3.162 \(\int \frac{(b x^2+c x^4)^3}{x^6} \, dx\)

Optimal. Leaf size=35 \[ b^2 c x^3+b^3 x+\frac{3}{5} b c^2 x^5+\frac{c^3 x^7}{7} \]

[Out]

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

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Rubi [A]  time = 0.0172534, antiderivative size = 35, 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, 194} \[ b^2 c x^3+b^3 x+\frac{3}{5} b c^2 x^5+\frac{c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^6,x]

[Out]

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

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 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0010966, size = 35, normalized size = 1. \[ b^2 c x^3+b^3 x+\frac{3}{5} b c^2 x^5+\frac{c^3 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^6,x]

[Out]

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

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Maple [A]  time = 0.041, size = 32, normalized size = 0.9 \begin{align*}{b}^{3}x+{b}^{2}c{x}^{3}+{\frac{3\,b{c}^{2}{x}^{5}}{5}}+{\frac{{c}^{3}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^6,x)

[Out]

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

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Maxima [A]  time = 1.06005, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, b c^{2} x^{5} + b^{2} c x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.41902, size = 66, normalized size = 1.89 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, b c^{2} x^{5} + b^{2} c x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.079728, size = 32, normalized size = 0.91 \begin{align*} b^{3} x + b^{2} c x^{3} + \frac{3 b c^{2} x^{5}}{5} + \frac{c^{3} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**6,x)

[Out]

b**3*x + b**2*c*x**3 + 3*b*c**2*x**5/5 + c**3*x**7/7

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Giac [A]  time = 1.29702, size = 42, normalized size = 1.2 \begin{align*} \frac{1}{7} \, c^{3} x^{7} + \frac{3}{5} \, b c^{2} x^{5} + b^{2} c x^{3} + b^{3} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^6,x, algorithm="giac")

[Out]

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