3.157 \(\int \frac{(b x^2+c x^4)^2}{x^{12}} \, dx\)

Optimal. Leaf size=30 \[ -\frac{b^2}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3} \]

[Out]

-b^2/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)

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Rubi [A]  time = 0.0159052, antiderivative size = 30, 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} \[ -\frac{b^2}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^12,x]

[Out]

-b^2/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^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^{12}} \, dx &=\int \frac{\left (b+c x^2\right )^2}{x^8} \, dx\\ &=\int \left (\frac{b^2}{x^8}+\frac{2 b c}{x^6}+\frac{c^2}{x^4}\right ) \, dx\\ &=-\frac{b^2}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0008539, size = 30, normalized size = 1. \[ -\frac{b^2}{7 x^7}-\frac{2 b c}{5 x^5}-\frac{c^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^12,x]

[Out]

-b^2/(7*x^7) - (2*b*c)/(5*x^5) - c^2/(3*x^3)

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Maple [A]  time = 0.047, size = 25, normalized size = 0.8 \begin{align*} -{\frac{{b}^{2}}{7\,{x}^{7}}}-{\frac{2\,bc}{5\,{x}^{5}}}-{\frac{{c}^{2}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^2/x^12,x)

[Out]

-1/7*b^2/x^7-2/5*b*c/x^5-1/3*c^2/x^3

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Maxima [A]  time = 1.08368, size = 35, normalized size = 1.17 \begin{align*} -\frac{35 \, c^{2} x^{4} + 42 \, b c x^{2} + 15 \, b^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^12,x, algorithm="maxima")

[Out]

-1/105*(35*c^2*x^4 + 42*b*c*x^2 + 15*b^2)/x^7

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Fricas [A]  time = 1.26561, size = 63, normalized size = 2.1 \begin{align*} -\frac{35 \, c^{2} x^{4} + 42 \, b c x^{2} + 15 \, b^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^12,x, algorithm="fricas")

[Out]

-1/105*(35*c^2*x^4 + 42*b*c*x^2 + 15*b^2)/x^7

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Sympy [A]  time = 0.429421, size = 27, normalized size = 0.9 \begin{align*} - \frac{15 b^{2} + 42 b c x^{2} + 35 c^{2} x^{4}}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**2/x**12,x)

[Out]

-(15*b**2 + 42*b*c*x**2 + 35*c**2*x**4)/(105*x**7)

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Giac [A]  time = 1.27573, size = 35, normalized size = 1.17 \begin{align*} -\frac{35 \, c^{2} x^{4} + 42 \, b c x^{2} + 15 \, b^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^2/x^12,x, algorithm="giac")

[Out]

-1/105*(35*c^2*x^4 + 42*b*c*x^2 + 15*b^2)/x^7