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

Optimal. Leaf size=87 \[ \frac{\left (4 a c^2+3 b\right ) \cosh ^{-1}(c x)}{8 c^5}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+3 b\right )}{8 c^4}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{4 c^2} \]

[Out]

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

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Rubi [A]  time = 0.261869, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{\left (4 a c^2+3 b\right ) \cosh ^{-1}(c x)}{8 c^5}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+3 b\right )}{8 c^4}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.7528, size = 80, normalized size = 0.92 \[ \frac{b x^{3} \sqrt{c x - 1} \sqrt{c x + 1}}{4 c^{2}} + \frac{x \left (4 a c^{2} + 3 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{8 c^{4}} + \frac{\left (4 a c^{2} + 3 b\right ) \operatorname{acosh}{\left (c x \right )}}{8 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**3*sqrt(c*x - 1)*sqrt(c*x + 1)/(4*c**2) + x*(4*a*c**2 + 3*b)*sqrt(c*x - 1)*s
qrt(c*x + 1)/(8*c**4) + (4*a*c**2 + 3*b)*acosh(c*x)/(8*c**5)

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Mathematica [A]  time = 0.0936331, size = 83, normalized size = 0.95 \[ \frac{c x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+b \left (2 c^2 x^2+3\right )\right )+\left (4 a c^2+3 b\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )}{8 c^5} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.028, size = 147, normalized size = 1.7 \[{\frac{{\it csgn} \left ( c \right ) }{8\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1} \left ( 2\,b{x}^{3}\sqrt{{c}^{2}{x}^{2}-1}{c}^{3}{\it csgn} \left ( c \right ) +4\,ax\sqrt{{c}^{2}{x}^{2}-1}{c}^{3}{\it csgn} \left ( c \right ) +3\,bx\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) c+4\,a\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ){c}^{2}+3\,b\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(2*b*x^3*(c^2*x^2-1)^(1/2)*c^3*csgn(c)+4*a*x*(c^
2*x^2-1)^(1/2)*c^3*csgn(c)+3*b*x*(c^2*x^2-1)^(1/2)*csgn(c)*c+4*a*ln((csgn(c)*(c^
2*x^2-1)^(1/2)+c*x)*csgn(c))*c^2+3*b*ln((csgn(c)*(c^2*x^2-1)^(1/2)+c*x)*csgn(c))
)*csgn(c)/(c^2*x^2-1)^(1/2)/c^5

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Maxima [A]  time = 1.38053, size = 177, normalized size = 2.03 \[ \frac{\sqrt{c^{2} x^{2} - 1} b x^{3}}{4 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a x}{2 \, c^{2}} + \frac{a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{2 \, \sqrt{c^{2}} c^{2}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} b x}{8 \, c^{4}} + \frac{3 \, b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{8 \, \sqrt{c^{2}} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.243741, size = 406, normalized size = 4.67 \[ -\frac{16 \, b c^{8} x^{8} + 32 \, a c^{8} x^{6} - 4 \,{\left (12 \, a c^{6} + 7 \, b c^{4}\right )} x^{4} + 4 \,{\left (4 \, a c^{4} + 3 \, b c^{2}\right )} x^{2} -{\left (16 \, b c^{7} x^{7} + 8 \,{\left (4 \, a c^{7} + b c^{5}\right )} x^{5} - 2 \,{\left (16 \, a c^{5} + 11 \, b c^{3}\right )} x^{3} +{\left (4 \, a c^{3} + 3 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} +{\left (8 \,{\left (4 \, a c^{6} + 3 \, b c^{4}\right )} x^{4} + 4 \, a c^{2} - 8 \,{\left (4 \, a c^{4} + 3 \, b c^{2}\right )} x^{2} - 4 \,{\left (2 \,{\left (4 \, a c^{5} + 3 \, b c^{3}\right )} x^{3} -{\left (4 \, a c^{3} + 3 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 3 \, b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{8 \,{\left (8 \, c^{9} x^{4} - 8 \, c^{7} x^{2} + c^{5} - 4 \,{\left (2 \, c^{8} x^{3} - c^{6} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(16*b*c^8*x^8 + 32*a*c^8*x^6 - 4*(12*a*c^6 + 7*b*c^4)*x^4 + 4*(4*a*c^4 + 3*
b*c^2)*x^2 - (16*b*c^7*x^7 + 8*(4*a*c^7 + b*c^5)*x^5 - 2*(16*a*c^5 + 11*b*c^3)*x
^3 + (4*a*c^3 + 3*b*c)*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (8*(4*a*c^6 + 3*b*c^4)*x
^4 + 4*a*c^2 - 8*(4*a*c^4 + 3*b*c^2)*x^2 - 4*(2*(4*a*c^5 + 3*b*c^3)*x^3 - (4*a*c
^3 + 3*b*c)*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + 3*b)*log(-c*x + sqrt(c*x + 1)*sqrt(
c*x - 1)))/(8*c^9*x^4 - 8*c^7*x^2 + c^5 - 4*(2*c^8*x^3 - c^6*x)*sqrt(c*x + 1)*sq
rt(c*x - 1))

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Sympy [A]  time = 86.6118, size = 212, normalized size = 2.44 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.249062, size = 151, normalized size = 1.74 \[ -\frac{{\left (4 \, a c^{18} + 5 \, b c^{16} -{\left (4 \, a c^{18} + 9 \, b c^{16} + 2 \,{\left ({\left (c x + 1\right )} b c^{16} - 3 \, b c^{16}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 2 \,{\left (4 \, a c^{18} + 3 \, b c^{16}\right )}{\rm ln}\left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{114688 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/114688*((4*a*c^18 + 5*b*c^16 - (4*a*c^18 + 9*b*c^16 + 2*((c*x + 1)*b*c^16 - 3
*b*c^16)*(c*x + 1))*(c*x + 1))*sqrt(c*x + 1)*sqrt(c*x - 1) + 2*(4*a*c^18 + 3*b*c
^16)*ln(abs(-sqrt(c*x + 1) + sqrt(c*x - 1))))/c