3.153 \(\int \frac{(b x^2+c x^4)^2}{x^8} \, dx\)

Optimal. Leaf size=23 \[ -\frac{b^2}{3 x^3}-\frac{2 b c}{x}+c^2 x \]

[Out]

-b^2/(3*x^3) - (2*b*c)/x + c^2*x

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Rubi [A]  time = 0.0180535, antiderivative size = 23, 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} \[ -\frac{b^2}{3 x^3}-\frac{2 b c}{x}+c^2 x \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^8,x]

[Out]

-b^2/(3*x^3) - (2*b*c)/x + c^2*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 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^8} \, dx &=\int \frac{\left (b+c x^2\right )^2}{x^4} \, dx\\ &=\int \left (c^2+\frac{b^2}{x^4}+\frac{2 b c}{x^2}\right ) \, dx\\ &=-\frac{b^2}{3 x^3}-\frac{2 b c}{x}+c^2 x\\ \end{align*}

Mathematica [A]  time = 0.0008095, size = 23, normalized size = 1. \[ -\frac{b^2}{3 x^3}-\frac{2 b c}{x}+c^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^8,x]

[Out]

-b^2/(3*x^3) - (2*b*c)/x + c^2*x

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Maple [A]  time = 0.046, size = 22, normalized size = 1. \begin{align*} -{\frac{{b}^{2}}{3\,{x}^{3}}}-2\,{\frac{bc}{x}}+{c}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^8,x)

[Out]

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

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Maxima [A]  time = 1.00879, size = 30, normalized size = 1.3 \begin{align*} c^{2} x - \frac{6 \, b c x^{2} + b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^8,x, algorithm="maxima")

[Out]

c^2*x - 1/3*(6*b*c*x^2 + b^2)/x^3

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Fricas [A]  time = 1.22724, size = 53, normalized size = 2.3 \begin{align*} \frac{3 \, c^{2} x^{4} - 6 \, b c x^{2} - b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^8,x, algorithm="fricas")

[Out]

1/3*(3*c^2*x^4 - 6*b*c*x^2 - b^2)/x^3

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Sympy [A]  time = 0.311306, size = 20, normalized size = 0.87 \begin{align*} c^{2} x - \frac{b^{2} + 6 b c x^{2}}{3 x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**8,x)

[Out]

c**2*x - (b**2 + 6*b*c*x**2)/(3*x**3)

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Giac [A]  time = 1.27357, size = 30, normalized size = 1.3 \begin{align*} c^{2} x - \frac{6 \, b c x^{2} + b^{2}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^8,x, algorithm="giac")

[Out]

c^2*x - 1/3*(6*b*c*x^2 + b^2)/x^3