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

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]

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

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Rubi [A]  time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, 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 12

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

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]

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 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 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]

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*}

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Mathematica [A]  time = 0.14, size = 117, normalized size = 0.94 \[ \frac {3 \sqrt {c^2 x^2-1} \left (6 a c^2+5 b\right ) \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )+c x \left (c^2 x^2-1\right ) \left (6 a c^2 \left (2 c^2 x^2+3\right )+b \left (8 c^4 x^4+10 c^2 x^2+15\right )\right )}{48 c^7 \sqrt {c x-1} \sqrt {c x+1}} \]

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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fricas [A]  time = 1.25, size = 96, normalized size = 0.77 \[ \frac {{\left (8 \, b c^{5} x^{5} + 2 \, {\left (6 \, a c^{5} + 5 \, b c^{3}\right )} x^{3} + 3 \, {\left (6 \, a c^{3} + 5 \, b c\right )} x\right )} \sqrt {c x + 1} \sqrt {c x - 1} - 3 \, {\left (6 \, a c^{2} + 5 \, b\right )} \log \left (-c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{48 \, c^{7}} \]

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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giac [A]  time = 0.23, size = 172, normalized size = 1.38 \[ \frac {{\left ({\left (2 \, {\left ({\left (c x + 1\right )} {\left (4 \, {\left (c x + 1\right )} {\left (\frac {{\left (c x + 1\right )} b}{c^{6}} - \frac {5 \, b}{c^{6}}\right )} + \frac {3 \, {\left (2 \, a c^{38} + 15 \, b c^{36}\right )}}{c^{42}}\right )} - \frac {18 \, a c^{38} + 55 \, b c^{36}}{c^{42}}\right )} {\left (c x + 1\right )} + \frac {54 \, a c^{38} + 85 \, b c^{36}}{c^{42}}\right )} {\left (c x + 1\right )} - \frac {3 \, {\left (10 \, a c^{38} + 11 \, b c^{36}\right )}}{c^{42}}\right )} \sqrt {c x + 1} \sqrt {c x - 1} - \frac {6 \, {\left (6 \, a c^{2} + 5 \, b\right )} \log \left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}{c^{6}}}{48 \, c} \]

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]

1/48*(((2*((c*x + 1)*(4*(c*x + 1)*((c*x + 1)*b/c^6 - 5*b/c^6) + 3*(2*a*c^38 + 15*b*c^36)/c^42) - (18*a*c^38 +
55*b*c^36)/c^42)*(c*x + 1) + (54*a*c^38 + 85*b*c^36)/c^42)*(c*x + 1) - 3*(10*a*c^38 + 11*b*c^36)/c^42)*sqrt(c*
x + 1)*sqrt(c*x - 1) - 6*(6*a*c^2 + 5*b)*log(sqrt(c*x + 1) - sqrt(c*x - 1))/c^6)/c

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maple [C]  time = 0.11, size = 191, normalized size = 1.53 \[ \frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (8 \sqrt {c^{2} x^{2}-1}\, b \,c^{5} x^{5} \mathrm {csgn}\relax (c )+12 \sqrt {c^{2} x^{2}-1}\, a \,c^{5} x^{3} \mathrm {csgn}\relax (c )+10 \sqrt {c^{2} x^{2}-1}\, b \,c^{3} x^{3} \mathrm {csgn}\relax (c )+18 \sqrt {c^{2} x^{2}-1}\, a \,c^{3} x \,\mathrm {csgn}\relax (c )+18 a \,c^{2} \ln \left (\left (c x +\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\relax (c )\right ) \mathrm {csgn}\relax (c )\right )+15 \sqrt {c^{2} x^{2}-1}\, b c x \,\mathrm {csgn}\relax (c )+15 b \ln \left (\left (c x +\sqrt {c^{2} x^{2}-1}\, \mathrm {csgn}\relax (c )\right ) \mathrm {csgn}\relax (c )\right )\right ) \mathrm {csgn}\relax (c )}{48 \sqrt {c^{2} x^{2}-1}\, c^{7}} \]

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 = 0.55, size = 153, normalized size = 1.22 \[ \frac {\sqrt {c^{2} x^{2} - 1} b x^{5}}{6 \, c^{2}} + \frac {\sqrt {c^{2} x^{2} - 1} a x^{3}}{4 \, c^{2}} + \frac {5 \, \sqrt {c^{2} x^{2} - 1} b x^{3}}{24 \, c^{4}} + \frac {3 \, \sqrt {c^{2} x^{2} - 1} a x}{8 \, c^{4}} + \frac {3 \, a \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{8 \, c^{5}} + \frac {5 \, \sqrt {c^{2} x^{2} - 1} b x}{16 \, c^{6}} + \frac {5 \, b \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{16 \, c^{7}} \]

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, 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)*c)/c^5 + 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)*c)/c^7

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mupad [B]  time = 32.63, size = 1154, normalized size = 9.23 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((23*a*((c*x - 1)^(1/2) - 1i)^3)/(2*((c*x + 1)^(1/2) - 1)^3) + (333*a*((c*x - 1)^(1/2) - 1i)^5)/(2*((c*x + 1)^
(1/2) - 1)^5) + (671*a*((c*x - 1)^(1/2) - 1i)^7)/(2*((c*x + 1)^(1/2) - 1)^7) + (671*a*((c*x - 1)^(1/2) - 1i)^9
)/(2*((c*x + 1)^(1/2) - 1)^9) + (333*a*((c*x - 1)^(1/2) - 1i)^11)/(2*((c*x + 1)^(1/2) - 1)^11) + (23*a*((c*x -
1)^(1/2) - 1i)^13)/(2*((c*x + 1)^(1/2) - 1)^13) - (3*a*((c*x - 1)^(1/2) - 1i)^15)/(2*((c*x + 1)^(1/2) - 1)^15
) - (3*a*((c*x - 1)^(1/2) - 1i))/(2*((c*x + 1)^(1/2) - 1)))/(c^5 - (8*c^5*((c*x - 1)^(1/2) - 1i)^2)/((c*x + 1)
^(1/2) - 1)^2 + (28*c^5*((c*x - 1)^(1/2) - 1i)^4)/((c*x + 1)^(1/2) - 1)^4 - (56*c^5*((c*x - 1)^(1/2) - 1i)^6)/
((c*x + 1)^(1/2) - 1)^6 + (70*c^5*((c*x - 1)^(1/2) - 1i)^8)/((c*x + 1)^(1/2) - 1)^8 - (56*c^5*((c*x - 1)^(1/2)
- 1i)^10)/((c*x + 1)^(1/2) - 1)^10 + (28*c^5*((c*x - 1)^(1/2) - 1i)^12)/((c*x + 1)^(1/2) - 1)^12 - (8*c^5*((c
*x - 1)^(1/2) - 1i)^14)/((c*x + 1)^(1/2) - 1)^14 + (c^5*((c*x - 1)^(1/2) - 1i)^16)/((c*x + 1)^(1/2) - 1)^16) -
((311*b*((c*x - 1)^(1/2) - 1i)^5)/(4*((c*x + 1)^(1/2) - 1)^5) - (175*b*((c*x - 1)^(1/2) - 1i)^3)/(12*((c*x +
1)^(1/2) - 1)^3) + (8361*b*((c*x - 1)^(1/2) - 1i)^7)/(4*((c*x + 1)^(1/2) - 1)^7) + (42259*b*((c*x - 1)^(1/2) -
1i)^9)/(6*((c*x + 1)^(1/2) - 1)^9) + (25295*b*((c*x - 1)^(1/2) - 1i)^11)/(2*((c*x + 1)^(1/2) - 1)^11) + (2529
5*b*((c*x - 1)^(1/2) - 1i)^13)/(2*((c*x + 1)^(1/2) - 1)^13) + (42259*b*((c*x - 1)^(1/2) - 1i)^15)/(6*((c*x + 1
)^(1/2) - 1)^15) + (8361*b*((c*x - 1)^(1/2) - 1i)^17)/(4*((c*x + 1)^(1/2) - 1)^17) + (311*b*((c*x - 1)^(1/2) -
1i)^19)/(4*((c*x + 1)^(1/2) - 1)^19) - (175*b*((c*x - 1)^(1/2) - 1i)^21)/(12*((c*x + 1)^(1/2) - 1)^21) + (5*b
*((c*x - 1)^(1/2) - 1i)^23)/(4*((c*x + 1)^(1/2) - 1)^23) + (5*b*((c*x - 1)^(1/2) - 1i))/(4*((c*x + 1)^(1/2) -
1)))/(c^7 - (12*c^7*((c*x - 1)^(1/2) - 1i)^2)/((c*x + 1)^(1/2) - 1)^2 + (66*c^7*((c*x - 1)^(1/2) - 1i)^4)/((c*
x + 1)^(1/2) - 1)^4 - (220*c^7*((c*x - 1)^(1/2) - 1i)^6)/((c*x + 1)^(1/2) - 1)^6 + (495*c^7*((c*x - 1)^(1/2) -
1i)^8)/((c*x + 1)^(1/2) - 1)^8 - (792*c^7*((c*x - 1)^(1/2) - 1i)^10)/((c*x + 1)^(1/2) - 1)^10 + (924*c^7*((c*
x - 1)^(1/2) - 1i)^12)/((c*x + 1)^(1/2) - 1)^12 - (792*c^7*((c*x - 1)^(1/2) - 1i)^14)/((c*x + 1)^(1/2) - 1)^14
+ (495*c^7*((c*x - 1)^(1/2) - 1i)^16)/((c*x + 1)^(1/2) - 1)^16 - (220*c^7*((c*x - 1)^(1/2) - 1i)^18)/((c*x +
1)^(1/2) - 1)^18 + (66*c^7*((c*x - 1)^(1/2) - 1i)^20)/((c*x + 1)^(1/2) - 1)^20 - (12*c^7*((c*x - 1)^(1/2) - 1i
)^22)/((c*x + 1)^(1/2) - 1)^22 + (c^7*((c*x - 1)^(1/2) - 1i)^24)/((c*x + 1)^(1/2) - 1)^24) + (3*a*atanh(((c*x
- 1)^(1/2) - 1i)/((c*x + 1)^(1/2) - 1)))/(2*c^5) + (5*b*atanh(((c*x - 1)^(1/2) - 1i)/((c*x + 1)^(1/2) - 1)))/(
4*c^7)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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