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

Optimal. Leaf size=67 \[ \frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left (2 a d^2+b c^2\right )}{c^4 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

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

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Rubi [A]  time = 0.239412, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ \frac{a}{c^2 x \sqrt{d x-c} \sqrt{c+d x}}-\frac{x \left (2 a d^2+b c^2\right )}{c^4 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.3995, size = 56, normalized size = 0.84 \[ \frac{a}{c^{2} x \sqrt{- c + d x} \sqrt{c + d x}} - \frac{x \left (2 a d^{2} + b c^{2}\right )}{c^{4} \sqrt{- c + d x} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a/(c**2*x*sqrt(-c + d*x)*sqrt(c + d*x)) - x*(2*a*d**2 + b*c**2)/(c**4*sqrt(-c +
d*x)*sqrt(c + d*x))

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Mathematica [A]  time = 0.0849212, size = 51, normalized size = 0.76 \[ \frac{a \left (c^2-2 d^2 x^2\right )-b c^2 x^2}{c^4 x \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.007, size = 48, normalized size = 0.7 \[{\frac{-2\,a{d}^{2}{x}^{2}-b{c}^{2}{x}^{2}+a{c}^{2}}{x{c}^{4}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-2*a*d^2*x^2-b*c^2*x^2+a*c^2)/(d*x+c)^(1/2)/x/c^4/(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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237032, size = 117, normalized size = 1.75 \[ \frac{b d x^{2} - \sqrt{d x + c} \sqrt{d x - c} b x + a d}{2 \, d^{4} x^{4} - 2 \, c^{2} d^{2} x^{2} -{\left (2 \, d^{3} x^{3} - c^{2} d x\right )} \sqrt{d x + c} \sqrt{d x - c}} \]

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^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 152.285, size = 165, normalized size = 2.46 \[ a \left (- \frac{d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{7}{4}, \frac{9}{4}, 1 & \frac{3}{2}, \frac{5}{2}, 3 \\\frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 3 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{4}} + \frac{i d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 1 & \\\frac{5}{4}, \frac{7}{4} & \frac{1}{2}, 1, 2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{4}}\right ) + b \left (- \frac{{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{3}{4}, \frac{5}{4}, 1 & \frac{1}{2}, \frac{3}{2}, 2 \\\frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d} + \frac{i{G_{6, 6}^{2, 6}\left (\begin{matrix} - \frac{1}{2}, 0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1 & \\\frac{1}{4}, \frac{3}{4} & - \frac{1}{2}, 0, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{2 \pi ^{\frac{3}{2}} c^{2} d}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277904, size = 296, normalized size = 4.42 \[ -\frac{{\left (b c^{2} + a d^{2}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{4} d} - \frac{2 \,{\left (b c^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + a d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, a c d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 4 \, b c^{4} + 12 \, a c^{2} d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} + 2 \, c{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 8 \, c^{3}\right )} c^{3} d} \]

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^2),x, algorithm="giac")

[Out]

-1/2*(b*c^2 + a*d^2)*sqrt(d*x + c)/(sqrt(d*x - c)*c^4*d) - 2*(b*c^2*(sqrt(d*x +
c) - sqrt(d*x - c))^4 + a*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*a*c*d^2*(sqr
t(d*x + c) - sqrt(d*x - c))^2 + 4*b*c^4 + 12*a*c^2*d^2)/(((sqrt(d*x + c) - sqrt(
d*x - c))^6 + 2*c*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2*(sqrt(d*x + c) - sqr
t(d*x - c))^2 + 8*c^3)*c^3*d)