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

Optimal. Leaf size=65 \[ -\frac{\frac{a}{c^2}+\frac{b}{d^2}}{\sqrt{d x-c} \sqrt{c+d x}}-\frac{a \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.266444, antiderivative size = 65, 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 \[ -\frac{\frac{a}{c^2}+\frac{b}{d^2}}{\sqrt{d x-c} \sqrt{c+d x}}-\frac{a \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{c^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.2202, size = 53, normalized size = 0.82 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{c^{3}} - \frac{\frac{a}{c^{2}} + \frac{b}{d^{2}}}{\sqrt{- c + d x} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.15873, size = 96, normalized size = 1.48 \[ \frac{c \sqrt{d x-c} \sqrt{c+d x} \left (a d^2+b c^2\right )+a d^2 \left (c^2-d^2 x^2\right ) \tan ^{-1}\left (\frac{c}{\sqrt{d x-c} \sqrt{c+d x}}\right )}{c^3 d^2 (c-d x) (c+d x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.034, size = 188, normalized size = 2.9 \[{\frac{1}{{c}^{2}{d}^{2}} \left ( \ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}a{d}^{4}-a{c}^{2}\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){d}^{2}-a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{d}^{2}\sqrt{-{c}^{2}}-b{c}^{2}\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/c^2*(ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^2*a*d^4-a*c^2*ln(-2*(c^
2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*d^2-a*(d^2*x^2-c^2)^(1/2)*d^2*(-c^2)^(1/2
)-b*c^2*(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/(-c^2)^(1/2)/(d^2*x^2-c^2)^(1/2)/d^2/(
d*x+c)^(1/2)/(d*x-c)^(1/2)

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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)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.242387, size = 223, normalized size = 3.43 \[ -\frac{{\left (b c^{3} + a c d^{2}\right )} \sqrt{d x + c} \sqrt{d x - c} -{\left (b c^{3} d + a c d^{3}\right )} x + 2 \,{\left (a d^{4} x^{2} - \sqrt{d x + c} \sqrt{d x - c} a d^{3} x - a c^{2} d^{2}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{c^{3} d^{4} x^{2} - \sqrt{d x + c} \sqrt{d x - c} c^{3} d^{3} x - c^{5} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 93.7092, size = 172, normalized size = 2.65 \[ a \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & 1, 2, \frac{5}{2} \\\frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2, \frac{5}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{3}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} 0, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, 1 & \\\frac{3}{4}, \frac{5}{4} & 0, \frac{1}{2}, \frac{3}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{3}}\right ) + b \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4}, 1 & 0, 1, \frac{3}{2} \\\frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{3}{2} & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c d^{2}} - \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} -1, - \frac{1}{2}, - \frac{1}{4}, 0, \frac{1}{4}, 1 & \\- \frac{1}{4}, \frac{1}{4} & -1, - \frac{1}{2}, \frac{1}{2}, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c d^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.252711, size = 155, normalized size = 2.38 \[ \frac{2 \, a \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{3}} - \frac{{\left (b c^{2} + a d^{2}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{3} d^{2}} + \frac{2 \,{\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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