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

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

[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 = 0.0848769, 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, Rules used = {460, 100, 12, 90, 52} \[ \frac{x^3 \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{24 c^4}+\frac{x \sqrt{c x-1} \sqrt{c x+1} \left (6 a c^2+5 b\right )}{16 c^6}+\frac{\left (6 a c^2+5 b\right ) \cosh ^{-1}(c x)}{16 c^7}+\frac{b x^5 \sqrt{c x-1} \sqrt{c x+1}}{6 c^2} \]

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)

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 100

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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^4 \left (a+b x^2\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx &=\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}-\frac{1}{6} \left (-6 a-\frac{5 b}{c^2}\right ) \int \frac{x^4}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx\\ &=\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \int \frac{3 x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{24 c^4}\\ &=\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \int \frac{x^2}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{8 c^4}\\ &=\frac{\left (5 b+6 a c^2\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{16 c^6}+\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \int \frac{1}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{16 c^6}\\ &=\frac{\left (5 b+6 a c^2\right ) x \sqrt{-1+c x} \sqrt{1+c x}}{16 c^6}+\frac{\left (5 b+6 a c^2\right ) x^3 \sqrt{-1+c x} \sqrt{1+c x}}{24 c^4}+\frac{b x^5 \sqrt{-1+c x} \sqrt{1+c x}}{6 c^2}+\frac{\left (5 b+6 a c^2\right ) \cosh ^{-1}(c x)}{16 c^7}\\ \end{align*}

Mathematica [A]  time = 0.103118, size = 117, normalized size = 0.94 \[ \frac{c x \left (c^2 x^2-1\right ) \left (6 a c^2 \left (2 c^2 x^2+3\right )+b \left (8 c^4 x^4+10 c^2 x^2+15\right )\right )+3 \sqrt{c^2 x^2-1} \left (6 a c^2+5 b\right ) \tanh ^{-1}\left (\frac{c x}{\sqrt{c^2 x^2-1}}\right )}{48 c^7 \sqrt{c x-1} \sqrt{c x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x*(-1 + c^2*x^2)*(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)*Sqrt[-1 +
c^2*x^2]*ArcTanh[(c*x)/Sqrt[-1 + c^2*x^2]])/(48*c^7*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

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Maple [C]  time = 0.077, size = 191, normalized size = 1.5 \begin{align*}{\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\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ){c}^{3}{x}^{3}b+18\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ){c}^{3}xa+15\,\sqrt{{c}^{2}{x}^{2}-1}{\it csgn} \left ( c \right ) cxb+18\,\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}+15\,\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^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*(c^2*x^2-1)^(1/2)*csgn(c)*c^3*x^3*b+18*(c^2*x^2-1)^(1/2)*csgn(c)*c^3*x*a+15*(c^2*x^2-1)^(1/2)*csgn(c)*c*x*
b+18*ln(((c^2*x^2-1)^(1/2)*csgn(c)+c*x)*csgn(c))*a*c^2+15*ln(((c^2*x^2-1)^(1/2)*csgn(c)+c*x)*csgn(c))*b)*csgn(
c)/c^7/(c^2*x^2-1)^(1/2)

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Maxima [A]  time = 0.961023, size = 231, normalized size = 1.85 \begin{align*} \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}} \end{align*}

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, 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*sqr
t(c^2*x^2 - 1)*a*x/c^4 + 3/8*a*log(2*c^2*x + 2*sqrt(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 + 5/16*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 = 1.54684, size = 224, normalized size = 1.79 \begin{align*} \frac{{\left (8 \, b c^{5} x^{5} + 2 \,{\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 (6 \, a c^{2} + 5 \, b\right )} \log \left (-c x + \sqrt{c x + 1} \sqrt{c x - 1}\right )}{48 \, c^{7}} \end{align*}

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, algorithm="fricas")

[Out]

1/48*((8*b*c^5*x^5 + 2*(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*(6*a*c
^2 + 5*b)*log(-c*x + sqrt(c*x + 1)*sqrt(c*x - 1)))/c^7

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Sympy [C]  time = 59.9497, size = 216, normalized size = 1.73 \begin{align*} \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}} \end{align*}

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 [A]  time = 1.23898, size = 205, normalized size = 1.64 \begin{align*} -\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 )} \log \left ({\left | -\sqrt{c x + 1} + \sqrt{c x - 1} \right |}\right )}{34603008 \, c} \end{align*}

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, 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^3
6 + 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)*log(abs(-sqrt(c*x + 1) + sqrt(c*x - 1))))/c