3.150 \(\int \frac{(b x^2+c x^4)^2}{x^5} \, dx\)

Optimal. Leaf size=23 \[ b^2 \log (x)+b c x^2+\frac{c^2 x^4}{4} \]

[Out]

b*c*x^2 + (c^2*x^4)/4 + b^2*Log[x]

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Rubi [A]  time = 0.0186775, antiderivative size = 23, 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} \[ b^2 \log (x)+b c x^2+\frac{c^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^5,x]

[Out]

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

Mathematica [A]  time = 0.0008704, size = 23, normalized size = 1. \[ b^2 \log (x)+b c x^2+\frac{c^2 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^5,x]

[Out]

b*c*x^2 + (c^2*x^4)/4 + b^2*Log[x]

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Maple [A]  time = 0.043, size = 22, normalized size = 1. \begin{align*} bc{x}^{2}+{\frac{{c}^{2}{x}^{4}}{4}}+{b}^{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^5,x)

[Out]

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

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Maxima [A]  time = 0.961073, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, c^{2} x^{4} + b c x^{2} + \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^5,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.27176, size = 49, normalized size = 2.13 \begin{align*} \frac{1}{4} \, c^{2} x^{4} + b c x^{2} + b^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^5,x, algorithm="fricas")

[Out]

1/4*c^2*x^4 + b*c*x^2 + b^2*log(x)

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Sympy [A]  time = 0.267114, size = 20, normalized size = 0.87 \begin{align*} b^{2} \log{\left (x \right )} + b c x^{2} + \frac{c^{2} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**5,x)

[Out]

b**2*log(x) + b*c*x**2 + c**2*x**4/4

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Giac [A]  time = 1.26192, size = 32, normalized size = 1.39 \begin{align*} \frac{1}{4} \, c^{2} x^{4} + b c x^{2} + \frac{1}{2} \, b^{2} \log \left (x^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^5,x, algorithm="giac")

[Out]

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