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

Optimal. Leaf size=67 \[ \frac{3 c \log \left (b+c x^2\right )}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{c}{b^3 \left (b+c x^2\right )}-\frac{1}{2 b^3 x^2}-\frac{c}{4 b^2 \left (b+c x^2\right )^2} \]

[Out]

-1/(2*b^3*x^2) - c/(4*b^2*(b + c*x^2)^2) - c/(b^3*(b + c*x^2)) - (3*c*Log[x])/b^
4 + (3*c*Log[b + c*x^2])/(2*b^4)

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Rubi [A]  time = 0.121368, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{3 c \log \left (b+c x^2\right )}{2 b^4}-\frac{3 c \log (x)}{b^4}-\frac{c}{b^3 \left (b+c x^2\right )}-\frac{1}{2 b^3 x^2}-\frac{c}{4 b^2 \left (b+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(b*x^2 + c*x^4)^3,x]

[Out]

-1/(2*b^3*x^2) - c/(4*b^2*(b + c*x^2)^2) - c/(b^3*(b + c*x^2)) - (3*c*Log[x])/b^
4 + (3*c*Log[b + c*x^2])/(2*b^4)

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Rubi in Sympy [A]  time = 19.1503, size = 66, normalized size = 0.99 \[ - \frac{c}{4 b^{2} \left (b + c x^{2}\right )^{2}} - \frac{c}{b^{3} \left (b + c x^{2}\right )} - \frac{1}{2 b^{3} x^{2}} - \frac{3 c \log{\left (x^{2} \right )}}{2 b^{4}} + \frac{3 c \log{\left (b + c x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(4*b**2*(b + c*x**2)**2) - c/(b**3*(b + c*x**2)) - 1/(2*b**3*x**2) - 3*c*log(
x**2)/(2*b**4) + 3*c*log(b + c*x**2)/(2*b**4)

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Mathematica [A]  time = 0.0976038, size = 59, normalized size = 0.88 \[ -\frac{\frac{b \left (2 b^2+9 b c x^2+6 c^2 x^4\right )}{x^2 \left (b+c x^2\right )^2}-6 c \log \left (b+c x^2\right )+12 c \log (x)}{4 b^4} \]

Antiderivative was successfully verified.

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

[Out]

-((b*(2*b^2 + 9*b*c*x^2 + 6*c^2*x^4))/(x^2*(b + c*x^2)^2) + 12*c*Log[x] - 6*c*Lo
g[b + c*x^2])/(4*b^4)

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Maple [A]  time = 0.019, size = 62, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{3}{x}^{2}}}-{\frac{c}{4\,{b}^{2} \left ( c{x}^{2}+b \right ) ^{2}}}-{\frac{c}{{b}^{3} \left ( c{x}^{2}+b \right ) }}-3\,{\frac{c\ln \left ( x \right ) }{{b}^{4}}}+{\frac{3\,c\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^4+b*x^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.676693, size = 104, normalized size = 1.55 \[ -\frac{6 \, c^{2} x^{4} + 9 \, b c x^{2} + 2 \, b^{2}}{4 \,{\left (b^{3} c^{2} x^{6} + 2 \, b^{4} c x^{4} + b^{5} x^{2}\right )}} + \frac{3 \, c \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac{3 \, c \log \left (x^{2}\right )}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(6*c^2*x^4 + 9*b*c*x^2 + 2*b^2)/(b^3*c^2*x^6 + 2*b^4*c*x^4 + b^5*x^2) + 3/2
*c*log(c*x^2 + b)/b^4 - 3/2*c*log(x^2)/b^4

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Fricas [A]  time = 0.25849, size = 161, normalized size = 2.4 \[ -\frac{6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3} - 6 \,{\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (c x^{2} + b\right ) + 12 \,{\left (c^{3} x^{6} + 2 \, b c^{2} x^{4} + b^{2} c x^{2}\right )} \log \left (x\right )}{4 \,{\left (b^{4} c^{2} x^{6} + 2 \, b^{5} c x^{4} + b^{6} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(6*b*c^2*x^4 + 9*b^2*c*x^2 + 2*b^3 - 6*(c^3*x^6 + 2*b*c^2*x^4 + b^2*c*x^2)*
log(c*x^2 + b) + 12*(c^3*x^6 + 2*b*c^2*x^4 + b^2*c*x^2)*log(x))/(b^4*c^2*x^6 + 2
*b^5*c*x^4 + b^6*x^2)

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Sympy [A]  time = 2.9944, size = 78, normalized size = 1.16 \[ - \frac{2 b^{2} + 9 b c x^{2} + 6 c^{2} x^{4}}{4 b^{5} x^{2} + 8 b^{4} c x^{4} + 4 b^{3} c^{2} x^{6}} - \frac{3 c \log{\left (x \right )}}{b^{4}} + \frac{3 c \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*b**2 + 9*b*c*x**2 + 6*c**2*x**4)/(4*b**5*x**2 + 8*b**4*c*x**4 + 4*b**3*c**2*
x**6) - 3*c*log(x)/b**4 + 3*c*log(b/c + x**2)/(2*b**4)

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GIAC/XCAS [A]  time = 0.275326, size = 89, normalized size = 1.33 \[ \frac{3 \, c{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} - \frac{3 \, c{\rm ln}\left ({\left | x \right |}\right )}{b^{4}} - \frac{6 \, b c^{2} x^{4} + 9 \, b^{2} c x^{2} + 2 \, b^{3}}{4 \,{\left (c x^{2} + b\right )}^{2} b^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/2*c*ln(abs(c*x^2 + b))/b^4 - 3*c*ln(abs(x))/b^4 - 1/4*(6*b*c^2*x^4 + 9*b^2*c*x
^2 + 2*b^3)/((c*x^2 + b)^2*b^4*x^2)