3.58 \(\int \frac{x^{12} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=133 \[ \frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}-\frac{b^3 x (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}-\frac{x^5 (2 b B-A c)}{5 c^3}+\frac{B x^7}{7 c^2} \]

[Out]

-((b^2*(4*b*B - 3*A*c)*x)/c^5) + (b*(3*b*B - 2*A*c)*x^3)/(3*c^4) - ((2*b*B - A*c
)*x^5)/(5*c^3) + (B*x^7)/(7*c^2) - (b^3*(b*B - A*c)*x)/(2*c^5*(b + c*x^2)) + (b^
(5/2)*(9*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(11/2))

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Rubi [A]  time = 0.307893, antiderivative size = 133, 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 \[ \frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}-\frac{b^3 x (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}-\frac{x^5 (2 b B-A c)}{5 c^3}+\frac{B x^7}{7 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

-((b^2*(4*b*B - 3*A*c)*x)/c^5) + (b*(3*b*B - 2*A*c)*x^3)/(3*c^4) - ((2*b*B - A*c
)*x^5)/(5*c^3) + (B*x^7)/(7*c^2) - (b^3*(b*B - A*c)*x)/(2*c^5*(b + c*x^2)) + (b^
(5/2)*(9*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(11/2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{7}}{7 c^{2}} - \frac{b^{\frac{5}{2}} \left (7 A c - 9 B b\right ) \operatorname{atan}{\left (\frac{\sqrt{c} x}{\sqrt{b}} \right )}}{2 c^{\frac{11}{2}}} + \frac{b^{3} x \left (A c - B b\right )}{2 c^{5} \left (b + c x^{2}\right )} - \frac{b x^{3} \left (2 A c - 3 B b\right )}{3 c^{4}} + \frac{x^{5} \left (A c - 2 B b\right )}{5 c^{3}} + \frac{\left (3 A c - 4 B b\right ) \int b^{2}\, dx}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

B*x**7/(7*c**2) - b**(5/2)*(7*A*c - 9*B*b)*atan(sqrt(c)*x/sqrt(b))/(2*c**(11/2))
+ b**3*x*(A*c - B*b)/(2*c**5*(b + c*x**2)) - b*x**3*(2*A*c - 3*B*b)/(3*c**4) +
x**5*(A*c - 2*B*b)/(5*c**3) + (3*A*c - 4*B*b)*Integral(b**2, x)/c**5

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Mathematica [A]  time = 0.18486, size = 134, normalized size = 1.01 \[ \frac{b^{5/2} (9 b B-7 A c) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b}}\right )}{2 c^{11/2}}-\frac{b^2 x (4 b B-3 A c)}{c^5}+\frac{x \left (A b^3 c-b^4 B\right )}{2 c^5 \left (b+c x^2\right )}+\frac{b x^3 (3 b B-2 A c)}{3 c^4}+\frac{x^5 (A c-2 b B)}{5 c^3}+\frac{B x^7}{7 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^12*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

-((b^2*(4*b*B - 3*A*c)*x)/c^5) + (b*(3*b*B - 2*A*c)*x^3)/(3*c^4) + ((-2*b*B + A*
c)*x^5)/(5*c^3) + (B*x^7)/(7*c^2) + ((-(b^4*B) + A*b^3*c)*x)/(2*c^5*(b + c*x^2))
+ (b^(5/2)*(9*b*B - 7*A*c)*ArcTan[(Sqrt[c]*x)/Sqrt[b]])/(2*c^(11/2))

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Maple [A]  time = 0.014, size = 155, normalized size = 1.2 \[{\frac{B{x}^{7}}{7\,{c}^{2}}}+{\frac{A{x}^{5}}{5\,{c}^{2}}}-{\frac{2\,B{x}^{5}b}{5\,{c}^{3}}}-{\frac{2\,Ab{x}^{3}}{3\,{c}^{3}}}+{\frac{B{x}^{3}{b}^{2}}{{c}^{4}}}+3\,{\frac{A{b}^{2}x}{{c}^{4}}}-4\,{\frac{Bx{b}^{3}}{{c}^{5}}}+{\frac{A{b}^{3}x}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{{b}^{4}xB}{2\,{c}^{5} \left ( c{x}^{2}+b \right ) }}-{\frac{7\,A{b}^{3}}{2\,{c}^{4}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}}+{\frac{9\,B{b}^{4}}{2\,{c}^{5}}\arctan \left ({cx{\frac{1}{\sqrt{bc}}}} \right ){\frac{1}{\sqrt{bc}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

1/7*B*x^7/c^2+1/5/c^2*A*x^5-2/5/c^3*B*x^5*b-2/3/c^3*A*x^3*b+1/c^4*B*x^3*b^2+3/c^
4*A*x*b^2-4/c^5*B*x*b^3+1/2*b^3/c^4*x/(c*x^2+b)*A-1/2*b^4/c^5*x/(c*x^2+b)*B-7/2*
b^3/c^4/(b*c)^(1/2)*arctan(c*x/(b*c)^(1/2))*A+9/2*b^4/c^5/(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^12/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218546, size = 1, normalized size = 0.01 \[ \left [\frac{60 \, B c^{4} x^{9} - 12 \,{\left (9 \, B b c^{3} - 7 \, A c^{4}\right )} x^{7} + 28 \,{\left (9 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{5} - 140 \,{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3} - 105 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c +{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{-\frac{b}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{b}{c}} - b}{c x^{2} + b}\right ) - 210 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} x}{420 \,{\left (c^{6} x^{2} + b c^{5}\right )}}, \frac{30 \, B c^{4} x^{9} - 6 \,{\left (9 \, B b c^{3} - 7 \, A c^{4}\right )} x^{7} + 14 \,{\left (9 \, B b^{2} c^{2} - 7 \, A b c^{3}\right )} x^{5} - 70 \,{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{3} + 105 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c +{\left (9 \, B b^{3} c - 7 \, A b^{2} c^{2}\right )} x^{2}\right )} \sqrt{\frac{b}{c}} \arctan \left (\frac{x}{\sqrt{\frac{b}{c}}}\right ) - 105 \,{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} x}{210 \,{\left (c^{6} x^{2} + b c^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^12/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

[1/420*(60*B*c^4*x^9 - 12*(9*B*b*c^3 - 7*A*c^4)*x^7 + 28*(9*B*b^2*c^2 - 7*A*b*c^
3)*x^5 - 140*(9*B*b^3*c - 7*A*b^2*c^2)*x^3 - 105*(9*B*b^4 - 7*A*b^3*c + (9*B*b^3
*c - 7*A*b^2*c^2)*x^2)*sqrt(-b/c)*log((c*x^2 - 2*c*x*sqrt(-b/c) - b)/(c*x^2 + b)
) - 210*(9*B*b^4 - 7*A*b^3*c)*x)/(c^6*x^2 + b*c^5), 1/210*(30*B*c^4*x^9 - 6*(9*B
*b*c^3 - 7*A*c^4)*x^7 + 14*(9*B*b^2*c^2 - 7*A*b*c^3)*x^5 - 70*(9*B*b^3*c - 7*A*b
^2*c^2)*x^3 + 105*(9*B*b^4 - 7*A*b^3*c + (9*B*b^3*c - 7*A*b^2*c^2)*x^2)*sqrt(b/c
)*arctan(x/sqrt(b/c)) - 105*(9*B*b^4 - 7*A*b^3*c)*x)/(c^6*x^2 + b*c^5)]

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Sympy [A]  time = 1.72402, size = 233, normalized size = 1.75 \[ \frac{B x^{7}}{7 c^{2}} - \frac{x \left (- A b^{3} c + B b^{4}\right )}{2 b c^{5} + 2 c^{6} x^{2}} - \frac{\sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right ) \log{\left (- \frac{c^{5} \sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right )}{- 7 A b^{2} c + 9 B b^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right ) \log{\left (\frac{c^{5} \sqrt{- \frac{b^{5}}{c^{11}}} \left (- 7 A c + 9 B b\right )}{- 7 A b^{2} c + 9 B b^{3}} + x \right )}}{4} - \frac{x^{5} \left (- A c + 2 B b\right )}{5 c^{3}} + \frac{x^{3} \left (- 2 A b c + 3 B b^{2}\right )}{3 c^{4}} - \frac{x \left (- 3 A b^{2} c + 4 B b^{3}\right )}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

B*x**7/(7*c**2) - x*(-A*b**3*c + B*b**4)/(2*b*c**5 + 2*c**6*x**2) - sqrt(-b**5/c
**11)*(-7*A*c + 9*B*b)*log(-c**5*sqrt(-b**5/c**11)*(-7*A*c + 9*B*b)/(-7*A*b**2*c
+ 9*B*b**3) + x)/4 + sqrt(-b**5/c**11)*(-7*A*c + 9*B*b)*log(c**5*sqrt(-b**5/c**
11)*(-7*A*c + 9*B*b)/(-7*A*b**2*c + 9*B*b**3) + x)/4 - x**5*(-A*c + 2*B*b)/(5*c*
*3) + x**3*(-2*A*b*c + 3*B*b**2)/(3*c**4) - x*(-3*A*b**2*c + 4*B*b**3)/c**5

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GIAC/XCAS [A]  time = 0.209625, size = 188, normalized size = 1.41 \[ \frac{{\left (9 \, B b^{4} - 7 \, A b^{3} c\right )} \arctan \left (\frac{c x}{\sqrt{b c}}\right )}{2 \, \sqrt{b c} c^{5}} - \frac{B b^{4} x - A b^{3} c x}{2 \,{\left (c x^{2} + b\right )} c^{5}} + \frac{15 \, B c^{12} x^{7} - 42 \, B b c^{11} x^{5} + 21 \, A c^{12} x^{5} + 105 \, B b^{2} c^{10} x^{3} - 70 \, A b c^{11} x^{3} - 420 \, B b^{3} c^{9} x + 315 \, A b^{2} c^{10} x}{105 \, c^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^12/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

1/2*(9*B*b^4 - 7*A*b^3*c)*arctan(c*x/sqrt(b*c))/(sqrt(b*c)*c^5) - 1/2*(B*b^4*x -
A*b^3*c*x)/((c*x^2 + b)*c^5) + 1/105*(15*B*c^12*x^7 - 42*B*b*c^11*x^5 + 21*A*c^
12*x^5 + 105*B*b^2*c^10*x^3 - 70*A*b*c^11*x^3 - 420*B*b^3*c^9*x + 315*A*b^2*c^10
*x)/c^14