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

Optimal. Leaf size=52 \[ \frac{\left (2 a d^2+c\right ) \cosh ^{-1}(d x)}{2 d^3}+\frac{\sqrt{d x-1} \sqrt{d x+1} (2 b+c x)}{2 d^2} \]

[Out]

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

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Rubi [B]  time = 0.181206, antiderivative size = 135, normalized size of antiderivative = 2.6, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{d^2 x^2-1} \left (2 a d^2+c\right ) \tanh ^{-1}\left (\frac{d x}{\sqrt{d^2 x^2-1}}\right )}{2 d^3 \sqrt{d x-1} \sqrt{d x+1}}-\frac{b \left (1-d^2 x^2\right )}{d^2 \sqrt{d x-1} \sqrt{d x+1}}-\frac{c x \left (1-d^2 x^2\right )}{2 d^2 \sqrt{d x-1} \sqrt{d x+1}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.0979, size = 87, normalized size = 1.67 \[ \frac{\left (2 b + c x\right ) \sqrt{d x - 1} \sqrt{d x + 1}}{2 d^{2}} + \frac{\left (2 a d^{2} + c\right ) \sqrt{d x - 1} \sqrt{d x + 1} \operatorname{atanh}{\left (\frac{d x}{\sqrt{d^{2} x^{2} - 1}} \right )}}{2 d^{3} \sqrt{d^{2} x^{2} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*b + c*x)*sqrt(d*x - 1)*sqrt(d*x + 1)/(2*d**2) + (2*a*d**2 + c)*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))

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Mathematica [A]  time = 0.0868635, size = 68, normalized size = 1.31 \[ \frac{\left (2 a d^2+c\right ) \log \left (d x+\sqrt{d x-1} \sqrt{d x+1}\right )+d \sqrt{d x-1} \sqrt{d x+1} (2 b+c x)}{2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.022, size = 120, normalized size = 2.3 \[{\frac{{\it csgn} \left ( d \right ) }{2\,{d}^{3}}\sqrt{dx-1}\sqrt{dx+1} \left ( cx\sqrt{{d}^{2}{x}^{2}-1}{\it csgn} \left ( d \right ) d+2\,\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) a{d}^{2}+2\,\sqrt{{d}^{2}{x}^{2}-1}b{\it csgn} \left ( d \right ) d+c\ln \left ( \left ({\it csgn} \left ( d \right ) \sqrt{{d}^{2}{x}^{2}-1}+dx \right ){\it csgn} \left ( d \right ) \right ) \right ){\frac{1}{\sqrt{{d}^{2}{x}^{2}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.32437, size = 142, normalized size = 2.73 \[ \frac{a \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{\sqrt{d^{2}}} + \frac{\sqrt{d^{2} x^{2} - 1} c x}{2 \, d^{2}} + \frac{\sqrt{d^{2} x^{2} - 1} b}{d^{2}} + \frac{c \log \left (2 \, d^{2} x + 2 \, \sqrt{d^{2} x^{2} - 1} \sqrt{d^{2}}\right )}{2 \, \sqrt{d^{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.222211, size = 259, normalized size = 4.98 \[ -\frac{2 \, c d^{4} x^{4} + 4 \, b d^{4} x^{3} - 2 \, c d^{2} x^{2} - 4 \, b d^{2} x -{\left (2 \, c d^{3} x^{3} + 4 \, b d^{3} x^{2} - c d x - 2 \, b d\right )} \sqrt{d x + 1} \sqrt{d x - 1} -{\left (2 \, a d^{2} + 2 \,{\left (2 \, a d^{3} + c d\right )} \sqrt{d x + 1} \sqrt{d x - 1} x - 2 \,{\left (2 \, a d^{4} + c d^{2}\right )} x^{2} + c\right )} \log \left (-d x + \sqrt{d x + 1} \sqrt{d x - 1}\right )}{2 \,{\left (2 \, d^{5} x^{2} - 2 \, \sqrt{d x + 1} \sqrt{d x - 1} d^{4} x - d^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 54.6317, size = 277, normalized size = 5.33 \[ \frac{a{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{b{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} & - \frac{1}{2}, - \frac{1}{2}, 0, 1 \\-1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} - \frac{i c{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{3}{2}, - \frac{5}{4}, -1, - \frac{3}{4}, - \frac{1}{2}, 1 & \\- \frac{5}{4}, - \frac{3}{4} & - \frac{3}{2}, -1, -1, 0 \end{matrix} \middle |{\frac{e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.257321, size = 104, normalized size = 2. \[ \frac{{\left ({\left (d x + 1\right )} c d^{4} + 2 \, b d^{5} - c d^{4}\right )} \sqrt{d x + 1} \sqrt{d x - 1} - 2 \,{\left (2 \, a d^{6} + c d^{4}\right )}{\rm ln}\left ({\left | -\sqrt{d x + 1} + \sqrt{d x - 1} \right |}\right )}{192 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*(((d*x + 1)*c*d^4 + 2*b*d^5 - c*d^4)*sqrt(d*x + 1)*sqrt(d*x - 1) - 2*(2*a*
d^6 + c*d^4)*ln(abs(-sqrt(d*x + 1) + sqrt(d*x - 1))))/d