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

Optimal. Leaf size=48 \[ -a \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.346262, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212 \[ -a \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.2141, size = 39, normalized size = 0.81 \[ - a \operatorname{atanh}{\left (\sqrt{- d^{2} x^{2} + 1} \right )} + \frac{b \operatorname{asin}{\left (d x \right )}}{d} - \frac{c \sqrt{- d^{2} x^{2} + 1}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0653181, size = 54, normalized size = 1.12 \[ -a \log \left (\sqrt{1-d^2 x^2}+1\right )+a \log (x)+\frac{b \sin ^{-1}(d x)}{d}-\frac{c \sqrt{1-d^2 x^2}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.027, size = 94, normalized size = 2. \[{\frac{{\it csgn} \left ( d \right ) }{{d}^{2}}\sqrt{-dx+1}\sqrt{dx+1} \left ( -{\it csgn} \left ( d \right ){\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) a{d}^{2}+\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) bd-{\it csgn} \left ( d \right ) \sqrt{-{d}^{2}{x}^{2}+1}c \right ){\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-d*x+1)^(1/2)*(d*x+1)^(1/2)/d^2*(-csgn(d)*arctanh(1/(-d^2*x^2+1)^(1/2))*a*d^2+a
rctan(csgn(d)*d*x/(-d^2*x^2+1)^(1/2))*b*d-csgn(d)*(-d^2*x^2+1)^(1/2)*c)*csgn(d)/
(-d^2*x^2+1)^(1/2)

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Maxima [A]  time = 1.53541, size = 89, normalized size = 1.85 \[ -a \log \left (\frac{2 \, \sqrt{-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{b \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{\sqrt{-d^{2} x^{2} + 1} c}{d^{2}} \]

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),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.235181, size = 171, normalized size = 3.56 \[ \frac{c d x^{2} - 2 \,{\left (\sqrt{d x + 1} \sqrt{-d x + 1} b - b\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right ) +{\left (\sqrt{d x + 1} \sqrt{-d x + 1} a d - a d\right )} \log \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{x}\right )}{\sqrt{d x + 1} \sqrt{-d x + 1} d - d} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 57.6914, size = 245, normalized size = 5.1 \[ \frac{i a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i b{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} - \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*a*meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2), (0,)), 1/(d*
*2*x**2))/(4*pi**(3/2)) - a*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4, 3/4),
(0, 1/2, 1/2, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) - I*b*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) + b*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) - I*c*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) - c*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)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

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),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError