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

Optimal. Leaf size=38 \[ -\frac{\log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left (b+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0347204, antiderivative size = 38, 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, Rules used = {1584, 266, 44} \[ -\frac{\log \left (b+c x^2\right )}{2 b^2}+\frac{\log (x)}{b^2}+\frac{1}{2 b \left (b+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(b*x^2 + c*x^4)^2,x]

[Out]

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

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^3}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{1}{x \left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (b+c x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{b^2 x}-\frac{c}{b (b+c x)^2}-\frac{c}{b^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{2 b \left (b+c x^2\right )}+\frac{\log (x)}{b^2}-\frac{\log \left (b+c x^2\right )}{2 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0127739, size = 33, normalized size = 0.87 \[ \frac{\frac{b}{b+c x^2}-\log \left (b+c x^2\right )+2 \log (x)}{2 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.054, size = 35, normalized size = 0.9 \begin{align*}{\frac{1}{2\,b \left ( c{x}^{2}+b \right ) }}+{\frac{\ln \left ( x \right ) }{{b}^{2}}}-{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c*x^4+b*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.965884, size = 50, normalized size = 1.32 \begin{align*} \frac{1}{2 \,{\left (b c x^{2} + b^{2}\right )}} - \frac{\log \left (c x^{2} + b\right )}{2 \, b^{2}} + \frac{\log \left (x^{2}\right )}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.4848, size = 108, normalized size = 2.84 \begin{align*} -\frac{{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) - 2 \,{\left (c x^{2} + b\right )} \log \left (x\right ) - b}{2 \,{\left (b^{2} c x^{2} + b^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.492261, size = 34, normalized size = 0.89 \begin{align*} \frac{1}{2 b^{2} + 2 b c x^{2}} + \frac{\log{\left (x \right )}}{b^{2}} - \frac{\log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.20935, size = 49, normalized size = 1.29 \begin{align*} -\frac{\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{2}} + \frac{\log \left ({\left | x \right |}\right )}{b^{2}} + \frac{1}{2 \,{\left (c x^{2} + b\right )} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(abs(c*x^2 + b))/b^2 + log(abs(x))/b^2 + 1/2/((c*x^2 + b)*b)