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

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

[Out]

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

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Rubi [A]  time = 0.321695, 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 \[ -\frac{a \sqrt{1-d^2 x^2}}{x}-b \tanh ^{-1}\left (\sqrt{1-d^2 x^2}\right )+\frac{c \sin ^{-1}(d x)}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.2794, size = 37, normalized size = 0.77 \[ - \frac{a \sqrt{- d^{2} x^{2} + 1}}{x} - b \operatorname{atanh}{\left (\sqrt{- d^{2} x^{2} + 1} \right )} + \frac{c \operatorname{asin}{\left (d x \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0851936, size = 54, normalized size = 1.12 \[ -\frac{a \sqrt{1-d^2 x^2}}{x}-b \log \left (\sqrt{1-d^2 x^2}+1\right )+b \log (x)+\frac{c \sin ^{-1}(d x)}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0., size = 97, normalized size = 2. \[{\frac{{\it csgn} \left ( d \right ) }{dx} \left ( -bx{\it csgn} \left ( d \right ) d{\it Artanh} \left ({\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}} \right ) -a\sqrt{-{d}^{2}{x}^{2}+1}{\it csgn} \left ( d \right ) d+c\arctan \left ({{\it csgn} \left ( d \right ) dx{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \right ) x \right ) \sqrt{-dx+1}\sqrt{dx+1}{\frac{1}{\sqrt{-{d}^{2}{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.49843, size = 89, normalized size = 1.85 \[ -b \log \left (\frac{2 \, \sqrt{-d^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{c \arcsin \left (\frac{d^{2} x}{\sqrt{d^{2}}}\right )}{\sqrt{d^{2}}} - \frac{\sqrt{-d^{2} x^{2} + 1} a}{x} \]

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^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.236247, size = 212, normalized size = 4.42 \[ \frac{a d^{3} x^{2} + \sqrt{d x + 1} \sqrt{-d x + 1} a d - a d - 2 \,{\left (\sqrt{d x + 1} \sqrt{-d x + 1} c x - c x\right )} \arctan \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{d x}\right ) +{\left (\sqrt{d x + 1} \sqrt{-d x + 1} b d x - b d x\right )} \log \left (\frac{\sqrt{d x + 1} \sqrt{-d x + 1} - 1}{x}\right )}{\sqrt{d x + 1} \sqrt{-d x + 1} d x - d x} \]

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^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 65.6606, size = 221, normalized size = 4.6 \[ \frac{i a d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{a d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} + \frac{i b{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{b{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}}} - \frac{i c{G_{6, 6}^{6, 2}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} & \frac{1}{2}, \frac{1}{2}, 1, 1 \\0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 0 & \end{matrix} \middle |{\frac{1}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} + \frac{c{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 1 & \\- \frac{1}{4}, \frac{1}{4} & - \frac{1}{2}, 0, 0, 0 \end{matrix} \middle |{\frac{e^{- 2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

I*a*d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), 1/(
d**2*x**2))/(4*pi**(3/2)) + a*d*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4,
5/4), (1/2, 1, 1, 0)), exp_polar(-2*I*pi)/(d**2*x**2))/(4*pi**(3/2)) + I*b*meij
erg(((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)) - b*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*c*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) + c*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)

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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^2),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError