*c^2*(2 + c^2*x^2) + b*(8 + 4*c^2*x^2 + 3*c^4*
x^4)))/(15*c^6)

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Maple [A]  time = 0.009, size = 57, normalized size = 0.6 \[{\frac{3\,b{x}^{4}{c}^{4}+5\,a{c}^{4}{x}^{2}+4\,b{c}^{2}{x}^{2}+10\,a{c}^{2}+8\,b}{15\,{c}^{6}}\sqrt{cx-1}\sqrt{cx+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)

[Out]

1/15*(c*x+1)^(1/2)*(c*x-1)^(1/2)*(3*b*c^4*x^4+5*a*c^4*x^2+4*b*c^2*x^2+10*a*c^2+8
*b)/c^6

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Maxima [A]  time = 1.39394, size = 128, normalized size = 1.24 \[ \frac{\sqrt{c^{2} x^{2} - 1} b x^{4}}{5 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a x^{2}}{3 \, c^{2}} + \frac{4 \, \sqrt{c^{2} x^{2} - 1} b x^{2}}{15 \, c^{4}} + \frac{2 \, \sqrt{c^{2} x^{2} - 1} a}{3 \, c^{4}} + \frac{8 \, \sqrt{c^{2} x^{2} - 1} b}{15 \, c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^3/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="maxima")

[Out]

1/5*sqrt(c^2*x^2 - 1)*b*x^4/c^2 + 1/3*sqrt(c^2*x^2 - 1)*a*x^2/c^2 + 4/15*sqrt(c^
2*x^2 - 1)*b*x^2/c^4 + 2/3*sqrt(c^2*x^2 - 1)*a/c^4 + 8/15*sqrt(c^2*x^2 - 1)*b/c^
6

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Fricas [A]  time = 0.253568, size = 333, normalized size = 3.23 \[ -\frac{48 \, b c^{10} x^{10} + 20 \,{\left (4 \, a c^{10} - b c^{8}\right )} x^{8} + 5 \,{\left (4 \, a c^{8} + 11 \, b c^{6}\right )} x^{6} - 5 \,{\left (43 \, a c^{6} + 35 \, b c^{4}\right )} x^{4} - 10 \, a c^{2} + 25 \,{\left (5 \, a c^{4} + 4 \, b c^{2}\right )} x^{2} -{\left (48 \, b c^{9} x^{9} + 4 \,{\left (20 \, a c^{9} + b c^{7}\right )} x^{7} + 3 \,{\left (20 \, a c^{7} + 21 \, b c^{5}\right )} x^{5} - 35 \,{\left (5 \, a c^{5} + 4 \, b c^{3}\right )} x^{3} + 10 \,{\left (5 \, a c^{3} + 4 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 8 \, b}{15 \,{\left (16 \, c^{11} x^{5} - 20 \, c^{9} x^{3} + 5 \, c^{7} x -{\left (16 \, c^{10} x^{4} - 12 \, c^{8} x^{2} + c^{6}\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^3/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="fricas")

[Out]

-1/15*(48*b*c^10*x^10 + 20*(4*a*c^10 - b*c^8)*x^8 + 5*(4*a*c^8 + 11*b*c^6)*x^6 -
5*(43*a*c^6 + 35*b*c^4)*x^4 - 10*a*c^2 + 25*(5*a*c^4 + 4*b*c^2)*x^2 - (48*b*c^9
*x^9 + 4*(20*a*c^9 + b*c^7)*x^7 + 3*(20*a*c^7 + 21*b*c^5)*x^5 - 35*(5*a*c^5 + 4*
b*c^3)*x^3 + 10*(5*a*c^3 + 4*b*c)*x)*sqrt(c*x + 1)*sqrt(c*x - 1) - 8*b)/(16*c^11
*x^5 - 20*c^9*x^3 + 5*c^7*x - (16*c^10*x^4 - 12*c^8*x^2 + c^6)*sqrt(c*x + 1)*sqr
t(c*x - 1))

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Sympy [A]  time = 89.8548, size = 216, normalized size = 2.1 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{9}{4}, - \frac{7}{4} & -2, -2, - \frac{3}{2}, 1 \\- \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{6}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 1 & \\- \frac{11}{4}, - \frac{9}{4} & -3, - \frac{5}{2}, - \frac{5}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

a*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3/2, -5/4, -1, -3/4, -1/2, 0), (
)), 1/(c**2*x**2))/(4*pi**(3/2)*c**4) + I*a*meijerg(((-2, -7/4, -3/2, -5/4, -1,
1), ()), ((-7/4, -5/4), (-2, -3/2, -3/2, 0)), exp_polar(2*I*pi)/(c**2*x**2))/(4*
pi**(3/2)*c**4) + b*meijerg(((-9/4, -7/4), (-2, -2, -3/2, 1)), ((-5/2, -9/4, -2,
-7/4, -3/2, 0), ()), 1/(c**2*x**2))/(4*pi**(3/2)*c**6) + I*b*meijerg(((-3, -11/
4, -5/2, -9/4, -2, 1), ()), ((-11/4, -9/4), (-3, -5/2, -5/2, 0)), exp_polar(2*I*
pi)/(c**2*x**2))/(4*pi**(3/2)*c**6)

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GIAC/XCAS [A]  time = 0.21793, size = 130, normalized size = 1.26 \[ \frac{{\left (15 \, a c^{27} + 15 \, b c^{25} -{\left (10 \, a c^{27} + 20 \, b c^{25} -{\left (5 \, a c^{27} + 22 \, b c^{25} + 3 \,{\left ({\left (c x + 1\right )} b c^{25} - 4 \, b c^{25}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1}}{276480 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^3/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="giac")

[Out]

1/276480*(15*a*c^27 + 15*b*c^25 - (10*a*c^27 + 20*b*c^25 - (5*a*c^27 + 22*b*c^25
+ 3*((c*x + 1)*b*c^25 - 4*b*c^25)*(c*x + 1))*(c*x + 1))*(c*x + 1))*sqrt(c*x + 1
)*sqrt(c*x - 1)/c