3.60 \(\int \frac{x^{10} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=110 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}+\frac{b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac{b x (3 b B-2 A c)}{c^4}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{B x^5}{5 c^2} \]

[Out]

(b*(3*b*B - 2*A*c)*x)/c^4 - ((2*b*B - A*c)*x^3)/(3*c^3) + (B*x^5)/(5*c^2) + (b^2
*(b*B - A*c)*x)/(2*c^4*(b + c*x^2)) - (b^(3/2)*(7*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x
)/Sqrt[b]])/(2*c^(9/2))

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Rubi [A]  time = 0.245605, antiderivative size = 110, 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 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}+\frac{b^2 x (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac{b x (3 b B-2 A c)}{c^4}-\frac{x^3 (2 b B-A c)}{3 c^3}+\frac{B x^5}{5 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(b*(3*b*B - 2*A*c)*x)/c^4 - ((2*b*B - A*c)*x^3)/(3*c^3) + (B*x^5)/(5*c^2) + (b^2
*(b*B - A*c)*x)/(2*c^4*(b + c*x^2)) - (b^(3/2)*(7*b*B - 5*A*c)*ArcTan[(Sqrt[c]*x
)/Sqrt[b]])/(2*c^(9/2))

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Rubi in Sympy [A]  time = 49.5373, size = 102, normalized size = 0.93 \[ \frac{B x^{5}}{5 c^{2}} + \frac{b^{\frac{3}{2}} \left (5 A c - 7 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{2 c^{\frac{9}{2}}} - \frac{b^{2} x \left (A c - B b\right )}{2 c^{4} \left (b + c x^{2}\right )} - \frac{b x \left (2 A c - 3 B b\right )}{c^{4}} + \frac{x^{3} \left (A c - 2 B b\right )}{3 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

B*x**5/(5*c**2) + b**(3/2)*(5*A*c - 7*B*b)*atan(sqrt(c)*x/sqrt(b))/(2*c**(9/2))
- b**2*x*(A*c - B*b)/(2*c**4*(b + c*x**2)) - b*x*(2*A*c - 3*B*b)/c**4 + x**3*(A*
c - 2*B*b)/(3*c**3)

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Mathematica [A]  time = 0.176229, size = 111, normalized size = 1.01 \[ -\frac{b^{3/2} (7 b B-5 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{9/2}}-\frac{x \left (A b^2 c-b^3 B\right )}{2 c^4 \left (b+c x^2\right )}+\frac{b x (3 b B-2 A c)}{c^4}+\frac{x^3 (A c-2 b B)}{3 c^3}+\frac{B x^5}{5 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^10*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(b*(3*b*B - 2*A*c)*x)/c^4 + ((-2*b*B + A*c)*x^3)/(3*c^3) + (B*x^5)/(5*c^2) - ((-
(b^3*B) + A*b^2*c)*x)/(2*c^4*(b + c*x^2)) - (b^(3/2)*(7*b*B - 5*A*c)*ArcTan[(Sqr
t[c]*x)/Sqrt[b]])/(2*c^(9/2))

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Maple [A]  time = 0.013, size = 132, normalized size = 1.2 \[{\frac{B{x}^{5}}{5\,{c}^{2}}}+{\frac{A{x}^{3}}{3\,{c}^{2}}}-{\frac{2\,B{x}^{3}b}{3\,{c}^{3}}}-2\,{\frac{Abx}{{c}^{3}}}+3\,{\frac{xB{b}^{2}}{{c}^{4}}}-{\frac{A{b}^{2}x}{2\,{c}^{3} \left ( c{x}^{2}+b \right ) }}+{\frac{Bx{b}^{3}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}+{\frac{5\,{b}^{2}A}{2\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}-{\frac{7\,B{b}^{3}}{2\,{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

1/5*B*x^5/c^2+1/3/c^2*A*x^3-2/3/c^3*B*x^3*b-2/c^3*A*x*b+3/c^4*x*B*b^2-1/2*b^2/c^
3*x/(c*x^2+b)*A+1/2*b^3/c^4*x/(c*x^2+b)*B+5/2*b^2/c^3/(b*c)^(1/2)*arctan(c*x/(b*
c)^(1/2))*A-7/2*b^3/c^4/(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^10/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224261, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, B c^{3} x^{7} - 4 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 20 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} - 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} + 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) + 30 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} x}{60 \,{\left (c^{5} x^{2} + b c^{4}\right )}}, \frac{6 \, B c^{3} x^{7} - 2 \,{\left (7 \, B b c^{2} - 5 \, A c^{3}\right )} x^{5} + 10 \,{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{3} - 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c +{\left (7 \, B b^{2} c - 5 \, A b c^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) + 15 \,{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} x}{30 \,{\left (c^{5} x^{2} + b c^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^10/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

[1/60*(12*B*c^3*x^7 - 4*(7*B*b*c^2 - 5*A*c^3)*x^5 + 20*(7*B*b^2*c - 5*A*b*c^2)*x
^3 - 15*(7*B*b^3 - 5*A*b^2*c + (7*B*b^2*c - 5*A*b*c^2)*x^2)*sqrt(-b/c)*log((c*x^
2 + 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)) + 30*(7*B*b^3 - 5*A*b^2*c)*x)/(c^5*x^2 +
b*c^4), 1/30*(6*B*c^3*x^7 - 2*(7*B*b*c^2 - 5*A*c^3)*x^5 + 10*(7*B*b^2*c - 5*A*b*
c^2)*x^3 - 15*(7*B*b^3 - 5*A*b^2*c + (7*B*b^2*c - 5*A*b*c^2)*x^2)*sqrt(b/c)*arct
an(x/sqrt(b/c)) + 15*(7*B*b^3 - 5*A*b^2*c)*x)/(c^5*x^2 + b*c^4)]

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Sympy [A]  time = 1.62263, size = 206, normalized size = 1.87 \[ \frac{B x^{5}}{5 c^{2}} + \frac{x \left (- A b^{2} c + B b^{3}\right )}{2 b c^{4} + 2 c^{5} x^{2}} + \frac{\sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right ) \log{\left (- \frac{c^{4} \sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right )}{- 5 A b c + 7 B b^{2}} + x \right )}}{4} - \frac{\sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right ) \log{\left (\frac{c^{4} \sqrt{- \frac{b^{3}}{c^{9}}} \left (- 5 A c + 7 B b\right )}{- 5 A b c + 7 B b^{2}} + x \right )}}{4} - \frac{x^{3} \left (- A c + 2 B b\right )}{3 c^{3}} + \frac{x \left (- 2 A b c + 3 B b^{2}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

B*x**5/(5*c**2) + x*(-A*b**2*c + B*b**3)/(2*b*c**4 + 2*c**5*x**2) + sqrt(-b**3/c
**9)*(-5*A*c + 7*B*b)*log(-c**4*sqrt(-b**3/c**9)*(-5*A*c + 7*B*b)/(-5*A*b*c + 7*
B*b**2) + x)/4 - sqrt(-b**3/c**9)*(-5*A*c + 7*B*b)*log(c**4*sqrt(-b**3/c**9)*(-5
*A*c + 7*B*b)/(-5*A*b*c + 7*B*b**2) + x)/4 - x**3*(-A*c + 2*B*b)/(3*c**3) + x*(-
2*A*b*c + 3*B*b**2)/c**4

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GIAC/XCAS [A]  time = 0.210145, size = 155, normalized size = 1.41 \[ -\frac{{\left (7 \, B b^{3} - 5 \, A b^{2} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{4}} + \frac{B b^{3} x - A b^{2} c x}{2 \,{\left (c x^{2} + b\right )} c^{4}} + \frac{3 \, B c^{8} x^{5} - 10 \, B b c^{7} x^{3} + 5 \, A c^{8} x^{3} + 45 \, B b^{2} c^{6} x - 30 \, A b c^{7} x}{15 \, c^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^10/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

-1/2*(7*B*b^3 - 5*A*b^2*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^4) + 1/2*(B*b^3*x
- A*b^2*c*x)/((c*x^2 + b)*c^4) + 1/15*(3*B*c^8*x^5 - 10*B*b*c^7*x^3 + 5*A*c^8*x^
3 + 45*B*b^2*c^6*x - 30*A*b*c^7*x)/c^10