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

Optimal. Leaf size=30 \[ \frac{b^2 x^3}{3}+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7} \]

[Out]

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

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Rubi [A]  time = 0.0172618, antiderivative size = 30, 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} \[ \frac{b^2 x^3}{3}+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)^2/x^2,x]

[Out]

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

Mathematica [A]  time = 0.000866, size = 30, normalized size = 1. \[ \frac{b^2 x^3}{3}+\frac{2}{5} b c x^5+\frac{c^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)^2/x^2,x]

[Out]

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

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Maple [A]  time = 0.04, size = 25, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}{x}^{3}}{3}}+{\frac{2\,bc{x}^{5}}{5}}+{\frac{{c}^{2}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.985917, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{2}{5} \, b c x^{5} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.20079, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{2}{5} \, b c x^{5} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.070299, size = 26, normalized size = 0.87 \begin{align*} \frac{b^{2} x^{3}}{3} + \frac{2 b c x^{5}}{5} + \frac{c^{2} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.2573, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, c^{2} x^{7} + \frac{2}{5} \, b c x^{5} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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