3.25.81 \(\int \frac {-175+248 x-49 x^2+(-248+346 x-98 x^2) \log (-1+x)}{-25+25 x} \, dx\)

Optimal. Leaf size=25 \[ \left (5-(2-x) \left (-2 (-3+x)+\frac {x}{25}\right )\right ) \log (-1+x) \]

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Rubi [A]  time = 0.10, antiderivative size = 26, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {6742, 698, 2395, 43} \begin {gather*} \frac {6801 \log (1-x)}{1225}-\frac {(124-49 x)^2 \log (x-1)}{1225} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-175 + 248*x - 49*x^2 + (-248 + 346*x - 98*x^2)*Log[-1 + x])/(-25 + 25*x),x]

[Out]

(6801*Log[1 - x])/1225 - ((124 - 49*x)^2*Log[-1 + x])/1225

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-175+248 x-49 x^2}{25 (-1+x)}-\frac {2}{25} (-124+49 x) \log (-1+x)\right ) \, dx\\ &=\frac {1}{25} \int \frac {-175+248 x-49 x^2}{-1+x} \, dx-\frac {2}{25} \int (-124+49 x) \log (-1+x) \, dx\\ &=-\frac {(124-49 x)^2 \log (-1+x)}{1225}+\frac {\int \frac {(-124+49 x)^2}{-1+x} \, dx}{1225}+\frac {1}{25} \int \left (199+\frac {24}{-1+x}-49 x\right ) \, dx\\ &=\frac {199 x}{25}-\frac {49 x^2}{50}+\frac {24}{25} \log (1-x)-\frac {(124-49 x)^2 \log (-1+x)}{1225}+\frac {\int \left (-9751+\frac {5625}{-1+x}+2401 x\right ) \, dx}{1225}\\ &=\frac {6801 \log (1-x)}{1225}-\frac {(124-49 x)^2 \log (-1+x)}{1225}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.03, size = 35, normalized size = 1.40 \begin {gather*} \frac {1}{25} \left (73 \log (1-x)-248 \log (-1+x)+248 x \log (-1+x)-49 x^2 \log (-1+x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-175 + 248*x - 49*x^2 + (-248 + 346*x - 98*x^2)*Log[-1 + x])/(-25 + 25*x),x]

[Out]

(73*Log[1 - x] - 248*Log[-1 + x] + 248*x*Log[-1 + x] - 49*x^2*Log[-1 + x])/25

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fricas [A]  time = 0.84, size = 16, normalized size = 0.64 \begin {gather*} -\frac {1}{25} \, {\left (49 \, x^{2} - 248 \, x + 175\right )} \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x^2+346*x-248)*log(x-1)-49*x^2+248*x-175)/(25*x-25),x, algorithm="fricas")

[Out]

-1/25*(49*x^2 - 248*x + 175)*log(x - 1)

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giac [A]  time = 0.15, size = 22, normalized size = 0.88 \begin {gather*} -\frac {1}{25} \, {\left (49 \, x^{2} - 248 \, x\right )} \log \left (x - 1\right ) - 7 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x^2+346*x-248)*log(x-1)-49*x^2+248*x-175)/(25*x-25),x, algorithm="giac")

[Out]

-1/25*(49*x^2 - 248*x)*log(x - 1) - 7*log(x - 1)

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maple [A]  time = 0.38, size = 22, normalized size = 0.88

method	result	size
risch	\(\left (-\frac {49}{25} x^{2}+\frac {248}{25} x \right ) \ln \left (x -1\right )-7 \ln \left (x -1\right )\)	\(22\)
norman	\(-7 \ln \left (x -1\right )+\frac {248 \ln \left (x -1\right ) x}{25}-\frac {49 \ln \left (x -1\right ) x^{2}}{25}\)	\(24\)
derivativedivides	\(-\frac {49 \ln \left (x -1\right ) \left (x -1\right )^{2}}{25}+6 \left (x -1\right ) \ln \left (x -1\right )+\frac {24 \ln \left (x -1\right )}{25}\)	\(28\)
default	\(-\frac {49 \ln \left (x -1\right ) \left (x -1\right )^{2}}{25}+6 \left (x -1\right ) \ln \left (x -1\right )+\frac {24 \ln \left (x -1\right )}{25}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-98*x^2+346*x-248)*ln(x-1)-49*x^2+248*x-175)/(25*x-25),x,method=_RETURNVERBOSE)

[Out]

(-49/25*x^2+248/25*x)*ln(x-1)-7*ln(x-1)

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maxima [B]  time = 0.59, size = 46, normalized size = 1.84 \begin {gather*} -\frac {49}{25} \, {\left (x^{2} + 2 \, x + 2 \, \log \left (x - 1\right )\right )} \log \left (x - 1\right ) + \frac {346}{25} \, {\left (x + \log \left (x - 1\right )\right )} \log \left (x - 1\right ) - \frac {248}{25} \, \log \left (x - 1\right )^{2} - 7 \, \log \left (x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x^2+346*x-248)*log(x-1)-49*x^2+248*x-175)/(25*x-25),x, algorithm="maxima")

[Out]

-49/25*(x^2 + 2*x + 2*log(x - 1))*log(x - 1) + 346/25*(x + log(x - 1))*log(x - 1) - 248/25*log(x - 1)^2 - 7*lo
g(x - 1)

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mupad [B]  time = 1.51, size = 16, normalized size = 0.64 \begin {gather*} -\frac {\ln \left (x-1\right )\,\left (49\,x^2-248\,x+175\right )}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - 1)*(98*x^2 - 346*x + 248) - 248*x + 49*x^2 + 175)/(25*x - 25),x)

[Out]

-(log(x - 1)*(49*x^2 - 248*x + 175))/25

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sympy [A]  time = 0.12, size = 22, normalized size = 0.88 \begin {gather*} \left (- \frac {49 x^{2}}{25} + \frac {248 x}{25}\right ) \log {\left (x - 1 \right )} - 7 \log {\left (x - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-98*x**2+346*x-248)*ln(x-1)-49*x**2+248*x-175)/(25*x-25),x)

[Out]

(-49*x**2/25 + 248*x/25)*log(x - 1) - 7*log(x - 1)

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