3.102.75 \(\int \frac {1-103 x^2+50 x^3-\log (x)}{x^2} \, dx\)

Optimal. Leaf size=27 \[ 3+e^3+25 (-2+x)^2-\frac {3 x^2-\log (x)}{x} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.56, number of steps used = 5, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {14, 2304} \begin {gather*} 25 x^2-103 x+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 103*x^2 + 50*x^3 - Log[x])/x^2,x]

[Out]

-103*x + 25*x^2 + Log[x]/x

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]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1-103 x^2+50 x^3}{x^2}-\frac {\log (x)}{x^2}\right ) \, dx\\ &=\int \frac {1-103 x^2+50 x^3}{x^2} \, dx-\int \frac {\log (x)}{x^2} \, dx\\ &=\frac {1}{x}+\frac {\log (x)}{x}+\int \left (-103+\frac {1}{x^2}+50 x\right ) \, dx\\ &=-103 x+25 x^2+\frac {\log (x)}{x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 15, normalized size = 0.56 \begin {gather*} -103 x+25 x^2+\frac {\log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 103*x^2 + 50*x^3 - Log[x])/x^2,x]

[Out]

-103*x + 25*x^2 + Log[x]/x

________________________________________________________________________________________

fricas [A]  time = 0.56, size = 17, normalized size = 0.63 \begin {gather*} \frac {25 \, x^{3} - 103 \, x^{2} + \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)+50*x^3-103*x^2+1)/x^2,x, algorithm="fricas")

[Out]

(25*x^3 - 103*x^2 + log(x))/x

________________________________________________________________________________________

giac [A]  time = 0.16, size = 15, normalized size = 0.56 \begin {gather*} 25 \, x^{2} - 103 \, x + \frac {\log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)+50*x^3-103*x^2+1)/x^2,x, algorithm="giac")

[Out]

25*x^2 - 103*x + log(x)/x

________________________________________________________________________________________

maple [A]  time = 0.02, size = 16, normalized size = 0.59

method	result	size
default	\(25 x^{2}-103 x +\frac {\ln \relax (x )}{x}\)	\(16\)
risch	\(25 x^{2}-103 x +\frac {\ln \relax (x )}{x}\)	\(16\)
norman	\(\frac {-103 x^{2}+25 x^{3}+\ln \relax (x )}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(x)+50*x^3-103*x^2+1)/x^2,x,method=_RETURNVERBOSE)

[Out]

25*x^2-103*x+ln(x)/x

________________________________________________________________________________________

maxima [A]  time = 0.38, size = 15, normalized size = 0.56 \begin {gather*} 25 \, x^{2} - 103 \, x + \frac {\log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(x)+50*x^3-103*x^2+1)/x^2,x, algorithm="maxima")

[Out]

25*x^2 - 103*x + log(x)/x

________________________________________________________________________________________

mupad [B]  time = 6.54, size = 14, normalized size = 0.52 \begin {gather*} x\,\left (25\,x-103\right )+\frac {\ln \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x) + 103*x^2 - 50*x^3 - 1)/x^2,x)

[Out]

x*(25*x - 103) + log(x)/x

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 12, normalized size = 0.44 \begin {gather*} 25 x^{2} - 103 x + \frac {\log {\relax (x )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(x)+50*x**3-103*x**2+1)/x**2,x)

[Out]

25*x**2 - 103*x + log(x)/x

________________________________________________________________________________________