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

Optimal. Leaf size=12 \[ b x+\frac{c x^3}{3} \]

[Out]

b*x + (c*x^3)/3

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Rubi [A]  time = 0.0044172, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {14} \[ b x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(b*x^2 + c*x^4)/x^2,x]

[Out]

b*x + (c*x^3)/3

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0003584, size = 12, normalized size = 1. \[ b x+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*x^2 + c*x^4)/x^2,x]

[Out]

b*x + (c*x^3)/3

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Maple [A]  time = 0.041, size = 11, normalized size = 0.9 \begin{align*} bx+{\frac{c{x}^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x+1/3*c*x^3

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Maxima [A]  time = 0.96898, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, c x^{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + b*x

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Fricas [A]  time = 1.11098, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{3} \, c x^{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + b*x

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Sympy [A]  time = 0.062623, size = 8, normalized size = 0.67 \begin{align*} b x + \frac{c x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + c*x**3/3

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Giac [A]  time = 1.25198, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{3} \, c x^{3} + b x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*c*x^3 + b*x