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

Optimal. Leaf size=87 \[ \frac{\sqrt{d x-1} \sqrt{d x+1} \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4}+\frac{b \cosh ^{-1}(d x)}{2 d^3}+\frac{c x^2 \sqrt{d x-1} \sqrt{d x+1}}{3 d^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.278479, antiderivative size = 151, normalized size of antiderivative = 1.74, number of steps used = 5, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{\left (1-d^2 x^2\right ) \left (2 \left (3 a d^2+2 c\right )+3 b d^2 x\right )}{6 d^4 \sqrt{d x-1} \sqrt{d x+1}}+\frac{b \sqrt{d^2 x^2-1} \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{2 d^3 \sqrt{d x-1} \sqrt{d x+1}}-\frac{c x^2 \left (1-d^2 x^2\right )}{3 d^2 \sqrt{d x-1} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(c*x^2*(1 - d^2*x^2))/(3*d^2*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) - ((2*(2*c + 3*a*d^2
) + 3*b*d^2*x)*(1 - d^2*x^2))/(6*d^4*Sqrt[-1 + d*x]*Sqrt[1 + d*x]) + (b*Sqrt[-1
+ d^2*x^2]*ArcTanh[(d*x)/Sqrt[-1 + d^2*x^2]])/(2*d^3*Sqrt[-1 + d*x]*Sqrt[1 + d*x
])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.6739, size = 119, normalized size = 1.37 \[ \frac{b \sqrt{d x - 1} \sqrt{d x + 1} \operatorname{atanh}{\left (\frac{d x}{\sqrt{d^{2} x^{2} - 1}} \right )}}{2 d^{3} \sqrt{d^{2} x^{2} - 1}} + \frac{c x^{2} \sqrt{d x - 1} \sqrt{d x + 1}}{3 d^{2}} + \frac{\sqrt{d x - 1} \sqrt{d x + 1} \left (6 a d^{2} + 3 b d^{2} x + 4 c\right )}{6 d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*sqrt(d*x - 1)*sqrt(d*x + 1)*atanh(d*x/sqrt(d**2*x**2 - 1))/(2*d**3*sqrt(d**2*x
**2 - 1)) + c*x**2*sqrt(d*x - 1)*sqrt(d*x + 1)/(3*d**2) + sqrt(d*x - 1)*sqrt(d*x
+ 1)*(6*a*d**2 + 3*b*d**2*x + 4*c)/(6*d**4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.12924, size = 80, normalized size = 0.92 \[ \frac{\sqrt{d x-1} \sqrt{d x+1} \left (3 d^2 (2 a+b x)+2 c \left (d^2 x^2+2\right )\right )+3 b d \log \left (d x+\sqrt{d x-1} \sqrt{d x+1}\right )}{6 d^4} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[-1 + d*x]*Sqrt[1 + d*x]*(3*d^2*(2*a + b*x) + 2*c*(2 + d^2*x^2)) + 3*b*d*Lo
g[d*x + Sqrt[-1 + d*x]*Sqrt[1 + d*x]])/(6*d^4)

_______________________________________________________________________________________

Maple [C]  time = 0., size = 137, normalized size = 1.6 \[{\frac{{\it csgn} \left ( d \right ) }{6\,{d}^{4}}\sqrt{dx-1}\sqrt{dx+1} \left ( 2\,{\it csgn} \left ( d \right ){x}^{2}c{d}^{2}\sqrt{{d}^{2}{x}^{2}-1}+3\,{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}xb{d}^{2}+6\,{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}a{d}^{2}+4\,{\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}c+3\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) bd \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(d*x-1)^(1/2)*(d*x+1)^(1/2)*(2*csgn(d)*x^2*c*d^2*(d^2*x^2-1)^(1/2)+3*csgn(d)
*(d^2*x^2-1)^(1/2)*x*b*d^2+6*csgn(d)*(d^2*x^2-1)^(1/2)*a*d^2+4*csgn(d)*(d^2*x^2-
1)^(1/2)*c+3*ln((csgn(d)*(d^2*x^2-1)^(1/2)+d*x)*csgn(d))*b*d)*csgn(d)/d^4/(d^2*x
^2-1)^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.35501, size = 147, normalized size = 1.69 \[ \frac{\sqrt{d^{2} x^{2} - 1} c x^{2}}{3 \, d^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} b x}{2 \, d^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} a}{d^{2}} + \frac{b \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} + \frac{2 \, \sqrt{d^{2} x^{2} - 1} c}{3 \, d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.236631, size = 370, normalized size = 4.25 \[ -\frac{8 \, c d^{6} x^{6} + 12 \, b d^{6} x^{5} - 15 \, b d^{4} x^{3} + 6 \,{\left (4 \, a d^{6} + c d^{4}\right )} x^{4} + 3 \, b d^{2} x + 6 \, a d^{2} - 6 \,{\left (5 \, a d^{4} + 3 \, c d^{2}\right )} x^{2} -{\left (8 \, c d^{5} x^{5} + 12 \, b d^{5} x^{4} - 9 \, b d^{3} x^{2} + 2 \,{\left (12 \, a d^{5} + 5 \, c d^{3}\right )} x^{3} - 6 \,{\left (3 \, a d^{3} + 2 \, c d\right )} x\right )} \sqrt{d x + 1} \sqrt{d x - 1} + 3 \,{\left (4 \, b d^{4} x^{3} - 3 \, b d^{2} x -{\left (4 \, b d^{3} x^{2} - b d\right )} \sqrt{d x + 1} \sqrt{d x - 1}\right )} \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right ) + 4 \, c}{6 \,{\left (4 \, d^{7} x^{3} - 3 \, d^{5} x -{\left (4 \, d^{6} x^{2} - d^{4}\right )} \sqrt{d x + 1} \sqrt{d x - 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(8*c*d^6*x^6 + 12*b*d^6*x^5 - 15*b*d^4*x^3 + 6*(4*a*d^6 + c*d^4)*x^4 + 3*b*
d^2*x + 6*a*d^2 - 6*(5*a*d^4 + 3*c*d^2)*x^2 - (8*c*d^5*x^5 + 12*b*d^5*x^4 - 9*b*
d^3*x^2 + 2*(12*a*d^5 + 5*c*d^3)*x^3 - 6*(3*a*d^3 + 2*c*d)*x)*sqrt(d*x + 1)*sqrt
(d*x - 1) + 3*(4*b*d^4*x^3 - 3*b*d^2*x - (4*b*d^3*x^2 - b*d)*sqrt(d*x + 1)*sqrt(
d*x - 1))*log(-d*x + sqrt(d*x + 1)*sqrt(d*x - 1)) + 4*c)/(4*d^7*x^3 - 3*d^5*x -
(4*d^6*x^2 - d^4)*sqrt(d*x + 1)*sqrt(d*x - 1))

_______________________________________________________________________________________

Sympy [A]  time = 107.638, size = 308, normalized size = 3.54 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} + \frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{5}{4}, - \frac{3}{4} & -1, -1, - \frac{1}{2}, 1 \\- \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} + \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} -2, - \frac{7}{4}, - \frac{3}{2}, - \frac{5}{4}, -1, 1 & \\- \frac{7}{4}, - \frac{5}{4} & -2, - \frac{3}{2}, - \frac{3}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(c*x**2+b*x+a)/(d*x-1)**(1/2)/(d*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/(
d**2*x**2))/(4*pi**(3/2)*d**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)/(d**2*x**2))/(4*pi**(3/2
)*d**2) + b*meijerg(((-3/4, -1/4), (-1/2, -1/2, 0, 1)), ((-1, -3/4, -1/2, -1/4,
0, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d**3) - I*b*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)/(d**2*x*
*2))/(4*pi**(3/2)*d**3) + c*meijerg(((-5/4, -3/4), (-1, -1, -1/2, 1)), ((-3/2, -
5/4, -1, -3/4, -1/2, 0), ()), 1/(d**2*x**2))/(4*pi**(3/2)*d**4) + I*c*meijerg(((
-2, -7/4, -3/2, -5/4, -1, 1), ()), ((-7/4, -5/4), (-2, -3/2, -3/2, 0)), exp_pola
r(2*I*pi)/(d**2*x**2))/(4*pi**(3/2)*d**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.281473, size = 130, normalized size = 1.49 \[ -\frac{6 \, b d^{10}{\rm ln}\left ({\left | -\sqrt{d x + 1} + \sqrt{d x - 1} \right |}\right ) -{\left (6 \, a d^{11} - 3 \, b d^{10} + 6 \, c d^{9} +{\left (2 \,{\left (d x + 1\right )} c d^{9} + 3 \, b d^{10} - 4 \, c d^{9}\right )}{\left (d x + 1\right )}\right )} \sqrt{d x + 1} \sqrt{d x - 1}}{3840 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3840*(6*b*d^10*ln(abs(-sqrt(d*x + 1) + sqrt(d*x - 1))) - (6*a*d^11 - 3*b*d^10
+ 6*c*d^9 + (2*(d*x + 1)*c*d^9 + 3*b*d^10 - 4*c*d^9)*(d*x + 1))*sqrt(d*x + 1)*s
qrt(d*x - 1))/d