3.267 \(\int \frac{a+b x^2}{x^4 \sqrt{-c+d x} \sqrt{c+d x}} \, dx\)

Optimal. Leaf size=75 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (2 a d^2+3 b c^2\right )}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.254355, antiderivative size = 75, 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 \[ \frac{\sqrt{d x-c} \sqrt{c+d x} \left (2 a d^2+3 b c^2\right )}{3 c^4 x}+\frac{a \sqrt{d x-c} \sqrt{c+d x}}{3 c^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.9889, size = 63, normalized size = 0.84 \[ \frac{a \sqrt{- c + d x} \sqrt{c + d x}}{3 c^{2} x^{3}} + \frac{\sqrt{- c + d x} \sqrt{c + d x} \left (2 a d^{2} + 3 b c^{2}\right )}{3 c^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)/x**4/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0741589, size = 56, normalized size = 0.75 \[ \sqrt{d x-c} \sqrt{c+d x} \left (\frac{2 a d^2+3 b c^2}{3 c^4 x}+\frac{a}{3 c^2 x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 49, normalized size = 0.7 \[{\frac{2\,a{d}^{2}{x}^{2}+3\,b{c}^{2}{x}^{2}+a{c}^{2}}{3\,{x}^{3}{c}^{4}}\sqrt{dx+c}\sqrt{dx-c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^4/(d*x-c)^(1/2)/(d*x+c)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.237519, size = 162, normalized size = 2.16 \[ \frac{6 \, b d^{2} x^{4} - a c^{2} - 3 \,{\left (b c^{2} - a d^{2}\right )} x^{2} - 3 \,{\left (2 \, b d x^{3} + a d x\right )} \sqrt{d x + c} \sqrt{d x - c}}{3 \,{\left (4 \, d^{3} x^{6} - 3 \, c^{2} d x^{4} -{\left (4 \, d^{2} x^{5} - c^{2} x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

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^4),x, algorithm="fricas")

[Out]

1/3*(6*b*d^2*x^4 - a*c^2 - 3*(b*c^2 - a*d^2)*x^2 - 3*(2*b*d*x^3 + a*d*x)*sqrt(d*
x + c)*sqrt(d*x - c))/(4*d^3*x^6 - 3*c^2*d*x^4 - (4*d^2*x^5 - c^2*x^3)*sqrt(d*x
+ c)*sqrt(d*x - c))

_______________________________________________________________________________________

Sympy [A]  time = 108.996, size = 170, normalized size = 2.27 \[ - \frac{a d^{3}{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{9}{4}, \frac{11}{4}, 1 & \frac{5}{2}, \frac{5}{2}, 3 \\2, \frac{9}{4}, \frac{5}{2}, \frac{11}{4}, 3 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} - \frac{i a d^{3}{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{3}{2}, \frac{7}{4}, 2, \frac{9}{4}, \frac{5}{2}, 1 & \\\frac{7}{4}, \frac{9}{4} & \frac{3}{2}, 2, 2, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{4}} - \frac{b d{G_{6, 6}^{5, 3}\left (\begin{matrix} \frac{5}{4}, \frac{7}{4}, 1 & \frac{3}{2}, \frac{3}{2}, 2 \\1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}, 2 & 0 \end{matrix} \middle |{\frac{c^{2}}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} - \frac{i b d{G_{6, 6}^{2, 6}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, 1 & \\\frac{3}{4}, \frac{5}{4} & \frac{1}{2}, 1, 1, 0 \end{matrix} \middle |{\frac{c^{2} e^{2 i \pi }}{d^{2} x^{2}}} \right )}}{4 \pi ^{\frac{3}{2}} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**4/(d*x-c)**(1/2)/(d*x+c)**(1/2),x)

[Out]

-a*d**3*meijerg(((9/4, 11/4, 1), (5/2, 5/2, 3)), ((2, 9/4, 5/2, 11/4, 3), (0,)),
c**2/(d**2*x**2))/(4*pi**(3/2)*c**4) - I*a*d**3*meijerg(((3/2, 7/4, 2, 9/4, 5/2
, 1), ()), ((7/4, 9/4), (3/2, 2, 2, 0)), c**2*exp_polar(2*I*pi)/(d**2*x**2))/(4*
pi**(3/2)*c**4) - b*d*meijerg(((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4
, 2), (0,)), c**2/(d**2*x**2))/(4*pi**(3/2)*c**2) - I*b*d*meijerg(((1/2, 3/4, 1,
5/4, 3/2, 1), ()), ((3/4, 5/4), (1/2, 1, 1, 0)), c**2*exp_polar(2*I*pi)/(d**2*x
**2))/(4*pi**(3/2)*c**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234151, size = 185, normalized size = 2.47 \[ \frac{8 \,{\left (3 \, b d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{8} + 24 \, b c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 24 \, a d^{4}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 48 \, b c^{4} d^{2} + 32 \, a c^{2} d^{4}\right )}}{3 \,{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{3} d} \]

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^4),x, algorithm="giac")

[Out]

8/3*(3*b*d^2*(sqrt(d*x + c) - sqrt(d*x - c))^8 + 24*b*c^2*d^2*(sqrt(d*x + c) - s
qrt(d*x - c))^4 + 24*a*d^4*(sqrt(d*x + c) - sqrt(d*x - c))^4 + 48*b*c^4*d^2 + 32
*a*c^2*d^4)/(((sqrt(d*x + c) - sqrt(d*x - c))^4 + 4*c^2)^3*d)