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

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

[Out]

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

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Rubi [A]  time = 0.0208536, 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} \[ \frac{3}{5} b^2 c x^5+\frac{b^3 x^3}{3}+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^3/x^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0019144, size = 43, normalized size = 1. \[ \frac{3}{5} b^2 c x^5+\frac{b^3 x^3}{3}+\frac{3}{7} b c^2 x^7+\frac{c^3 x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.043, size = 36, normalized size = 0.8 \begin{align*}{\frac{{b}^{3}{x}^{3}}{3}}+{\frac{3\,{b}^{2}c{x}^{5}}{5}}+{\frac{3\,b{c}^{2}{x}^{7}}{7}}+{\frac{{c}^{3}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05931, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{9} \, c^{3} x^{9} + \frac{3}{7} \, b c^{2} x^{7} + \frac{3}{5} \, b^{2} c x^{5} + \frac{1}{3} \, b^{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^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.199, size = 80, normalized size = 1.86 \begin{align*} \frac{1}{9} \, c^{3} x^{9} + \frac{3}{7} \, b c^{2} x^{7} + \frac{3}{5} \, b^{2} c x^{5} + \frac{1}{3} \, b^{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^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.074509, size = 39, normalized size = 0.91 \begin{align*} \frac{b^{3} x^{3}}{3} + \frac{3 b^{2} c x^{5}}{5} + \frac{3 b c^{2} x^{7}}{7} + \frac{c^{3} x^{9}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b**3*x**3/3 + 3*b**2*c*x**5/5 + 3*b*c**2*x**7/7 + c**3*x**9/9

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Giac [A]  time = 1.29417, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{9} \, c^{3} x^{9} + \frac{3}{7} \, b c^{2} x^{7} + \frac{3}{5} \, b^{2} c x^{5} + \frac{1}{3} \, b^{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^4,x, algorithm="giac")

[Out]

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