3.79 \(\int \frac{x^8 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=90 \[ \frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{3/2} c^{5/2}}-\frac{x (5 b B-A c)}{8 b c^2 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 c^2 \left (b+c x^2\right )^2} \]

[Out]

((b*B - A*c)*x)/(4*c^2*(b + c*x^2)^2) - ((5*b*B - A*c)*x)/(8*b*c^2*(b + c*x^2))
+ ((3*b*B + A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(3/2)*c^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.180611, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{3/2} c^{5/2}}-\frac{x (5 b B-A c)}{8 b c^2 \left (b+c x^2\right )}+\frac{x (b B-A c)}{4 c^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

((b*B - A*c)*x)/(4*c^2*(b + c*x^2)^2) - ((5*b*B - A*c)*x)/(8*b*c^2*(b + c*x^2))
+ ((3*b*B + A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(3/2)*c^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 23.6155, size = 78, normalized size = 0.87 \[ - \frac{x \left (A c - B b\right )}{4 c^{2} \left (b + c x^{2}\right )^{2}} + \frac{x \left (A c - 5 B b\right )}{8 b c^{2} \left (b + c x^{2}\right )} + \frac{\left (A c + 3 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{3}{2}} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

-x*(A*c - B*b)/(4*c**2*(b + c*x**2)**2) + x*(A*c - 5*B*b)/(8*b*c**2*(b + c*x**2)
) + (A*c + 3*B*b)*atan(sqrt(c)*x/sqrt(b))/(8*b**(3/2)*c**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.137071, size = 83, normalized size = 0.92 \[ \frac{\frac{(A c+3 b B) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{b^{3/2}}+\frac{\sqrt{c} x \left (-b c \left (A+5 B x^2\right )+A c^2 x^2-3 b^2 B\right )}{b \left (b+c x^2\right )^2}}{8 c^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

((Sqrt[c]*x*(-3*b^2*B + A*c^2*x^2 - b*c*(A + 5*B*x^2)))/(b*(b + c*x^2)^2) + ((3*
b*B + A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/b^(3/2))/(8*c^(5/2))

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 89, normalized size = 1. \[{\frac{1}{ \left ( c{x}^{2}+b \right ) ^{2}} \left ({\frac{ \left ( Ac-5\,Bb \right ){x}^{3}}{8\,bc}}-{\frac{ \left ( Ac+3\,Bb \right ) x}{8\,{c}^{2}}} \right ) }+{\frac{A}{8\,bc}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

(1/8*(A*c-5*B*b)/b/c*x^3-1/8*(A*c+3*B*b)/c^2*x)/(c*x^2+b)^2+1/8/c/b/(b*c)^(1/2)*
arctan(c*x/(b*c)^(1/2))*A+3/8/c^2/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*B

_______________________________________________________________________________________

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)*x^8/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.223054, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 3 \, B b^{3} + A b^{2} c + 2 \,{\left (3 \, B b^{2} c + A b c^{2}\right )} x^{2}\right )} \log \left (\frac{2 \, b c x +{\left (c x^{2} - b\right )} \sqrt{-b c}}{c x^{2} + b}\right ) - 2 \,{\left ({\left (5 \, B b c - A c^{2}\right )} x^{3} +{\left (3 \, B b^{2} + A b c\right )} x\right )} \sqrt{-b c}}{16 \,{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \sqrt{-b c}}, \frac{{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 3 \, B b^{3} + A b^{2} c + 2 \,{\left (3 \, B b^{2} c + A b c^{2}\right )} x^{2}\right )} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) -{\left ({\left (5 \, B b c - A c^{2}\right )} x^{3} +{\left (3 \, B b^{2} + A b c\right )} x\right )} \sqrt{b c}}{8 \,{\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )} \sqrt{b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^8/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

[1/16*(((3*B*b*c^2 + A*c^3)*x^4 + 3*B*b^3 + A*b^2*c + 2*(3*B*b^2*c + A*b*c^2)*x^
2)*log((2*b*c*x + (c*x^2 - b)*sqrt(-b*c))/(c*x^2 + b)) - 2*((5*B*b*c - A*c^2)*x^
3 + (3*B*b^2 + A*b*c)*x)*sqrt(-b*c))/((b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*sqrt
(-b*c)), 1/8*(((3*B*b*c^2 + A*c^3)*x^4 + 3*B*b^3 + A*b^2*c + 2*(3*B*b^2*c + A*b*
c^2)*x^2)*arctan(sqrt(b*c)*x/b) - ((5*B*b*c - A*c^2)*x^3 + (3*B*b^2 + A*b*c)*x)*
sqrt(b*c))/((b*c^4*x^4 + 2*b^2*c^3*x^2 + b^3*c^2)*sqrt(b*c))]

_______________________________________________________________________________________

Sympy [A]  time = 1.70804, size = 153, normalized size = 1.7 \[ - \frac{\sqrt{- \frac{1}{b^{3} c^{5}}} \left (A c + 3 B b\right ) \log{\left (- b^{2} c^{2} \sqrt{- \frac{1}{b^{3} c^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{b^{3} c^{5}}} \left (A c + 3 B b\right ) \log{\left (b^{2} c^{2} \sqrt{- \frac{1}{b^{3} c^{5}}} + x \right )}}{16} - \frac{x^{3} \left (- A c^{2} + 5 B b c\right ) + x \left (A b c + 3 B b^{2}\right )}{8 b^{3} c^{2} + 16 b^{2} c^{3} x^{2} + 8 b c^{4} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

-sqrt(-1/(b**3*c**5))*(A*c + 3*B*b)*log(-b**2*c**2*sqrt(-1/(b**3*c**5)) + x)/16
+ sqrt(-1/(b**3*c**5))*(A*c + 3*B*b)*log(b**2*c**2*sqrt(-1/(b**3*c**5)) + x)/16
- (x**3*(-A*c**2 + 5*B*b*c) + x*(A*b*c + 3*B*b**2))/(8*b**3*c**2 + 16*b**2*c**3*
x**2 + 8*b*c**4*x**4)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217001, size = 105, normalized size = 1.17 \[ \frac{{\left (3 \, B b + A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b c^{2}} - \frac{5 \, B b c x^{3} - A c^{2} x^{3} + 3 \, B b^{2} x + A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^8/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

1/8*(3*B*b + A*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b*c^2) - 1/8*(5*B*b*c*x^3 - A
*c^2*x^3 + 3*B*b^2*x + A*b*c*x)/((c*x^2 + b)^2*b*c^2)