3.170 \(\int \frac{(b x^2+c x^4)^3}{x^{14}} \, dx\)

Optimal. Leaf size=39 \[ -\frac{3 b^2 c}{5 x^5}-\frac{b^3}{7 x^7}-\frac{b c^2}{x^3}-\frac{c^3}{x} \]

[Out]

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

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Rubi [A]  time = 0.0213671, antiderivative size = 39, 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} \[ -\frac{3 b^2 c}{5 x^5}-\frac{b^3}{7 x^7}-\frac{b c^2}{x^3}-\frac{c^3}{x} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^14,x]

[Out]

-b^3/(7*x^7) - (3*b^2*c)/(5*x^5) - (b*c^2)/x^3 - 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^{14}} \, dx &=\int \frac{\left (b+c x^2\right )^3}{x^8} \, dx\\ &=\int \left (\frac{b^3}{x^8}+\frac{3 b^2 c}{x^6}+\frac{3 b c^2}{x^4}+\frac{c^3}{x^2}\right ) \, dx\\ &=-\frac{b^3}{7 x^7}-\frac{3 b^2 c}{5 x^5}-\frac{b c^2}{x^3}-\frac{c^3}{x}\\ \end{align*}

Mathematica [A]  time = 0.0038905, size = 39, normalized size = 1. \[ -\frac{3 b^2 c}{5 x^5}-\frac{b^3}{7 x^7}-\frac{b c^2}{x^3}-\frac{c^3}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^14,x]

[Out]

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

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Maple [A]  time = 0.049, size = 36, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{7\,{x}^{7}}}-{\frac{3\,{b}^{2}c}{5\,{x}^{5}}}-{\frac{b{c}^{2}}{{x}^{3}}}-{\frac{{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^14,x)

[Out]

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

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Maxima [A]  time = 0.973394, size = 50, normalized size = 1.28 \begin{align*} -\frac{35 \, c^{3} x^{6} + 35 \, b c^{2} x^{4} + 21 \, b^{2} c x^{2} + 5 \, b^{3}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^14,x, algorithm="maxima")

[Out]

-1/35*(35*c^3*x^6 + 35*b*c^2*x^4 + 21*b^2*c*x^2 + 5*b^3)/x^7

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Fricas [A]  time = 1.46214, size = 84, normalized size = 2.15 \begin{align*} -\frac{35 \, c^{3} x^{6} + 35 \, b c^{2} x^{4} + 21 \, b^{2} c x^{2} + 5 \, b^{3}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^14,x, algorithm="fricas")

[Out]

-1/35*(35*c^3*x^6 + 35*b*c^2*x^4 + 21*b^2*c*x^2 + 5*b^3)/x^7

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Sympy [A]  time = 0.425772, size = 39, normalized size = 1. \begin{align*} - \frac{5 b^{3} + 21 b^{2} c x^{2} + 35 b c^{2} x^{4} + 35 c^{3} x^{6}}{35 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**14,x)

[Out]

-(5*b**3 + 21*b**2*c*x**2 + 35*b*c**2*x**4 + 35*c**3*x**6)/(35*x**7)

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Giac [A]  time = 1.28919, size = 50, normalized size = 1.28 \begin{align*} -\frac{35 \, c^{3} x^{6} + 35 \, b c^{2} x^{4} + 21 \, b^{2} c x^{2} + 5 \, b^{3}}{35 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^14,x, algorithm="giac")

[Out]

-1/35*(35*c^3*x^6 + 35*b*c^2*x^4 + 21*b^2*c*x^2 + 5*b^3)/x^7