\(\int \frac {x (a+b x^2)}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\) [351]

Optimal result

Integrand size = 27, antiderivative size = 65 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\left (2 b+3 a c^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3 c^4}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {471, 75} \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\sqrt {c x-1} \sqrt {c x+1} \left (3 a c^2+2 b\right )}{3 c^4}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{3 c^2} \]

[In]

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

[Out]

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

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 471

Int[((e_.)*(x_))^(m_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^(p_.)*((c_) + (d_.)
*(x_)^(n_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*(a1 + b1*x^(n/2))^(p + 1)*((a2 + b2*x^(n/2))^(p + 1)/(b1*b2*e*(
m + n*(p + 1) + 1))), x] - Dist[(a1*a2*d*(m + 1) - b1*b2*c*(m + n*(p + 1) + 1))/(b1*b2*(m + n*(p + 1) + 1)), I
nt[(e*x)^m*(a1 + b1*x^(n/2))^p*(a2 + b2*x^(n/2))^p, x], x] /; FreeQ[{a1, b1, a2, b2, c, d, e, m, n, p}, x] &&
EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && NeQ[m + n*(p + 1) + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 c^2}-\frac {1}{3} \left (-3 a-\frac {2 b}{c^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx \\ & = \frac {\left (2 b+3 a c^2\right ) \sqrt {-1+c x} \sqrt {1+c x}}{3 c^4}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{3 c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.66 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (3 a c^2+b \left (2+c^2 x^2\right )\right )}{3 c^4} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 4.07 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.58

method	result	size
gosper	\(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (b \,c^{2} x^{2}+3 c^{2} a +2 b \right )}{3 c^{4}}\)	\(38\)
default	\(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (b \,c^{2} x^{2}+3 c^{2} a +2 b \right )}{3 c^{4}}\)	\(38\)
risch	\(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (b \,c^{2} x^{2}+3 c^{2} a +2 b \right )}{3 c^{4}}\)	\(38\)

[In]

int(x*(b*x^2+a)/(c*x-1)^(1/2)/(c*x+1)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.57 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {{\left (b c^{2} x^{2} + 3 \, a c^{2} + 2 \, b\right )} \sqrt {c x + 1} \sqrt {c x - 1}}{3 \, c^{4}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 4.08 (sec) , antiderivative size = 202, normalized size of antiderivative = 3.11 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {a {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{4} & 0, 0, \frac {1}{2}, 1 \\- \frac {1}{2}, - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} + \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} -1, - \frac {3}{4}, - \frac {1}{2}, - \frac {1}{4}, 0, 1 & \\- \frac {3}{4}, - \frac {1}{4} & -1, - \frac {1}{2}, - \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{2}} + \frac {b {G_{6, 6}^{6, 2}\left (\begin {matrix} - \frac {5}{4}, - \frac {3}{4} & -1, -1, - \frac {1}{2}, 1 \\- \frac {3}{2}, - \frac {5}{4}, -1, - \frac {3}{4}, - \frac {1}{2}, 0 & \end {matrix} \middle | {\frac {1}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{4}} + \frac {i b {G_{6, 6}^{2, 6}\left (\begin {matrix} -2, - \frac {7}{4}, - \frac {3}{2}, - \frac {5}{4}, -1, 1 & \\- \frac {7}{4}, - \frac {5}{4} & -2, - \frac {3}{2}, - \frac {3}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{c^{2} x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}} c^{4}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.83 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\sqrt {c^{2} x^{2} - 1} b x^{2}}{3 \, c^{2}} + \frac {\sqrt {c^{2} x^{2} - 1} a}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1} b}{3 \, c^{4}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.91 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\sqrt {c x + 1} \sqrt {c x - 1} {\left ({\left (c x + 1\right )} {\left (\frac {{\left (c x + 1\right )} b}{c^{3}} - \frac {2 \, b}{c^{3}}\right )} + \frac {3 \, {\left (a c^{11} + b c^{9}\right )}}{c^{12}}\right )}}{3 \, c} \]

[In]

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

[Out]

1/3*sqrt(c*x + 1)*sqrt(c*x - 1)*((c*x + 1)*((c*x + 1)*b/c^3 - 2*b/c^3) + 3*(a*c^11 + b*c^9)/c^12)/c

Mupad [B] (verification not implemented)

Time = 6.57 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.02 \[ \int \frac {x \left (a+b x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx=\frac {\sqrt {c\,x-1}\,\left (\frac {3\,a\,c^2+2\,b}{3\,c^4}+\frac {b\,x^3}{3\,c}+\frac {b\,x^2}{3\,c^2}+\frac {x\,\left (3\,a\,c^3+2\,b\,c\right )}{3\,c^4}\right )}{\sqrt {c\,x+1}} \]

[In]

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

[Out]

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