\(\int \frac {a x^2+b x^3}{x^2} \, dx\) [207]

Optimal result

Integrand size = 15, antiderivative size = 12 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=a x+\frac {b x^2}{2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14} \[ \int \frac {a x^2+b x^3}{x^2} \, dx=a x+\frac {b x^2}{2} \]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=a x+\frac {b x^2}{2} \]

[In]

Integrate[(a*x^2 + b*x^3)/x^2,x]

[Out]

a*x + (b*x^2)/2

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(\frac {x \left (b x +2 a \right )}{2}\)	\(11\)
default	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
risch	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
parallelrisch	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
parts	\(a x +\frac {1}{2} b \,x^{2}\)	\(11\)
norman	\(\frac {a \,x^{2}+\frac {1}{2} b \,x^{3}}{x}\)	\(17\)

[In]

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

[Out]

1/2*x*(b*x+2*a)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

1/2*b*x^2 + a*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.67 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=a x + \frac {b x^{2}}{2} \]

[In]

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

[Out]

a*x + b*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

1/2*b*x^2 + a*x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=\frac {1}{2} \, b x^{2} + a x \]

[In]

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

[Out]

1/2*b*x^2 + a*x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {a x^2+b x^3}{x^2} \, dx=\frac {b\,x^2}{2}+a\,x \]

[In]

int((a*x^2 + b*x^3)/x^2,x)

[Out]

a*x + (b*x^2)/2