3.249 \(\int \frac{x^4 \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\)

Optimal. Leaf size=125 \[ \frac{\left (6 a c^2+5 b\right ) \cosh ^{-1}(c x)}{16 c^7}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{16 c^6}+\frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{24 c^4}+\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{6 c^2} \]

[Out]

((5*b + 6*a*c^2)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c^6) + ((5*b + 6*a*c^2)*x^3
*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^4) + (b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(
6*c^2) + ((5*b + 6*a*c^2)*ArcCosh[c*x])/(16*c^7)

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Rubi [A]  time = 0.33463, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\left (6 a c^2+5 b\right ) \cosh ^{-1}(c x)}{16 c^7}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{16 c^6}+\frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{24 c^4}+\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{6 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^4*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

((5*b + 6*a*c^2)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*c^6) + ((5*b + 6*a*c^2)*x^3
*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(24*c^4) + (b*x^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(
6*c^2) + ((5*b + 6*a*c^2)*ArcCosh[c*x])/(16*c^7)

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Rubi in Sympy [A]  time = 17.229, size = 116, normalized size = 0.93 \[ \frac{b x^{5} \sqrt{c x - 1} \sqrt{c x + 1}}{6 c^{2}} + \frac{x^{3} \left (6 a c^{2} + 5 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{24 c^{4}} + \frac{x \left (6 a c^{2} + 5 b\right ) \sqrt{c x - 1} \sqrt{c x + 1}}{16 c^{6}} + \frac{\left (6 a c^{2} + 5 b\right ) \operatorname{acosh}{\left (c x \right )}}{16 c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

b*x**5*sqrt(c*x - 1)*sqrt(c*x + 1)/(6*c**2) + x**3*(6*a*c**2 + 5*b)*sqrt(c*x - 1
)*sqrt(c*x + 1)/(24*c**4) + x*(6*a*c**2 + 5*b)*sqrt(c*x - 1)*sqrt(c*x + 1)/(16*c
**6) + (6*a*c**2 + 5*b)*acosh(c*x)/(16*c**7)

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Mathematica [A]  time = 0.13668, size = 102, normalized size = 0.82 \[ \frac{3 \left (6 a c^2+5 b\right ) \log \left (c x+\sqrt{c x-1} \sqrt{c x+1}\right )+c x \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2 \left (2 c^2 x^2+3\right )+b \left (8 c^4 x^4+10 c^2 x^2+15\right )\right )}{48 c^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^4*(a + b*x^2))/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]),x]

[Out]

(c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(6*a*c^2*(3 + 2*c^2*x^2) + b*(15 + 10*c^2*x^2
+ 8*c^4*x^4)) + 3*(5*b + 6*a*c^2)*Log[c*x + Sqrt[-1 + c*x]*Sqrt[1 + c*x]])/(48*c
^7)

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Maple [C]  time = 0.05, size = 191, normalized size = 1.5 \[{\frac{{\it csgn} \left ( c \right ) }{48\,{c}^{7}}\sqrt{cx-1}\sqrt{cx+1} \left ( 8\,{\it csgn} \left ( c \right ){x}^{5}b{c}^{5}\sqrt{{c}^{2}{x}^{2}-1}+12\,{\it csgn} \left ( c \right ){x}^{3}a{c}^{5}\sqrt{{c}^{2}{x}^{2}-1}+10\,b{x}^{3}\sqrt{{c}^{2}{x}^{2}-1}{c}^{3}{\it csgn} \left ( c \right ) +18\,ax\sqrt{{c}^{2}{x}^{2}-1}{c}^{3}{\it csgn} \left ( c \right ) +15\,bx\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) c+18\,a\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ){c}^{2}+15\,b\ln \left ( \left ({\it csgn} \left ( c \right ) \sqrt{{c}^{2}{x}^{2}-1}+cx \right ){\it csgn} \left ( c \right ) \right ) \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(b*x^2+a)/(c*x-1)^(1/2)/(c*x+1)^(1/2),x)

[Out]

1/48*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(8*csgn(c)*x^5*b*c^5*(c^2*x^2-1)^(1/2)+12*csgn(
c)*x^3*a*c^5*(c^2*x^2-1)^(1/2)+10*b*x^3*(c^2*x^2-1)^(1/2)*c^3*csgn(c)+18*a*x*(c^
2*x^2-1)^(1/2)*c^3*csgn(c)+15*b*x*(c^2*x^2-1)^(1/2)*csgn(c)*c+18*a*ln((csgn(c)*(
c^2*x^2-1)^(1/2)+c*x)*csgn(c))*c^2+15*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^7

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Maxima [A]  time = 1.48597, size = 231, normalized size = 1.85 \[ \frac{\sqrt{c^{2} x^{2} - 1} b x^{5}}{6 \, c^{2}} + \frac{\sqrt{c^{2} x^{2} - 1} a x^{3}}{4 \, c^{2}} + \frac{5 \, \sqrt{c^{2} x^{2} - 1} b x^{3}}{24 \, c^{4}} + \frac{3 \, \sqrt{c^{2} x^{2} - 1} a x}{8 \, c^{4}} + \frac{3 \, a \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{8 \, \sqrt{c^{2}} c^{4}} + \frac{5 \, \sqrt{c^{2} x^{2} - 1} b x}{16 \, c^{6}} + \frac{5 \, b \log \left (2 \, c^{2} x + 2 \, \sqrt{c^{2} x^{2} - 1} \sqrt{c^{2}}\right )}{16 \, \sqrt{c^{2}} c^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^4/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="maxima")

[Out]

1/6*sqrt(c^2*x^2 - 1)*b*x^5/c^2 + 1/4*sqrt(c^2*x^2 - 1)*a*x^3/c^2 + 5/24*sqrt(c^
2*x^2 - 1)*b*x^3/c^4 + 3/8*sqrt(c^2*x^2 - 1)*a*x/c^4 + 3/8*a*log(2*c^2*x + 2*sqr
t(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^4) + 5/16*sqrt(c^2*x^2 - 1)*b*x/c^6 + 5/1
6*b*log(2*c^2*x + 2*sqrt(c^2*x^2 - 1)*sqrt(c^2))/(sqrt(c^2)*c^6)

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Fricas [A]  time = 0.270148, size = 581, normalized size = 4.65 \[ -\frac{256 \, b c^{12} x^{12} + 192 \,{\left (2 \, a c^{12} - b c^{10}\right )} x^{10} - 48 \,{\left (4 \, a c^{10} - 3 \, b c^{8}\right )} x^{8} - 4 \,{\left (174 \, a c^{8} + 157 \, b c^{6}\right )} x^{6} + 102 \,{\left (6 \, a c^{6} + 5 \, b c^{4}\right )} x^{4} - 18 \,{\left (6 \, a c^{4} + 5 \, b c^{2}\right )} x^{2} -{\left (256 \, b c^{11} x^{11} + 144 \, b c^{7} x^{7} + 64 \,{\left (6 \, a c^{11} - b c^{9}\right )} x^{9} - 4 \,{\left (162 \, a c^{7} + 137 \, b c^{5}\right )} x^{5} + 52 \,{\left (6 \, a c^{5} + 5 \, b c^{3}\right )} x^{3} - 3 \,{\left (6 \, a c^{3} + 5 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 3 \,{\left (32 \,{\left (6 \, a c^{8} + 5 \, b c^{6}\right )} x^{6} - 48 \,{\left (6 \, a c^{6} + 5 \, b c^{4}\right )} x^{4} - 6 \, a c^{2} + 18 \,{\left (6 \, a c^{4} + 5 \, b c^{2}\right )} x^{2} - 2 \,{\left (16 \,{\left (6 \, a c^{7} + 5 \, b c^{5}\right )} x^{5} - 16 \,{\left (6 \, a c^{5} + 5 \, b c^{3}\right )} x^{3} + 3 \,{\left (6 \, a c^{3} + 5 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} - 5 \, b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{48 \,{\left (32 \, c^{13} x^{6} - 48 \, c^{11} x^{4} + 18 \, c^{9} x^{2} - c^{7} - 2 \,{\left (16 \, c^{12} x^{5} - 16 \, c^{10} x^{3} + 3 \, c^{8} x\right )} \sqrt{c x + 1} \sqrt{c x - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^4/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="fricas")

[Out]

-1/48*(256*b*c^12*x^12 + 192*(2*a*c^12 - b*c^10)*x^10 - 48*(4*a*c^10 - 3*b*c^8)*
x^8 - 4*(174*a*c^8 + 157*b*c^6)*x^6 + 102*(6*a*c^6 + 5*b*c^4)*x^4 - 18*(6*a*c^4
+ 5*b*c^2)*x^2 - (256*b*c^11*x^11 + 144*b*c^7*x^7 + 64*(6*a*c^11 - b*c^9)*x^9 -
4*(162*a*c^7 + 137*b*c^5)*x^5 + 52*(6*a*c^5 + 5*b*c^3)*x^3 - 3*(6*a*c^3 + 5*b*c)
*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + 3*(32*(6*a*c^8 + 5*b*c^6)*x^6 - 48*(6*a*c^6 +
5*b*c^4)*x^4 - 6*a*c^2 + 18*(6*a*c^4 + 5*b*c^2)*x^2 - 2*(16*(6*a*c^7 + 5*b*c^5)*
x^5 - 16*(6*a*c^5 + 5*b*c^3)*x^3 + 3*(6*a*c^3 + 5*b*c)*x)*sqrt(c*x + 1)*sqrt(c*x
- 1) - 5*b)*log(-c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/(32*c^13*x^6 - 48*c^11*x^4
+ 18*c^9*x^2 - c^7 - 2*(16*c^12*x^5 - 16*c^10*x^3 + 3*c^8*x)*sqrt(c*x + 1)*sqrt
(c*x - 1))

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Sympy [A]  time = 134.074, size = 216, normalized size = 1.73 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{7}{4}, - \frac{5}{4} & - \frac{3}{2}, - \frac{3}{2}, -1, 1 \\-2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{5}{2}, - \frac{9}{4}, -2, - \frac{7}{4}, - \frac{3}{2}, 1 & \\- \frac{9}{4}, - \frac{7}{4} & - \frac{5}{2}, -2, -2, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{5}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{11}{4}, - \frac{9}{4} & - \frac{5}{2}, - \frac{5}{2}, -2, 1 \\-3, - \frac{11}{4}, - \frac{5}{2}, - \frac{9}{4}, -2, 0 & \end{matrix} \middle |{\frac{1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{7}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{7}{2}, - \frac{13}{4}, -3, - \frac{11}{4}, - \frac{5}{2}, 1 & \\- \frac{13}{4}, - \frac{11}{4} & - \frac{7}{2}, -3, -3, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(b*x**2+a)/(c*x-1)**(1/2)/(c*x+1)**(1/2),x)

[Out]

a*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*a*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) + b*meijerg(((-11/4, -9/4), (-5/2, -5/2, -2, 1)), ((-3, -11/4, -
5/2, -9/4, -2, 0), ()), 1/(c**2*x**2))/(4*pi**(3/2)*c**7) - I*b*meijerg(((-7/2,
-13/4, -3, -11/4, -5/2, 1), ()), ((-13/4, -11/4), (-7/2, -3, -3, 0)), exp_polar(
2*I*pi)/(c**2*x**2))/(4*pi**(3/2)*c**7)

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GIAC/XCAS [A]  time = 0.246501, size = 205, normalized size = 1.64 \[ -\frac{{\left (30 \, a c^{38} + 33 \, b c^{36} -{\left (54 \, a c^{38} + 85 \, b c^{36} - 2 \,{\left (18 \, a c^{38} + 55 \, b c^{36} -{\left (6 \, a c^{38} + 45 \, b c^{36} + 4 \,{\left ({\left (c x + 1\right )} b c^{36} - 5 \, b c^{36}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )}{\left (c x + 1\right )}\right )} \sqrt{c x + 1} \sqrt{c x - 1} + 6 \,{\left (6 \, a c^{38} + 5 \, b c^{36}\right )}{\rm ln}\left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{34603008 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)*x^4/(sqrt(c*x + 1)*sqrt(c*x - 1)),x, algorithm="giac")

[Out]

-1/34603008*((30*a*c^38 + 33*b*c^36 - (54*a*c^38 + 85*b*c^36 - 2*(18*a*c^38 + 55
*b*c^36 - (6*a*c^38 + 45*b*c^36 + 4*((c*x + 1)*b*c^36 - 5*b*c^36)*(c*x + 1))*(c*
x + 1))*(c*x + 1))*(c*x + 1))*sqrt(c*x + 1)*sqrt(c*x - 1) + 6*(6*a*c^38 + 5*b*c^
36)*ln(abs(-sqrt(c*x + 1) + sqrt(c*x - 1))))/c