3.175 \(\int \frac{x^9}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=53 \[ \frac{b^2 x^2}{2 c^3}-\frac{b^3 \log \left (b+c x^2\right )}{2 c^4}-\frac{b x^4}{4 c^2}+\frac{x^6}{6 c} \]

[Out]

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

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Rubi [A]  time = 0.0452063, antiderivative size = 53, 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} \[ \frac{b^2 x^2}{2 c^3}-\frac{b^3 \log \left (b+c x^2\right )}{2 c^4}-\frac{b x^4}{4 c^2}+\frac{x^6}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[x^9/(b*x^2 + c*x^4),x]

[Out]

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

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^9}{b x^2+c x^4} \, dx &=\int \frac{x^7}{b+c x^2} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{b+c x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{c^3}-\frac{b x}{c^2}+\frac{x^2}{c}-\frac{b^3}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=\frac{b^2 x^2}{2 c^3}-\frac{b x^4}{4 c^2}+\frac{x^6}{6 c}-\frac{b^3 \log \left (b+c x^2\right )}{2 c^4}\\ \end{align*}

Mathematica [A]  time = 0.0059786, size = 53, normalized size = 1. \[ \frac{b^2 x^2}{2 c^3}-\frac{b^3 \log \left (b+c x^2\right )}{2 c^4}-\frac{b x^4}{4 c^2}+\frac{x^6}{6 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(b*x^2 + c*x^4),x]

[Out]

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

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Maple [A]  time = 0.045, size = 46, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}{x}^{2}}{2\,{c}^{3}}}-{\frac{b{x}^{4}}{4\,{c}^{2}}}+{\frac{{x}^{6}}{6\,c}}-{\frac{{b}^{3}\ln \left ( c{x}^{2}+b \right ) }{2\,{c}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(c*x^4+b*x^2),x)

[Out]

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

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Maxima [A]  time = 1.03044, size = 62, normalized size = 1.17 \begin{align*} -\frac{b^{3} \log \left (c x^{2} + b\right )}{2 \, c^{4}} + \frac{2 \, c^{2} x^{6} - 3 \, b c x^{4} + 6 \, b^{2} x^{2}}{12 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+b*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.48985, size = 99, normalized size = 1.87 \begin{align*} \frac{2 \, c^{3} x^{6} - 3 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} - 6 \, b^{3} \log \left (c x^{2} + b\right )}{12 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+b*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.332092, size = 44, normalized size = 0.83 \begin{align*} - \frac{b^{3} \log{\left (b + c x^{2} \right )}}{2 c^{4}} + \frac{b^{2} x^{2}}{2 c^{3}} - \frac{b x^{4}}{4 c^{2}} + \frac{x^{6}}{6 c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(c*x**4+b*x**2),x)

[Out]

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

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Giac [A]  time = 1.28292, size = 63, normalized size = 1.19 \begin{align*} -\frac{b^{3} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac{2 \, c^{2} x^{6} - 3 \, b c x^{4} + 6 \, b^{2} x^{2}}{12 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(c*x^4+b*x^2),x, algorithm="giac")

[Out]

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