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

Optimal. Leaf size=15 \[ \frac{b x^2}{2}-\frac{a}{x} \]

[Out]

-(a/x) + (b*x^2)/2

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Rubi [A]  time = 0.0045238, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{b x^2}{2}-\frac{a}{x} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.0009421, size = 15, normalized size = 1. \[ \frac{b x^2}{2}-\frac{a}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)/x^2,x]

[Out]

-(a/x) + (b*x^2)/2

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Maple [A]  time = 0.003, size = 14, normalized size = 0.9 \begin{align*} -{\frac{a}{x}}+{\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)

[Out]

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

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Maxima [A]  time = 0.959778, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \, b x^{2} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 - a/x

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Fricas [A]  time = 1.84135, size = 28, normalized size = 1.87 \begin{align*} \frac{b x^{3} - 2 \, a}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*x^3 - 2*a)/x

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Sympy [A]  time = 0.29559, size = 8, normalized size = 0.53 \begin{align*} - \frac{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)

[Out]

-a/x + b*x**2/2

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Giac [A]  time = 1.12264, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{2} \, b x^{2} - \frac{a}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*x^2 - a/x