3.151 \(\int \frac{(b x^2+c x^4)^2}{x^6} \, dx\)

Optimal. Leaf size=24 \[ -\frac{b^2}{x}+2 b c x+\frac{c^2 x^3}{3} \]

[Out]

-(b^2/x) + 2*b*c*x + (c^2*x^3)/3

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Rubi [A]  time = 0.0167236, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 270} \[ -\frac{b^2}{x}+2 b c x+\frac{c^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^6,x]

[Out]

-(b^2/x) + 2*b*c*x + (c^2*x^3)/3

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (b x^2+c x^4\right )^2}{x^6} \, dx &=\int \frac{\left (b+c x^2\right )^2}{x^2} \, dx\\ &=\int \left (2 b c+\frac{b^2}{x^2}+c^2 x^2\right ) \, dx\\ &=-\frac{b^2}{x}+2 b c x+\frac{c^2 x^3}{3}\\ \end{align*}

Mathematica [A]  time = 0.0006806, size = 24, normalized size = 1. \[ -\frac{b^2}{x}+2 b c x+\frac{c^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^6,x]

[Out]

-(b^2/x) + 2*b*c*x + (c^2*x^3)/3

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Maple [A]  time = 0.046, size = 23, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{x}}+2\,bcx+{\frac{{c}^{2}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^6,x)

[Out]

-b^2/x+2*b*c*x+1/3*c^2*x^3

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Maxima [A]  time = 0.963049, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac{b^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^6,x, algorithm="maxima")

[Out]

1/3*c^2*x^3 + 2*b*c*x - b^2/x

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Fricas [A]  time = 1.21534, size = 50, normalized size = 2.08 \begin{align*} \frac{c^{2} x^{4} + 6 \, b c x^{2} - 3 \, b^{2}}{3 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^6,x, algorithm="fricas")

[Out]

1/3*(c^2*x^4 + 6*b*c*x^2 - 3*b^2)/x

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Sympy [A]  time = 0.265041, size = 19, normalized size = 0.79 \begin{align*} - \frac{b^{2}}{x} + 2 b c x + \frac{c^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**6,x)

[Out]

-b**2/x + 2*b*c*x + c**2*x**3/3

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Giac [A]  time = 1.25199, size = 30, normalized size = 1.25 \begin{align*} \frac{1}{3} \, c^{2} x^{3} + 2 \, b c x - \frac{b^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^6,x, algorithm="giac")

[Out]

1/3*c^2*x^3 + 2*b*c*x - b^2/x