3.166 \(\int \frac{(b x^2+c x^4)^3}{x^{10}} \, dx\)

Optimal. Leaf size=37 \[ -\frac{3 b^2 c}{x}-\frac{b^3}{3 x^3}+3 b c^2 x+\frac{c^3 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0193349, antiderivative size = 37, 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} \[ -\frac{3 b^2 c}{x}-\frac{b^3}{3 x^3}+3 b c^2 x+\frac{c^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^10,x]

[Out]

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

Mathematica [A]  time = 0.003986, size = 37, normalized size = 1. \[ -\frac{3 b^2 c}{x}-\frac{b^3}{3 x^3}+3 b c^2 x+\frac{c^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^10,x]

[Out]

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

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Maple [A]  time = 0.048, size = 34, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{3\,{x}^{3}}}-3\,{\frac{{b}^{2}c}{x}}+3\,b{c}^{2}x+{\frac{{c}^{3}{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^10,x)

[Out]

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

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Maxima [A]  time = 0.954365, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{3} \, c^{3} x^{3} + 3 \, b c^{2} x - \frac{9 \, b^{2} c x^{2} + b^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^10,x, algorithm="maxima")

[Out]

1/3*c^3*x^3 + 3*b*c^2*x - 1/3*(9*b^2*c*x^2 + b^3)/x^3

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Fricas [A]  time = 1.43342, size = 72, normalized size = 1.95 \begin{align*} \frac{c^{3} x^{6} + 9 \, b c^{2} x^{4} - 9 \, b^{2} c x^{2} - b^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^10,x, algorithm="fricas")

[Out]

1/3*(c^3*x^6 + 9*b*c^2*x^4 - 9*b^2*c*x^2 - b^3)/x^3

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Sympy [A]  time = 0.323275, size = 34, normalized size = 0.92 \begin{align*} 3 b c^{2} x + \frac{c^{3} x^{3}}{3} - \frac{b^{3} + 9 b^{2} 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)**3/x**10,x)

[Out]

3*b*c**2*x + c**3*x**3/3 - (b**3 + 9*b**2*c*x**2)/(3*x**3)

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Giac [A]  time = 1.30085, size = 46, normalized size = 1.24 \begin{align*} \frac{1}{3} \, c^{3} x^{3} + 3 \, b c^{2} x - \frac{9 \, b^{2} c x^{2} + b^{3}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^10,x, algorithm="giac")

[Out]

1/3*c^3*x^3 + 3*b*c^2*x - 1/3*(9*b^2*c*x^2 + b^3)/x^3