3.168 \(\int \frac{(b x^2+c x^4)^3}{x^{12}} \, dx\)

Optimal. Leaf size=34 \[ -\frac{b^2 c}{x^3}-\frac{b^3}{5 x^5}-\frac{3 b c^2}{x}+c^3 x \]

[Out]

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

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Rubi [A]  time = 0.0202517, antiderivative size = 34, 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} \[ -\frac{b^2 c}{x^3}-\frac{b^3}{5 x^5}-\frac{3 b c^2}{x}+c^3 x \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^12,x]

[Out]

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

Mathematica [A]  time = 0.0053235, size = 34, normalized size = 1. \[ -\frac{b^2 c}{x^3}-\frac{b^3}{5 x^5}-\frac{3 b c^2}{x}+c^3 x \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^12,x]

[Out]

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

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Maple [A]  time = 0.048, size = 33, normalized size = 1. \begin{align*} -{\frac{{b}^{3}}{5\,{x}^{5}}}-{\frac{{b}^{2}c}{{x}^{3}}}-3\,{\frac{b{c}^{2}}{x}}+{c}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^12,x)

[Out]

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

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Maxima [A]  time = 0.962633, size = 45, normalized size = 1.32 \begin{align*} c^{3} x - \frac{15 \, b c^{2} x^{4} + 5 \, b^{2} c x^{2} + b^{3}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^12,x, algorithm="maxima")

[Out]

c^3*x - 1/5*(15*b*c^2*x^4 + 5*b^2*c*x^2 + b^3)/x^5

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Fricas [A]  time = 1.46828, size = 76, normalized size = 2.24 \begin{align*} \frac{5 \, c^{3} x^{6} - 15 \, b c^{2} x^{4} - 5 \, b^{2} c x^{2} - b^{3}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^12,x, algorithm="fricas")

[Out]

1/5*(5*c^3*x^6 - 15*b*c^2*x^4 - 5*b^2*c*x^2 - b^3)/x^5

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Sympy [A]  time = 0.382389, size = 32, normalized size = 0.94 \begin{align*} c^{3} x - \frac{b^{3} + 5 b^{2} c x^{2} + 15 b c^{2} x^{4}}{5 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**12,x)

[Out]

c**3*x - (b**3 + 5*b**2*c*x**2 + 15*b*c**2*x**4)/(5*x**5)

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Giac [A]  time = 1.26849, size = 45, normalized size = 1.32 \begin{align*} c^{3} x - \frac{15 \, b c^{2} x^{4} + 5 \, b^{2} c x^{2} + b^{3}}{5 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^12,x, algorithm="giac")

[Out]

c^3*x - 1/5*(15*b*c^2*x^4 + 5*b^2*c*x^2 + b^3)/x^5