3.195 \(\int \frac{x^7}{(b x^2+c x^4)^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{b}{2 c^2 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^2} \]

[Out]

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

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Rubi [A]  time = 0.0319761, antiderivative size = 33, 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, 43} \[ \frac{b}{2 c^2 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Int[x^7/(b*x^2 + c*x^4)^2,x]

[Out]

b/(2*c^2*(b + c*x^2)) + Log[b + c*x^2]/(2*c^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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^7}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac{x^3}{\left (b+c x^2\right )^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{b}{c (b+c x)^2}+\frac{1}{c (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b}{2 c^2 \left (b+c x^2\right )}+\frac{\log \left (b+c x^2\right )}{2 c^2}\\ \end{align*}

Mathematica [A]  time = 0.007953, size = 27, normalized size = 0.82 \[ \frac{\frac{b}{b+c x^2}+\log \left (b+c x^2\right )}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^7/(b*x^2 + c*x^4)^2,x]

[Out]

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

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Maple [A]  time = 0.051, size = 30, normalized size = 0.9 \begin{align*}{\frac{b}{2\,{c}^{2} \left ( c{x}^{2}+b \right ) }}+{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7/(c*x^4+b*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.965196, size = 43, normalized size = 1.3 \begin{align*} \frac{b}{2 \,{\left (c^{3} x^{2} + b c^{2}\right )}} + \frac{\log \left (c x^{2} + b\right )}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+b*x^2)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49813, size = 76, normalized size = 2.3 \begin{align*} \frac{{\left (c x^{2} + b\right )} \log \left (c x^{2} + b\right ) + b}{2 \,{\left (c^{3} x^{2} + b c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+b*x^2)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.36137, size = 29, normalized size = 0.88 \begin{align*} \frac{b}{2 b c^{2} + 2 c^{3} x^{2}} + \frac{\log{\left (b + c x^{2} \right )}}{2 c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7/(c*x**4+b*x**2)**2,x)

[Out]

b/(2*b*c**2 + 2*c**3*x**2) + log(b + c*x**2)/(2*c**2)

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Giac [A]  time = 1.20305, size = 43, normalized size = 1.3 \begin{align*} -\frac{x^{2}}{2 \,{\left (c x^{2} + b\right )} c} + \frac{\log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7/(c*x^4+b*x^2)^2,x, algorithm="giac")

[Out]

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