3.1.5 \(\int \frac {a x^2+b x^3+c x^4}{x^2} \, dx\) [5]

Optimal. Leaf size=20 \[ a x+\frac {b x^2}{2}+\frac {c x^3}{3} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {14} \begin {gather*} a x+\frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a*x^2 + b*x^3 + c*x^4)/x^2,x]

[Out]

a*x + (b*x^2)/2 + (c*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+c x^4}{x^2} \, dx &=\int \left (a+b x+c x^2\right ) \, dx\\ &=a x+\frac {b x^2}{2}+\frac {c x^3}{3}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} a x+\frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a*x^2 + b*x^3 + c*x^4)/x^2,x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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Maple [A]
time = 0.01, size = 17, normalized size = 0.85

method	result	size
default	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\)	\(17\)
risch	\(a x +\frac {1}{2} b \,x^{2}+\frac {1}{3} c \,x^{3}\)	\(17\)
gosper	\(\frac {x \left (2 c \,x^{2}+3 b x +6 a \right )}{6}\)	\(18\)
norman	\(\frac {a \,x^{2}+\frac {1}{2} b \,x^{3}+\frac {1}{3} c \,x^{4}}{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^3+a*x^2)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)/x^2,x, algorithm="maxima")

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x

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Fricas [A]
time = 0.32, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)/x^2,x, algorithm="fricas")

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x

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Sympy [A]
time = 0.01, size = 15, normalized size = 0.75 \begin {gather*} a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**3+a*x**2)/x**2,x)

[Out]

a*x + b*x**2/2 + c*x**3/3

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Giac [A]
time = 3.31, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^3+a*x^2)/x^2,x, algorithm="giac")

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x

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Mupad [B]
time = 0.03, size = 16, normalized size = 0.80 \begin {gather*} \frac {c\,x^3}{3}+\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 + c*x^4)/x^2,x)

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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