3.172 \(\int \frac{(b x^2+c x^4)^3}{x^{16}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{3 b^2 c}{7 x^7}-\frac{b^3}{9 x^9}-\frac{3 b c^2}{5 x^5}-\frac{c^3}{3 x^3} \]

[Out]

-b^3/(9*x^9) - (3*b^2*c)/(7*x^7) - (3*b*c^2)/(5*x^5) - c^3/(3*x^3)

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Rubi [A]  time = 0.0213949, antiderivative size = 43, 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} \[ -\frac{3 b^2 c}{7 x^7}-\frac{b^3}{9 x^9}-\frac{3 b c^2}{5 x^5}-\frac{c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^16,x]

[Out]

-b^3/(9*x^9) - (3*b^2*c)/(7*x^7) - (3*b*c^2)/(5*x^5) - c^3/(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 )^3}{x^{16}} \, dx &=\int \frac{\left (b+c x^2\right )^3}{x^{10}} \, dx\\ &=\int \left (\frac{b^3}{x^{10}}+\frac{3 b^2 c}{x^8}+\frac{3 b c^2}{x^6}+\frac{c^3}{x^4}\right ) \, dx\\ &=-\frac{b^3}{9 x^9}-\frac{3 b^2 c}{7 x^7}-\frac{3 b c^2}{5 x^5}-\frac{c^3}{3 x^3}\\ \end{align*}

Mathematica [A]  time = 0.0044876, size = 43, normalized size = 1. \[ -\frac{3 b^2 c}{7 x^7}-\frac{b^3}{9 x^9}-\frac{3 b c^2}{5 x^5}-\frac{c^3}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^16,x]

[Out]

-b^3/(9*x^9) - (3*b^2*c)/(7*x^7) - (3*b*c^2)/(5*x^5) - c^3/(3*x^3)

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Maple [A]  time = 0.049, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{b}^{3}}{9\,{x}^{9}}}-{\frac{3\,{b}^{2}c}{7\,{x}^{7}}}-{\frac{3\,b{c}^{2}}{5\,{x}^{5}}}-{\frac{{c}^{3}}{3\,{x}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^16,x)

[Out]

-1/9*b^3/x^9-3/7*b^2*c/x^7-3/5*b*c^2/x^5-1/3*c^3/x^3

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Maxima [A]  time = 0.971687, size = 50, normalized size = 1.16 \begin{align*} -\frac{105 \, c^{3} x^{6} + 189 \, b c^{2} x^{4} + 135 \, b^{2} c x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^16,x, algorithm="maxima")

[Out]

-1/315*(105*c^3*x^6 + 189*b*c^2*x^4 + 135*b^2*c*x^2 + 35*b^3)/x^9

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Fricas [A]  time = 1.45717, size = 90, normalized size = 2.09 \begin{align*} -\frac{105 \, c^{3} x^{6} + 189 \, b c^{2} x^{4} + 135 \, b^{2} c x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^16,x, algorithm="fricas")

[Out]

-1/315*(105*c^3*x^6 + 189*b*c^2*x^4 + 135*b^2*c*x^2 + 35*b^3)/x^9

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Sympy [A]  time = 0.487765, size = 39, normalized size = 0.91 \begin{align*} - \frac{35 b^{3} + 135 b^{2} c x^{2} + 189 b c^{2} x^{4} + 105 c^{3} x^{6}}{315 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**16,x)

[Out]

-(35*b**3 + 135*b**2*c*x**2 + 189*b*c**2*x**4 + 105*c**3*x**6)/(315*x**9)

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Giac [A]  time = 1.25488, size = 50, normalized size = 1.16 \begin{align*} -\frac{105 \, c^{3} x^{6} + 189 \, b c^{2} x^{4} + 135 \, b^{2} c x^{2} + 35 \, b^{3}}{315 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^16,x, algorithm="giac")

[Out]

-1/315*(105*c^3*x^6 + 189*b*c^2*x^4 + 135*b^2*c*x^2 + 35*b^3)/x^9