3.3.6 \(\int \frac {a x^2+b x^3}{x} \, dx\) [206]

Optimal. Leaf size=17 \[ \frac {a x^2}{2}+\frac {b x^3}{3} \]

[Out]

1/2*a*x^2+1/3*b*x^3

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 17, 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*} \frac {a x^2}{2}+\frac {b x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3)/x,x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {a x^2}{2}+\frac {b x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3)/x,x]

[Out]

(a*x^2)/2 + (b*x^3)/3

________________________________________________________________________________________

Maple [A]
time = 0.02, size = 14, normalized size = 0.82

method	result	size
gosper	\(\frac {x^{2} \left (2 b x +3 a \right )}{6}\)	\(14\)
default	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}\)	\(14\)
norman	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}\)	\(14\)
risch	\(\frac {1}{2} a \,x^{2}+\frac {1}{3} b \,x^{3}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*x^2+1/3*b*x^3

________________________________________________________________________________________

Maxima [A]
time = 0.28, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*x^3 + 1/2*a*x^2

________________________________________________________________________________________

Fricas [A]
time = 1.14, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*x^3 + 1/2*a*x^2

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**2/2 + b*x**3/3

________________________________________________________________________________________

Giac [A]
time = 1.12, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, b x^{3} + \frac {1}{2} \, a x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b*x^3 + 1/2*a*x^2

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {x^2\,\left (3\,a+2\,b\,x\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2 + b*x^3)/x,x)

[Out]

(x^2*(3*a + 2*b*x))/6

________________________________________________________________________________________