3.53.24 \(\int \frac {-125+250 x^2+374 x^3}{125 x+187 x^2} \, dx\)

Optimal. Leaf size=18 \[ x^2-\log \left (\frac {x}{5+\frac {187 x}{25}}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 14, normalized size of antiderivative = 0.78, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1593, 1620} \begin {gather*} x^2-\log (x)+\log (187 x+125) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-125 + 250*x^2 + 374*x^3)/(125*x + 187*x^2),x]

[Out]

x^2 - Log[x] + Log[125 + 187*x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-125+250 x^2+374 x^3}{x (125+187 x)} \, dx\\ &=\int \left (-\frac {1}{x}+2 x+\frac {187}{125+187 x}\right ) \, dx\\ &=x^2-\log (x)+\log (125+187 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 0.78 \begin {gather*} x^2-\log (x)+\log (125+187 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-125 + 250*x^2 + 374*x^3)/(125*x + 187*x^2),x]

[Out]

x^2 - Log[x] + Log[125 + 187*x]

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fricas [A]  time = 1.39, size = 14, normalized size = 0.78 \begin {gather*} x^{2} + \log \left (187 \, x + 125\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((374*x^3+250*x^2-125)/(187*x^2+125*x),x, algorithm="fricas")

[Out]

x^2 + log(187*x + 125) - log(x)

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giac [A]  time = 0.21, size = 16, normalized size = 0.89 \begin {gather*} x^{2} + \log \left ({\left | 187 \, x + 125 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((374*x^3+250*x^2-125)/(187*x^2+125*x),x, algorithm="giac")

[Out]

x^2 + log(abs(187*x + 125)) - log(abs(x))

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maple [A]  time = 0.18, size = 15, normalized size = 0.83

method	result	size
default	\(x^{2}+\ln \left (187 x +125\right )-\ln \relax (x )\)	\(15\)
norman	\(x^{2}+\ln \left (187 x +125\right )-\ln \relax (x )\)	\(15\)
risch	\(x^{2}+\ln \left (187 x +125\right )-\ln \relax (x )\)	\(15\)
meijerg	\(\ln \left (1+\frac {187 x}{125}\right )-\ln \relax (x )-\ln \left (11\right )-\ln \left (17\right )+3 \ln \relax (5)-\frac {125 x \left (-\frac {561 x}{125}+6\right )}{561}+\frac {250 x}{187}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((374*x^3+250*x^2-125)/(187*x^2+125*x),x,method=_RETURNVERBOSE)

[Out]

x^2+ln(187*x+125)-ln(x)

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maxima [A]  time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} x^{2} + \log \left (187 \, x + 125\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((374*x^3+250*x^2-125)/(187*x^2+125*x),x, algorithm="maxima")

[Out]

x^2 + log(187*x + 125) - log(x)

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mupad [B]  time = 3.26, size = 12, normalized size = 0.67 \begin {gather*} 2\,\mathrm {atanh}\left (\frac {374\,x}{125}+1\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((250*x^2 + 374*x^3 - 125)/(125*x + 187*x^2),x)

[Out]

2*atanh((374*x)/125 + 1) + x^2

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sympy [A]  time = 0.08, size = 12, normalized size = 0.67 \begin {gather*} x^{2} - \log {\relax (x )} + \log {\left (x + \frac {125}{187} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((374*x**3+250*x**2-125)/(187*x**2+125*x),x)

[Out]

x**2 - log(x) + log(x + 125/187)

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