3.62.47 \(\int \frac {50+11 x-400 x^2-72 x^3+6 x \log (x)}{5 x} \, dx\)

Optimal. Leaf size=22 \[ 3+x-2 \left (5+\frac {3 x}{5}\right ) \left (4 x^2-\log (x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 14, 2295} \begin {gather*} -\frac {24 x^3}{5}-40 x^2+x+\frac {6}{5} x \log (x)+10 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(50 + 11*x - 400*x^2 - 72*x^3 + 6*x*Log[x])/(5*x),x]

[Out]

x - 40*x^2 - (24*x^3)/5 + 10*Log[x] + (6*x*Log[x])/5

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {50+11 x-400 x^2-72 x^3+6 x \log (x)}{x} \, dx\\ &=\frac {1}{5} \int \left (\frac {50+11 x-400 x^2-72 x^3}{x}+6 \log (x)\right ) \, dx\\ &=\frac {1}{5} \int \frac {50+11 x-400 x^2-72 x^3}{x} \, dx+\frac {6}{5} \int \log (x) \, dx\\ &=-\frac {6 x}{5}+\frac {6}{5} x \log (x)+\frac {1}{5} \int \left (11+\frac {50}{x}-400 x-72 x^2\right ) \, dx\\ &=x-40 x^2-\frac {24 x^3}{5}+10 \log (x)+\frac {6}{5} x \log (x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 25, normalized size = 1.14 \begin {gather*} x-40 x^2-\frac {24 x^3}{5}+10 \log (x)+\frac {6}{5} x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 + 11*x - 400*x^2 - 72*x^3 + 6*x*Log[x])/(5*x),x]

[Out]

x - 40*x^2 - (24*x^3)/5 + 10*Log[x] + (6*x*Log[x])/5

________________________________________________________________________________________

fricas [A]  time = 0.52, size = 21, normalized size = 0.95 \begin {gather*} -\frac {24}{5} \, x^{3} - 40 \, x^{2} + \frac {2}{5} \, {\left (3 \, x + 25\right )} \log \relax (x) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(6*x*log(x)-72*x^3-400*x^2+11*x+50)/x,x, algorithm="fricas")

[Out]

-24/5*x^3 - 40*x^2 + 2/5*(3*x + 25)*log(x) + x

________________________________________________________________________________________

giac [A]  time = 0.14, size = 21, normalized size = 0.95 \begin {gather*} -\frac {24}{5} \, x^{3} - 40 \, x^{2} + \frac {6}{5} \, x \log \relax (x) + x + 10 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(6*x*log(x)-72*x^3-400*x^2+11*x+50)/x,x, algorithm="giac")

[Out]

-24/5*x^3 - 40*x^2 + 6/5*x*log(x) + x + 10*log(x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 22, normalized size = 1.00

method	result	size
default	\(\frac {6 x \ln \relax (x )}{5}+x -\frac {24 x^{3}}{5}-40 x^{2}+10 \ln \relax (x )\)	\(22\)
norman	\(\frac {6 x \ln \relax (x )}{5}+x -\frac {24 x^{3}}{5}-40 x^{2}+10 \ln \relax (x )\)	\(22\)
risch	\(\frac {6 x \ln \relax (x )}{5}+x -\frac {24 x^{3}}{5}-40 x^{2}+10 \ln \relax (x )\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(6*x*ln(x)-72*x^3-400*x^2+11*x+50)/x,x,method=_RETURNVERBOSE)

[Out]

6/5*x*ln(x)+x-24/5*x^3-40*x^2+10*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 21, normalized size = 0.95 \begin {gather*} -\frac {24}{5} \, x^{3} - 40 \, x^{2} + \frac {6}{5} \, x \log \relax (x) + x + 10 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(6*x*log(x)-72*x^3-400*x^2+11*x+50)/x,x, algorithm="maxima")

[Out]

-24/5*x^3 - 40*x^2 + 6/5*x*log(x) + x + 10*log(x)

________________________________________________________________________________________

mupad [B]  time = 4.04, size = 21, normalized size = 0.95 \begin {gather*} x+10\,\ln \relax (x)+\frac {6\,x\,\ln \relax (x)}{5}-40\,x^2-\frac {24\,x^3}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((11*x)/5 + (6*x*log(x))/5 - 80*x^2 - (72*x^3)/5 + 10)/x,x)

[Out]

x + 10*log(x) + (6*x*log(x))/5 - 40*x^2 - (24*x^3)/5

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 26, normalized size = 1.18 \begin {gather*} - \frac {24 x^{3}}{5} - 40 x^{2} + \frac {6 x \log {\relax (x )}}{5} + x + 10 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(6*x*ln(x)-72*x**3-400*x**2+11*x+50)/x,x)

[Out]

-24*x**3/5 - 40*x**2 + 6*x*log(x)/5 + x + 10*log(x)

________________________________________________________________________________________