3.161 \(\int \frac{(b x^2+c x^4)^3}{x^5} \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (b+c x^2\right )^4}{8 c} \]

[Out]

(b + c*x^2)^4/(8*c)

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Rubi [A]  time = 0.0088485, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {1584, 261} \[ \frac{\left (b+c x^2\right )^4}{8 c} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(b + c*x^2)^4/(8*c)

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0023785, size = 16, normalized size = 1. \[ \frac{\left (b+c x^2\right )^4}{8 c} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^5,x]

[Out]

(b + c*x^2)^4/(8*c)

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Maple [B]  time = 0.042, size = 36, normalized size = 2.3 \begin{align*}{\frac{{c}^{3}{x}^{8}}{8}}+{\frac{b{c}^{2}{x}^{6}}{2}}+{\frac{3\,{b}^{2}c{x}^{4}}{4}}+{\frac{{b}^{3}{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^3/x^5,x)

[Out]

1/8*c^3*x^8+1/2*b*c^2*x^6+3/4*b^2*c*x^4+1/2*b^3*x^2

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Maxima [B]  time = 0.971654, size = 47, normalized size = 2.94 \begin{align*} \frac{1}{8} \, c^{3} x^{8} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{4} \, b^{2} c x^{4} + \frac{1}{2} \, b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^5,x, algorithm="maxima")

[Out]

1/8*c^3*x^8 + 1/2*b*c^2*x^6 + 3/4*b^2*c*x^4 + 1/2*b^3*x^2

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Fricas [B]  time = 1.15507, size = 80, normalized size = 5. \begin{align*} \frac{1}{8} \, c^{3} x^{8} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{4} \, b^{2} c x^{4} + \frac{1}{2} \, b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^5,x, algorithm="fricas")

[Out]

1/8*c^3*x^8 + 1/2*b*c^2*x^6 + 3/4*b^2*c*x^4 + 1/2*b^3*x^2

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Sympy [B]  time = 0.075562, size = 37, normalized size = 2.31 \begin{align*} \frac{b^{3} x^{2}}{2} + \frac{3 b^{2} c x^{4}}{4} + \frac{b c^{2} x^{6}}{2} + \frac{c^{3} x^{8}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**3/x**5,x)

[Out]

b**3*x**2/2 + 3*b**2*c*x**4/4 + b*c**2*x**6/2 + c**3*x**8/8

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Giac [B]  time = 1.20831, size = 47, normalized size = 2.94 \begin{align*} \frac{1}{8} \, c^{3} x^{8} + \frac{1}{2} \, b c^{2} x^{6} + \frac{3}{4} \, b^{2} c x^{4} + \frac{1}{2} \, b^{3} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^3/x^5,x, algorithm="giac")

[Out]

1/8*c^3*x^8 + 1/2*b*c^2*x^6 + 3/4*b^2*c*x^4 + 1/2*b^3*x^2