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

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

[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 = 0.0694708, 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, Rules used = {460, 90, 52} \[ \frac{x \sqrt{c x-1} \sqrt{c x+1} \left (4 a c^2+3 b\right )}{8 c^4}+\frac{\left (4 a c^2+3 b\right ) \cosh ^{-1}(c x)}{8 c^5}+\frac{b x^3 \sqrt{c x-1} \sqrt{c x+1}}{4 c^2} \]

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)

Rule 460

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]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 52

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ArcCosh[(b*x)/a]/b, x] /; FreeQ[{a,
b, c, d}, x] && EqQ[a + c, 0] && EqQ[b - d, 0] && GtQ[a, 0]

Rubi steps

\begin{align*} \int \frac{x^2 \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{4 c^2}-\frac{1}{4} \left (-4 a-\frac{3 b}{c^2}\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{\left (3 b+4 a c^2\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{8 c^4}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{4 c^2}+\frac{\left (3 b+4 a c^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^4}\\ &=\frac{\left (3 b+4 a c^2\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{8 c^4}+\frac{b x^3 \sqrt{-1+c x} \sqrt{1+c x}}{4 c^2}+\frac{\left (3 b+4 a c^2\right ) \cosh ^{-1}(c x)}{8 c^5}\\ \end{align*}

Mathematica [A]  time = 0.0664952, size = 98, normalized size = 1.13 \[ \frac{c x \left (c^2 x^2-1\right ) \left (4 a c^2+b \left (2 c^2 x^2+3\right )\right )+\sqrt{c^2 x^2-1} \left (4 a c^2+3 b\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{8 c^5 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.018, size = 147, normalized size = 1.7 \begin{align*}{\frac{{\it csgn} \left ( c \right ) }{8\,{c}^{5}}\sqrt{cx-1}\sqrt{cx+1} \left ( 2\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ){c}^{3}{x}^{3}b+4\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ){c}^{3}xa+3\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) cxb+4\,\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) a{c}^{2}+3\,\ln \left ( \left ( \sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) +cx \right ){\it csgn} \left ( c \right ) \right ) b \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}-1}}}} \end{align*}

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

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Maxima [A]  time = 0.966629, size = 177, normalized size = 2.03 \begin{align*} \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}} \end{align*}

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, 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))/(s
qrt(c^2)*c^4)

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Fricas [A]  time = 1.5332, size = 180, normalized size = 2.07 \begin{align*} \frac{{\left (2 \, b c^{3} x^{3} +{\left (4 \, a c^{3} + 3 \, b c\right )} x\right )} \sqrt{c x + 1} \sqrt{c x - 1} -{\left (4 \, a c^{2} + 3 \, b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{8 \, c^{5}} \end{align*}

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, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 40.417, size = 212, normalized size = 2.44 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.22821, size = 151, normalized size = 1.74 \begin{align*} -\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 )} \log \left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{114688 \, c} \end{align*}

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, 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)*log(abs(-sqrt(c*x + 1) + sqrt(c*x - 1))))/c