3.59 \(\int \frac{x^{11} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=105 \[ -\frac{b^3 (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 (4 b B-3 A c) \log \left (b+c x^2\right )}{2 c^5}+\frac{b x^2 (3 b B-2 A c)}{2 c^4}-\frac{x^4 (2 b B-A c)}{4 c^3}+\frac{B x^6}{6 c^2} \]

[Out]

(b*(3*b*B - 2*A*c)*x^2)/(2*c^4) - ((2*b*B - A*c)*x^4)/(4*c^3) + (B*x^6)/(6*c^2)
- (b^3*(b*B - A*c))/(2*c^5*(b + c*x^2)) - (b^2*(4*b*B - 3*A*c)*Log[b + c*x^2])/(
2*c^5)

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Rubi [A]  time = 0.292811, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{b^3 (b B-A c)}{2 c^5 \left (b+c x^2\right )}-\frac{b^2 (4 b B-3 A c) \log \left (b+c x^2\right )}{2 c^5}+\frac{b x^2 (3 b B-2 A c)}{2 c^4}-\frac{x^4 (2 b B-A c)}{4 c^3}+\frac{B x^6}{6 c^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^11*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(b*(3*b*B - 2*A*c)*x^2)/(2*c^4) - ((2*b*B - A*c)*x^4)/(4*c^3) + (B*x^6)/(6*c^2)
- (b^3*(b*B - A*c))/(2*c^5*(b + c*x^2)) - (b^2*(4*b*B - 3*A*c)*Log[b + c*x^2])/(
2*c^5)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B x^{6}}{6 c^{2}} + \frac{b^{3} \left (A c - B b\right )}{2 c^{5} \left (b + c x^{2}\right )} + \frac{b^{2} \left (3 A c - 4 B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{5}} + \frac{\left (A c - 2 B b\right ) \int ^{x^{2}} x\, dx}{2 c^{3}} - \frac{\left (2 A c - 3 B b\right ) \int ^{x^{2}} b\, dx}{2 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

B*x**6/(6*c**2) + b**3*(A*c - B*b)/(2*c**5*(b + c*x**2)) + b**2*(3*A*c - 4*B*b)*
log(b + c*x**2)/(2*c**5) + (A*c - 2*B*b)*Integral(x, (x, x**2))/(2*c**3) - (2*A*
c - 3*B*b)*Integral(b, (x, x**2))/(2*c**4)

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Mathematica [A]  time = 0.125469, size = 93, normalized size = 0.89 \[ \frac{\frac{6 b^3 (A c-b B)}{b+c x^2}+6 b^2 (3 A c-4 b B) \log \left (b+c x^2\right )+3 c^2 x^4 (A c-2 b B)+6 b c x^2 (3 b B-2 A c)+2 B c^3 x^6}{12 c^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^11*(A + B*x^2))/(b*x^2 + c*x^4)^2,x]

[Out]

(6*b*c*(3*b*B - 2*A*c)*x^2 + 3*c^2*(-2*b*B + A*c)*x^4 + 2*B*c^3*x^6 + (6*b^3*(-(
b*B) + A*c))/(b + c*x^2) + 6*b^2*(-4*b*B + 3*A*c)*Log[b + c*x^2])/(12*c^5)

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Maple [A]  time = 0.016, size = 122, normalized size = 1.2 \[{\frac{B{x}^{6}}{6\,{c}^{2}}}+{\frac{A{x}^{4}}{4\,{c}^{2}}}-{\frac{B{x}^{4}b}{2\,{c}^{3}}}-{\frac{Ab{x}^{2}}{{c}^{3}}}+{\frac{3\,B{b}^{2}{x}^{2}}{2\,{c}^{4}}}+{\frac{A{b}^{3}}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}-{\frac{B{b}^{4}}{2\,{c}^{5} \left ( c{x}^{2}+b \right ) }}+{\frac{3\,{b}^{2}\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{4}}}-2\,{\frac{{b}^{3}\ln \left ( c{x}^{2}+b \right ) B}{{c}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11*(B*x^2+A)/(c*x^4+b*x^2)^2,x)

[Out]

1/6*B*x^6/c^2+1/4/c^2*A*x^4-1/2/c^3*B*x^4*b-1/c^3*A*x^2*b+3/2/c^4*B*b^2*x^2+1/2*
b^3/c^4/(c*x^2+b)*A-1/2*b^4/c^5/(c*x^2+b)*B+3/2*b^2/c^4*ln(c*x^2+b)*A-2*b^3/c^5*
ln(c*x^2+b)*B

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Maxima [A]  time = 1.37907, size = 144, normalized size = 1.37 \[ -\frac{B b^{4} - A b^{3} c}{2 \,{\left (c^{6} x^{2} + b c^{5}\right )}} + \frac{2 \, B c^{2} x^{6} - 3 \,{\left (2 \, B b c - A c^{2}\right )} x^{4} + 6 \,{\left (3 \, B b^{2} - 2 \, A b c\right )} x^{2}}{12 \, c^{4}} - \frac{{\left (4 \, B b^{3} - 3 \, A b^{2} c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^11/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

-1/2*(B*b^4 - A*b^3*c)/(c^6*x^2 + b*c^5) + 1/12*(2*B*c^2*x^6 - 3*(2*B*b*c - A*c^
2)*x^4 + 6*(3*B*b^2 - 2*A*b*c)*x^2)/c^4 - 1/2*(4*B*b^3 - 3*A*b^2*c)*log(c*x^2 +
b)/c^5

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Fricas [A]  time = 0.209238, size = 200, normalized size = 1.9 \[ \frac{2 \, B c^{4} x^{8} -{\left (4 \, B b c^{3} - 3 \, A c^{4}\right )} x^{6} - 6 \, B b^{4} + 6 \, A b^{3} c + 3 \,{\left (4 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{4} + 6 \,{\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} - 6 \,{\left (4 \, B b^{4} - 3 \, A b^{3} c +{\left (4 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x^{2} + b\right )}{12 \,{\left (c^{6} x^{2} + b c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^11/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

1/12*(2*B*c^4*x^8 - (4*B*b*c^3 - 3*A*c^4)*x^6 - 6*B*b^4 + 6*A*b^3*c + 3*(4*B*b^2
*c^2 - 3*A*b*c^3)*x^4 + 6*(3*B*b^3*c - 2*A*b^2*c^2)*x^2 - 6*(4*B*b^4 - 3*A*b^3*c
+ (4*B*b^3*c - 3*A*b^2*c^2)*x^2)*log(c*x^2 + b))/(c^6*x^2 + b*c^5)

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Sympy [A]  time = 1.63636, size = 102, normalized size = 0.97 \[ \frac{B x^{6}}{6 c^{2}} - \frac{b^{2} \left (- 3 A c + 4 B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{5}} - \frac{- A b^{3} c + B b^{4}}{2 b c^{5} + 2 c^{6} x^{2}} - \frac{x^{4} \left (- A c + 2 B b\right )}{4 c^{3}} + \frac{x^{2} \left (- 2 A b c + 3 B b^{2}\right )}{2 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11*(B*x**2+A)/(c*x**4+b*x**2)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211884, size = 182, normalized size = 1.73 \[ -\frac{{\left (4 \, B b^{3} - 3 \, A b^{2} c\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{5}} + \frac{2 \, B c^{4} x^{6} - 6 \, B b c^{3} x^{4} + 3 \, A c^{4} x^{4} + 18 \, B b^{2} c^{2} x^{2} - 12 \, A b c^{3} x^{2}}{12 \, c^{6}} + \frac{4 \, B b^{3} c x^{2} - 3 \, A b^{2} c^{2} x^{2} + 3 \, B b^{4} - 2 \, A b^{3} c}{2 \,{\left (c x^{2} + b\right )} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^11/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

-1/2*(4*B*b^3 - 3*A*b^2*c)*ln(abs(c*x^2 + b))/c^5 + 1/12*(2*B*c^4*x^6 - 6*B*b*c^
3*x^4 + 3*A*c^4*x^4 + 18*B*b^2*c^2*x^2 - 12*A*b*c^3*x^2)/c^6 + 1/2*(4*B*b^3*c*x^
2 - 3*A*b^2*c^2*x^2 + 3*B*b^4 - 2*A*b^3*c)/((c*x^2 + b)*c^5)