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

Optimal. Leaf size=56 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{c}+\frac{b \sqrt{d x-c} \sqrt{c+d x}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.241805, antiderivative size = 56, 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 \[ \frac{a \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{c}+\frac{b \sqrt{d x-c} \sqrt{c+d x}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8825, size = 44, normalized size = 0.79 \[ \frac{a \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{c} + \frac{b \sqrt{- c + d x} \sqrt{c + d x}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0996939, size = 55, normalized size = 0.98 \[ \frac{b \sqrt{d x-c} \sqrt{c+d x}}{d^2}-\frac{a \tan ^{-1}\left (\frac{c}{\sqrt{d x-c} \sqrt{c+d x}}\right )}{c} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.026, size = 108, normalized size = 1.9 \[{\frac{1}{{d}^{2}} \left ( -\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ) a{d}^{2}+b\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ) \sqrt{dx-c}\sqrt{dx+c}{\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*a*d^2+b*(-c^2)^(1/2)*(d^2*x^2-
c^2)^(1/2))*(d*x-c)^(1/2)*(d*x+c)^(1/2)/(d^2*x^2-c^2)^(1/2)/(-c^2)^(1/2)/d^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),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.240987, size = 176, normalized size = 3.14 \[ -\frac{b c d^{2} x^{2} - \sqrt{d x + c} \sqrt{d x - c} b c d x - b c^{3} - 2 \,{\left (a d^{3} x - \sqrt{d x + c} \sqrt{d x - c} a d^{2}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{c d^{3} x - \sqrt{d x + c} \sqrt{d x - c} c d^{2}} \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 44.7243, size = 178, normalized size = 3.18 \[ - \frac{a{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & 1, 1, \frac{3}{2} \\\frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{i a{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, 1 & \\\frac{1}{4}, \frac{3}{4} & 0, \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c} + \frac{b c{G_{6, 6}^{6, 2}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{4} & 0, 0, \frac{1}{2}, 1 \\- \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, \frac{1}{2}, 0 & \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} + \frac{i b c{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{3}{4}, - \frac{1}{2}, - \frac{1}{4}, 0, 1 & \\- \frac{3}{4}, - \frac{1}{4} & -1, - \frac{1}{2}, - \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.220305, size = 74, normalized size = 1.32 \[ -\frac{2 \, a \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c} + \frac{\sqrt{d x + c} \sqrt{d x - c} b}{d^{2}} \]

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),x, algorithm="giac")

[Out]

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