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

Optimal. Leaf size=24 \[ -\frac{b^2}{4 x^4}-\frac{b c}{x^2}+c^2 \log (x) \]

[Out]

-b^2/(4*x^4) - (b*c)/x^2 + c^2*Log[x]

________________________________________________________________________________________

Rubi [A]  time = 0.0185294, antiderivative size = 24, 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} \[ -\frac{b^2}{4 x^4}-\frac{b c}{x^2}+c^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^9,x]

[Out]

-b^2/(4*x^4) - (b*c)/x^2 + c^2*Log[x]

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{\left (b x^2+c x^4\right )^2}{x^9} \, dx &=\int \frac{\left (b+c x^2\right )^2}{x^5} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+c x)^2}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{b^2}{x^3}+\frac{2 b c}{x^2}+\frac{c^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^2}{4 x^4}-\frac{b c}{x^2}+c^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0007747, size = 24, normalized size = 1. \[ -\frac{b^2}{4 x^4}-\frac{b c}{x^2}+c^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^9,x]

[Out]

-b^2/(4*x^4) - (b*c)/x^2 + c^2*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.047, size = 23, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{4\,{x}^{4}}}-{\frac{bc}{{x}^{2}}}+{c}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.982372, size = 35, normalized size = 1.46 \begin{align*} \frac{1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac{4 \, b c x^{2} + b^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.20648, size = 62, normalized size = 2.58 \begin{align*} \frac{4 \, c^{2} x^{4} \log \left (x\right ) - 4 \, b c x^{2} - b^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.356161, size = 22, normalized size = 0.92 \begin{align*} c^{2} \log{\left (x \right )} - \frac{b^{2} + 4 b c x^{2}}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.32315, size = 46, normalized size = 1.92 \begin{align*} \frac{1}{2} \, c^{2} \log \left (x^{2}\right ) - \frac{3 \, c^{2} x^{4} + 4 \, b c x^{2} + b^{2}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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