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

Optimal. Leaf size=117 \[ -\frac{\left (3 a d^2+2 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{2 c^5}-\frac{3 a d^2+2 b c^2}{2 c^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{2 c^2 x^2 \sqrt{d x-c} \sqrt{c+d x}} \]

[Out]

-(2*b*c^2 + 3*a*d^2)/(2*c^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + a/(2*c^2*x^2*Sqrt[-c
+ d*x]*Sqrt[c + d*x]) - ((2*b*c^2 + 3*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*
x])/c])/(2*c^5)

_______________________________________________________________________________________

Rubi [A]  time = 0.347629, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ -\frac{\left (3 a d^2+2 b c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d x-c} \sqrt{c+d x}}{c}\right )}{2 c^5}-\frac{3 a d^2+2 b c^2}{2 c^4 \sqrt{d x-c} \sqrt{c+d x}}+\frac{a}{2 c^2 x^2 \sqrt{d x-c} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

-(2*b*c^2 + 3*a*d^2)/(2*c^4*Sqrt[-c + d*x]*Sqrt[c + d*x]) + a/(2*c^2*x^2*Sqrt[-c
+ d*x]*Sqrt[c + d*x]) - ((2*b*c^2 + 3*a*d^2)*ArcTan[(Sqrt[-c + d*x]*Sqrt[c + d*
x])/c])/(2*c^5)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 21.4516, size = 100, normalized size = 0.85 \[ \frac{a}{2 c^{2} x^{2} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{3 a d^{2} + 2 b c^{2}}{2 c^{4} \sqrt{- c + d x} \sqrt{c + d x}} - \frac{\left (3 a d^{2} + 2 b c^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{- c + d x} \sqrt{c + d x}}{c} \right )}}{2 c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a/(2*c**2*x**2*sqrt(-c + d*x)*sqrt(c + d*x)) - (3*a*d**2 + 2*b*c**2)/(2*c**4*sqr
t(-c + d*x)*sqrt(c + d*x)) - (3*a*d**2 + 2*b*c**2)*atan(sqrt(-c + d*x)*sqrt(c +
d*x)/c)/(2*c**5)

_______________________________________________________________________________________

Mathematica [C]  time = 0.289895, size = 126, normalized size = 1.08 \[ \frac{\frac{a \left (c^3-3 c d^2 x^2\right )-2 b c^3 x^2}{x^2 \sqrt{d x-c} \sqrt{c+d x}}+i \left (3 a d^2+2 b c^2\right ) \log \left (\frac{-4 c^4 \sqrt{d x-c} \sqrt{c+d x}+4 i c^5}{3 a d^2 x+2 b c^2 x}\right )}{2 c^5} \]

Antiderivative was successfully verified.

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

[Out]

((-2*b*c^3*x^2 + a*(c^3 - 3*c*d^2*x^2))/(x^2*Sqrt[-c + d*x]*Sqrt[c + d*x]) + I*(
2*b*c^2 + 3*a*d^2)*Log[((4*I)*c^5 - 4*c^4*Sqrt[-c + d*x]*Sqrt[c + d*x])/(2*b*c^2
*x + 3*a*d^2*x)])/(2*c^5)

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 315, normalized size = 2.7 \[{\frac{1}{2\,{c}^{4}{x}^{2}} \left ( 3\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}a{d}^{4}+2\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{4}b{c}^{2}{d}^{2}-3\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}a{c}^{2}{d}^{2}-2\,\ln \left ( -2\,{\frac{{c}^{2}-\sqrt{-{c}^{2}}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}{x}} \right ){x}^{2}b{c}^{4}-3\,a{d}^{2}\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{x}^{2}\sqrt{-{c}^{2}}-2\,b\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{c}^{2}{x}^{2}\sqrt{-{c}^{2}}+a\sqrt{{d}^{2}{x}^{2}-{c}^{2}}{c}^{2}\sqrt{-{c}^{2}} \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\frac{1}{\sqrt{{d}^{2}{x}^{2}-{c}^{2}}}}{\frac{1}{\sqrt{dx+c}}}{\frac{1}{\sqrt{dx-c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2/c^4*(3*ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^4*a*d^4+2*ln(-2*(c^
2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/2))/x)*x^4*b*c^2*d^2-3*ln(-2*(c^2-(-c^2)^(1/2)*(
d^2*x^2-c^2)^(1/2))/x)*x^2*a*c^2*d^2-2*ln(-2*(c^2-(-c^2)^(1/2)*(d^2*x^2-c^2)^(1/
2))/x)*x^2*b*c^4-3*a*d^2*(d^2*x^2-c^2)^(1/2)*x^2*(-c^2)^(1/2)-2*b*(d^2*x^2-c^2)^
(1/2)*c^2*x^2*(-c^2)^(1/2)+a*(d^2*x^2-c^2)^(1/2)*c^2*(-c^2)^(1/2))/(-c^2)^(1/2)/
x^2/(d^2*x^2-c^2)^(1/2)/(d*x+c)^(1/2)/(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)/((d*x + c)^(3/2)*(d*x - c)^(3/2)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.254717, size = 474, normalized size = 4.05 \[ \frac{3 \, a c^{5} d x + 4 \,{\left (2 \, b c^{3} d^{3} + 3 \, a c d^{5}\right )} x^{5} -{\left (6 \, b c^{5} d + 13 \, a c^{3} d^{3}\right )} x^{3} -{\left (a c^{5} + 4 \,{\left (2 \, b c^{3} d^{2} + 3 \, a c d^{4}\right )} x^{4} -{\left (2 \, b c^{5} + 7 \, a c^{3} d^{2}\right )} x^{2}\right )} \sqrt{d x + c} \sqrt{d x - c} - 2 \,{\left (4 \,{\left (2 \, b c^{2} d^{4} + 3 \, a d^{6}\right )} x^{6} - 5 \,{\left (2 \, b c^{4} d^{2} + 3 \, a c^{2} d^{4}\right )} x^{4} +{\left (2 \, b c^{6} + 3 \, a c^{4} d^{2}\right )} x^{2} -{\left (4 \,{\left (2 \, b c^{2} d^{3} + 3 \, a d^{5}\right )} x^{5} - 3 \,{\left (2 \, b c^{4} d + 3 \, a c^{2} d^{3}\right )} x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )} \arctan \left (-\frac{d x - \sqrt{d x + c} \sqrt{d x - c}}{c}\right )}{2 \,{\left (4 \, c^{5} d^{4} x^{6} - 5 \, c^{7} d^{2} x^{4} + c^{9} x^{2} -{\left (4 \, c^{5} d^{3} x^{5} - 3 \, c^{7} d x^{3}\right )} \sqrt{d x + c} \sqrt{d x - c}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(3*a*c^5*d*x + 4*(2*b*c^3*d^3 + 3*a*c*d^5)*x^5 - (6*b*c^5*d + 13*a*c^3*d^3)*
x^3 - (a*c^5 + 4*(2*b*c^3*d^2 + 3*a*c*d^4)*x^4 - (2*b*c^5 + 7*a*c^3*d^2)*x^2)*sq
rt(d*x + c)*sqrt(d*x - c) - 2*(4*(2*b*c^2*d^4 + 3*a*d^6)*x^6 - 5*(2*b*c^4*d^2 +
3*a*c^2*d^4)*x^4 + (2*b*c^6 + 3*a*c^4*d^2)*x^2 - (4*(2*b*c^2*d^3 + 3*a*d^5)*x^5
- 3*(2*b*c^4*d + 3*a*c^2*d^3)*x^3)*sqrt(d*x + c)*sqrt(d*x - c))*arctan(-(d*x - s
qrt(d*x + c)*sqrt(d*x - c))/c))/(4*c^5*d^4*x^6 - 5*c^7*d^2*x^4 + c^9*x^2 - (4*c^
5*d^3*x^5 - 3*c^7*d*x^3)*sqrt(d*x + c)*sqrt(d*x - c))

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.295925, size = 285, normalized size = 2.44 \[ \frac{{\left (2 \, b c^{2} + 3 \, a d^{2}\right )} \arctan \left (\frac{{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}}{2 \, c}\right )}{c^{5}} - \frac{{\left (b c^{2} + a d^{2}\right )} \sqrt{d x + c}}{2 \, \sqrt{d x - c} c^{5}} + \frac{2 \,{\left (b c^{2} + a d^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2} + 2 \, c\right )} c^{4}} + \frac{2 \,{\left (a d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{6} - 4 \, a c^{2} d^{2}{\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{2}\right )}}{{\left ({\left (\sqrt{d x + c} - \sqrt{d x - c}\right )}^{4} + 4 \, c^{2}\right )}^{2} c^{4}} \]

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^3),x, algorithm="giac")

[Out]

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