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

Optimal. Leaf size=96 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]

[Out]

-(A/(b^3*x)) + ((b*B - A*c)*x)/(4*b^2*(b + c*x^2)^2) + ((3*b*B - 7*A*c)*x)/(8*b^
3*(b + c*x^2)) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2)*Sqrt[c
])

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Rubi [A]  time = 0.279585, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(b^3*x)) + ((b*B - A*c)*x)/(4*b^2*(b + c*x^2)^2) + ((3*b*B - 7*A*c)*x)/(8*b^
3*(b + c*x^2)) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2)*Sqrt[c
])

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Rubi in Sympy [A]  time = 30.6541, size = 88, normalized size = 0.92 \[ - \frac{A}{b^{3} x} - \frac{x \left (A c - B b\right )}{4 b^{2} \left (b + c x^{2}\right )^{2}} - \frac{x \left (7 A c - 3 B b\right )}{8 b^{3} \left (b + c x^{2}\right )} - \frac{3 \left (5 A c - B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{8 b^{\frac{7}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/(b**3*x) - x*(A*c - B*b)/(4*b**2*(b + c*x**2)**2) - x*(7*A*c - 3*B*b)/(8*b**3
*(b + c*x**2)) - 3*(5*A*c - B*b)*atan(sqrt(c)*x/sqrt(b))/(8*b**(7/2)*sqrt(c))

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Mathematica [A]  time = 0.107797, size = 96, normalized size = 1. \[ \frac{3 (b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{8 b^{7/2} \sqrt{c}}+\frac{x (3 b B-7 A c)}{8 b^3 \left (b+c x^2\right )}-\frac{A}{b^3 x}+\frac{x (b B-A c)}{4 b^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

-(A/(b^3*x)) + ((b*B - A*c)*x)/(4*b^2*(b + c*x^2)^2) + ((3*b*B - 7*A*c)*x)/(8*b^
3*(b + c*x^2)) + (3*(b*B - 5*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(8*b^(7/2)*Sqrt[c
])

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Maple [A]  time = 0.016, size = 125, normalized size = 1.3 \[ -{\frac{A}{{b}^{3}x}}-{\frac{7\,A{x}^{3}{c}^{2}}{8\,{b}^{3} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{3\,Bc{x}^{3}}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{9\,Axc}{8\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}+{\frac{5\,Bx}{8\,b \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{15\,Ac}{8\,{b}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{3\,B}{8\,{b}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A/b^3/x-7/8/b^3/(c*x^2+b)^2*A*x^3*c^2+3/8/b^2/(c*x^2+b)^2*B*x^3*c-9/8/b^2/(c*x^
2+b)^2*A*x*c+5/8/b/(c*x^2+b)^2*x*B-15/8/b^3/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*
A*c+3/8/b^2/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*B

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.222152, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \log \left (-\frac{2 \, b c x -{\left (c x^{2} - b\right )} \sqrt{-b c}}{c x^{2} + b}\right ) - 2 \,{\left (3 \,{\left (B b c - 5 \, A c^{2}\right )} x^{4} - 8 \, A b^{2} + 5 \,{\left (B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt{-b c}}{16 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{-b c}}, \frac{3 \,{\left ({\left (B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 2 \,{\left (B b^{2} c - 5 \, A b c^{2}\right )} x^{3} +{\left (B b^{3} - 5 \, A b^{2} c\right )} x\right )} \arctan \left (\frac{\sqrt{b c} x}{b}\right ) +{\left (3 \,{\left (B b c - 5 \, A c^{2}\right )} x^{4} - 8 \, A b^{2} + 5 \,{\left (B b^{2} - 5 \, A b c\right )} x^{2}\right )} \sqrt{b c}}{8 \,{\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )} \sqrt{b c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.71002, size = 194, normalized size = 2.02 \[ - \frac{3 \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right ) \log{\left (- \frac{3 b^{4} \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right ) \log{\left (\frac{3 b^{4} \sqrt{- \frac{1}{b^{7} c}} \left (- 5 A c + B b\right )}{- 15 A c + 3 B b} + x \right )}}{16} + \frac{- 8 A b^{2} + x^{4} \left (- 15 A c^{2} + 3 B b c\right ) + x^{2} \left (- 25 A b c + 5 B b^{2}\right )}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(-1/(b**7*c))*(-5*A*c + B*b)*log(-3*b**4*sqrt(-1/(b**7*c))*(-5*A*c + B*b)
/(-15*A*c + 3*B*b) + x)/16 + 3*sqrt(-1/(b**7*c))*(-5*A*c + B*b)*log(3*b**4*sqrt(
-1/(b**7*c))*(-5*A*c + B*b)/(-15*A*c + 3*B*b) + x)/16 + (-8*A*b**2 + x**4*(-15*A
*c**2 + 3*B*b*c) + x**2*(-25*A*b*c + 5*B*b**2))/(8*b**5*x + 16*b**4*c*x**3 + 8*b
**3*c**2*x**5)

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GIAC/XCAS [A]  time = 0.213915, size = 111, normalized size = 1.16 \[ \frac{3 \,{\left (B b - 5 \, A c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{8 \, \sqrt{b c} b^{3}} - \frac{A}{b^{3} x} + \frac{3 \, B b c x^{3} - 7 \, A c^{2} x^{3} + 5 \, B b^{2} x - 9 \, A b c x}{8 \,{\left (c x^{2} + b\right )}^{2} b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/8*(B*b - 5*A*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*b^3) - A/(b^3*x) + 1/8*(3*B*b
*c*x^3 - 7*A*c^2*x^3 + 5*B*b^2*x - 9*A*b*c*x)/((c*x^2 + b)^2*b^3)