3.266 \(\int \frac{a+b x^2}{x^3 \sqrt{-c+d x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=76 \[ \frac{\left (a d^2+2 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{2 c^3}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{2 c^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.260603, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097 \[ \frac{\left (a d^2+2 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{2 c^3}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{2 c^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/(x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 13.3616, size = 63, normalized size = 0.83 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{2 c^{2} x^{2}} + \frac{\left (a d^{2} + 2 b c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{2 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**3/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

a*sqrt(-c + d*x)*sqrt(c + d*x)/(2*c**2*x**2) + (a*d**2 + 2*b*c**2)*atan(sqrt(-c
+ d*x)*sqrt(c + d*x)/c)/(2*c**3)

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Mathematica [C]  time = 0.139132, size = 103, normalized size = 1.36 \[ \frac{a c \sqrt{d x-c} \sqrt{c+d x}-i x^2 \left (a d^2+2 b c^2\right ) \log \left (\frac{4 c^2 \left (\sqrt{d x-c} \sqrt{c+d x}-i c\right )}{x \left (a d^2+2 b c^2\right )}\right )}{2 c^3 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/(x^3*Sqrt[-c + d*x]*Sqrt[c + d*x]),x]

[Out]

(a*c*Sqrt[-c + d*x]*Sqrt[c + d*x] - I*(2*b*c^2 + a*d^2)*x^2*Log[(4*c^2*((-I)*c +
Sqrt[-c + d*x]*Sqrt[c + d*x]))/((2*b*c^2 + a*d^2)*x)])/(2*c^3*x^2)

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Maple [B]  time = 0.029, size = 158, normalized size = 2.1 \[ -{\frac{1}{2\,{c}^{2}{x}^{2}}\sqrt{dx-c}\sqrt{dx+c} \left ( \ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}a{d}^{2}+2\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}b{c}^{2}-a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}\sqrt{-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^3/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(sqrt(d*x + c)*sqrt(d*x - c)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.245142, size = 279, normalized size = 3.67 \[ -\frac{2 \, a c d^{3} x^{3} - 2 \, a c^{3} d x -{\left (2 \, a c d^{2} x^{2} - a c^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} + 2 \,{\left (2 \,{\left (2 \, b c^{2} d + a d^{3}\right )} \sqrt{d x + c} \sqrt{d x - c} x^{3} - 2 \,{\left (2 \, b c^{2} d^{2} + a d^{4}\right )} x^{4} +{\left (2 \, b c^{4} + a c^{2} d^{2}\right )} x^{2}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{2 \,{\left (2 \, c^{3} d^{2} x^{4} - 2 \, \sqrt{d x + c} \sqrt{d x - c} c^{3} d x^{3} - c^{5} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(sqrt(d*x + c)*sqrt(d*x - c)*x^3),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 68.5575, size = 162, normalized size = 2.13 \[ - \frac{a d^{2}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & 2, 2, \frac{5}{2} \\\frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} + \frac{i a d^{2}{G_{6, 6}^{2, 6}\left (\begin{matrix} 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, 1 & \\\frac{5}{4}, \frac{7}{4} & 1, \frac{3}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{3}} - \frac{b{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{i b{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**3/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

-a*d**2*meijerg(((7/4, 9/4, 1), (2, 2, 5/2)), ((3/2, 7/4, 2, 9/4, 5/2), (0,)), c
**2/(d**2*x**2))/(4*pi**(3/2)*c**3) + I*a*d**2*meijerg(((1, 5/4, 3/2, 7/4, 2, 1)
, ()), ((5/4, 7/4), (1, 3/2, 3/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*pi
**(3/2)*c**3) - b*meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2)
, (0,)), c**2/(d**2*x**2))/(4*pi**(3/2)*c) + I*b*meijerg(((0, 1/4, 1/2, 3/4, 1,
1), ()), ((1/4, 3/4), (0, 1/2, 1/2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*
pi**(3/2)*c)

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GIAC/XCAS [A]  time = 0.229743, size = 190, normalized size = 2.5 \[ -\frac{\frac{{\left (2 \, b c^{2} d + a d^{3}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} + \frac{2 \,{\left (a d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 4 \, a c^{2} d^{3}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{2} c^{2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)/(sqrt(d*x + c)*sqrt(d*x - c)*x^3),x, algorithm="giac")

[Out]

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