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

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

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Rubi [A]  time = 0.00, antiderivative size = 12, normalized size of antiderivative = 1.00, 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} \begin {gather*} a x+\frac {b x^2}{2} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 12, normalized size = 1.00 \begin {gather*} a x+\frac {b x^2}{2} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 12, normalized size = 1.00 \begin {gather*} a x+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a*x^2 + b*x^3)/x^2,x]

[Out]

a*x + (b*x^2)/2

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fricas [A]  time = 0.50, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

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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giac [A]  time = 0.15, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

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

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maple [A]  time = 0.05, size = 11, normalized size = 0.92 \begin {gather*} \frac {1}{2} b \,x^{2}+a x \end {gather*}

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 = 1.34, size = 10, normalized size = 0.83 \begin {gather*} \frac {1}{2} \, b x^{2} + a x \end {gather*}

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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mupad [B]  time = 0.02, size = 10, normalized size = 0.83 \begin {gather*} \frac {b\,x^2}{2}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2 + b*x^3)/x^2,x)

[Out]

a*x + (b*x^2)/2

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sympy [A]  time = 0.06, size = 8, normalized size = 0.67 \begin {gather*} a x + \frac {b x^{2}}{2} \end {gather*}

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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