3.207 \(\int \frac{a x^2+b x^3}{x^2} \, dx\)

Optimal. Leaf size=12 \[ a x+\frac{b x^2}{2} \]

[Out]

a*x + (b*x^2)/2

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Rubi [A]  time = 0.0035858, 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} \[ a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3)/x^2,x]

[Out]

a*x + (b*x^2)/2

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{a x^2+b x^3}{x^2} \, dx &=\int (a+b x) \, dx\\ &=a x+\frac{b x^2}{2}\\ \end{align*}

Mathematica [A]  time = 0.000439, size = 12, normalized size = 1. \[ a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3)/x^2,x]

[Out]

a*x + (b*x^2)/2

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Maple [A]  time = 0.001, size = 11, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{2}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x+1/2*b*x^2

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Maxima [A]  time = 0.974904, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Fricas [A]  time = 0.801145, size = 23, normalized size = 1.92 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x

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Sympy [A]  time = 0.100523, size = 8, normalized size = 0.67 \begin{align*} a x + \frac{b x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a*x**2)/x**2,x)

[Out]

a*x + b*x**2/2

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Giac [A]  time = 1.13288, size = 14, normalized size = 1.17 \begin{align*} \frac{1}{2} \, b x^{2} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 + a*x