3.74 \(\int \frac{x^{13} \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=111 \[ -\frac{b^3 (b B-A c)}{4 c^5 \left (b+c x^2\right )^2}+\frac{b^2 (4 b B-3 A c)}{2 c^5 \left (b+c x^2\right )}+\frac{3 b (2 b B-A c) \log \left (b+c x^2\right )}{2 c^5}-\frac{x^2 (3 b B-A c)}{2 c^4}+\frac{B x^4}{4 c^3} \]

[Out]

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

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Rubi [A]  time = 0.309433, antiderivative size = 111, 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 \[ -\frac{b^3 (b B-A c)}{4 c^5 \left (b+c x^2\right )^2}+\frac{b^2 (4 b B-3 A c)}{2 c^5 \left (b+c x^2\right )}+\frac{3 b (2 b B-A c) \log \left (b+c x^2\right )}{2 c^5}-\frac{x^2 (3 b B-A c)}{2 c^4}+\frac{B x^4}{4 c^3} \]

Antiderivative was successfully verified.

[In]  Int[(x^13*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

-((3*b*B - A*c)*x^2)/(2*c^4) + (B*x^4)/(4*c^3) - (b^3*(b*B - A*c))/(4*c^5*(b + c
*x^2)^2) + (b^2*(4*b*B - 3*A*c))/(2*c^5*(b + c*x^2)) + (3*b*(2*b*B - 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. \[ \frac{B \int ^{x^{2}} x\, dx}{2 c^{3}} + \frac{b^{3} \left (A c - B b\right )}{4 c^{5} \left (b + c x^{2}\right )^{2}} - \frac{b^{2} \left (3 A c - 4 B b\right )}{2 c^{5} \left (b + c x^{2}\right )} - \frac{3 b \left (A c - 2 B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{5}} + \left (\frac{A c}{2} - \frac{3 B b}{2}\right ) \int ^{x^{2}} \frac{1}{c^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

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

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Mathematica [A]  time = 0.112259, size = 94, normalized size = 0.85 \[ \frac{\frac{b^3 (A c-b B)}{\left (b+c x^2\right )^2}+\frac{2 b^2 (4 b B-3 A c)}{b+c x^2}+2 c x^2 (A c-3 b B)+6 b (2 b B-A c) \log \left (b+c x^2\right )+B c^2 x^4}{4 c^5} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^13*(A + B*x^2))/(b*x^2 + c*x^4)^3,x]

[Out]

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

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Maple [A]  time = 0.018, size = 134, normalized size = 1.2 \[{\frac{B{x}^{4}}{4\,{c}^{3}}}-{\frac{3\,Bb{x}^{2}}{2\,{c}^{4}}}+{\frac{A{x}^{2}}{2\,{c}^{3}}}-{\frac{3\,{b}^{2}A}{2\,{c}^{4} \left ( c{x}^{2}+b \right ) }}+2\,{\frac{B{b}^{3}}{{c}^{5} \left ( c{x}^{2}+b \right ) }}+{\frac{A{b}^{3}}{4\,{c}^{4} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{B{b}^{4}}{4\,{c}^{5} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{3\,b\ln \left ( c{x}^{2}+b \right ) A}{2\,{c}^{4}}}+3\,{\frac{{b}^{2}\ln \left ( c{x}^{2}+b \right ) B}{{c}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13*(B*x^2+A)/(c*x^4+b*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 1.38037, size = 157, normalized size = 1.41 \[ \frac{7 \, B b^{4} - 5 \, A b^{3} c + 2 \,{\left (4 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x^{2}}{4 \,{\left (c^{7} x^{4} + 2 \, b c^{6} x^{2} + b^{2} c^{5}\right )}} + \frac{B c x^{4} - 2 \,{\left (3 \, B b - A c\right )} x^{2}}{4 \, c^{4}} + \frac{3 \,{\left (2 \, B b^{2} - A b c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^13/(c*x^4 + b*x^2)^3,x, algorithm="maxima")

[Out]

1/4*(7*B*b^4 - 5*A*b^3*c + 2*(4*B*b^3*c - 3*A*b^2*c^2)*x^2)/(c^7*x^4 + 2*b*c^6*x
^2 + b^2*c^5) + 1/4*(B*c*x^4 - 2*(3*B*b - A*c)*x^2)/c^4 + 3/2*(2*B*b^2 - A*b*c)*
log(c*x^2 + b)/c^5

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Fricas [A]  time = 0.226408, size = 242, normalized size = 2.18 \[ \frac{B c^{4} x^{8} - 2 \,{\left (2 \, B b c^{3} - A c^{4}\right )} x^{6} + 7 \, B b^{4} - 5 \, A b^{3} c -{\left (11 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} x^{4} + 2 \,{\left (B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 6 \,{\left (2 \, B b^{4} - A b^{3} c +{\left (2 \, B b^{2} c^{2} - A b c^{3}\right )} x^{4} + 2 \,{\left (2 \, B b^{3} c - A b^{2} c^{2}\right )} x^{2}\right )} \log \left (c x^{2} + b\right )}{4 \,{\left (c^{7} x^{4} + 2 \, b c^{6} x^{2} + b^{2} c^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^13/(c*x^4 + b*x^2)^3,x, algorithm="fricas")

[Out]

1/4*(B*c^4*x^8 - 2*(2*B*b*c^3 - A*c^4)*x^6 + 7*B*b^4 - 5*A*b^3*c - (11*B*b^2*c^2
- 4*A*b*c^3)*x^4 + 2*(B*b^3*c - 2*A*b^2*c^2)*x^2 + 6*(2*B*b^4 - A*b^3*c + (2*B*
b^2*c^2 - A*b*c^3)*x^4 + 2*(2*B*b^3*c - A*b^2*c^2)*x^2)*log(c*x^2 + b))/(c^7*x^4
+ 2*b*c^6*x^2 + b^2*c^5)

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Sympy [A]  time = 3.0285, size = 116, normalized size = 1.05 \[ \frac{B x^{4}}{4 c^{3}} + \frac{3 b \left (- A c + 2 B b\right ) \log{\left (b + c x^{2} \right )}}{2 c^{5}} + \frac{- 5 A b^{3} c + 7 B b^{4} + x^{2} \left (- 6 A b^{2} c^{2} + 8 B b^{3} c\right )}{4 b^{2} c^{5} + 8 b c^{6} x^{2} + 4 c^{7} x^{4}} - \frac{x^{2} \left (- A c + 3 B b\right )}{2 c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13*(B*x**2+A)/(c*x**4+b*x**2)**3,x)

[Out]

B*x**4/(4*c**3) + 3*b*(-A*c + 2*B*b)*log(b + c*x**2)/(2*c**5) + (-5*A*b**3*c + 7
*B*b**4 + x**2*(-6*A*b**2*c**2 + 8*B*b**3*c))/(4*b**2*c**5 + 8*b*c**6*x**2 + 4*c
**7*x**4) - x**2*(-A*c + 3*B*b)/(2*c**4)

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GIAC/XCAS [A]  time = 0.217114, size = 178, normalized size = 1.6 \[ \frac{3 \,{\left (2 \, B b^{2} - A b c\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{5}} + \frac{B c^{3} x^{4} - 6 \, B b c^{2} x^{2} + 2 \, A c^{3} x^{2}}{4 \, c^{6}} - \frac{18 \, B b^{2} c^{2} x^{4} - 9 \, A b c^{3} x^{4} + 28 \, B b^{3} c x^{2} - 12 \, A b^{2} c^{2} x^{2} + 11 \, B b^{4} - 4 \, A b^{3} c}{4 \,{\left (c x^{2} + b\right )}^{2} c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x^13/(c*x^4 + b*x^2)^3,x, algorithm="giac")

[Out]

3/2*(2*B*b^2 - A*b*c)*ln(abs(c*x^2 + b))/c^5 + 1/4*(B*c^3*x^4 - 6*B*b*c^2*x^2 +
2*A*c^3*x^2)/c^6 - 1/4*(18*B*b^2*c^2*x^4 - 9*A*b*c^3*x^4 + 28*B*b^3*c*x^2 - 12*A
*b^2*c^2*x^2 + 11*B*b^4 - 4*A*b^3*c)/((c*x^2 + b)^2*c^5)