3.101.31 \(\int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+(-190+73 x-7 x^2) \log (\frac {-38+7 x}{-5+x})}{190-73 x+7 x^2} \, dx\)

Optimal. Leaf size=29 \[ -4-x+(9-x)^2 x-x \log \left (7+\frac {3}{5-x}\right ) \]

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Rubi [A]  time = 0.18, antiderivative size = 48, normalized size of antiderivative = 1.66, number of steps used = 24, number of rules used = 7, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6728, 616, 31, 632, 703, 701, 2486} \begin {gather*} x^3-18 x^2+80 x-5 \log (38-7 x)+(5-x) \log \left (\frac {38-7 x}{5-x}\right )+5 \log (5-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15200 - 12683*x + 3758*x^2 - 471*x^3 + 21*x^4 + (-190 + 73*x - 7*x^2)*Log[(-38 + 7*x)/(-5 + x)])/(190 - 7
3*x + 7*x^2),x]

[Out]

80*x - 18*x^2 + x^3 - 5*Log[38 - 7*x] + (5 - x)*Log[(38 - 7*x)/(5 - x)] + 5*Log[5 - x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 616

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, Int[1/Simp
[b/2 - q/2 + c*x, x], x], x] - Dist[c/q, Int[1/Simp[b/2 + q/2 + c*x, x], x], x]] /; FreeQ[{a, b, c}, x] && NeQ
[b^2 - 4*a*c, 0] && PosQ[b^2 - 4*a*c] && PerfectSquareQ[b^2 - 4*a*c]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 701

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)
^m, a + b*x + c*x^2, x], x] /; FreeQ[{a, b, c, d, e}, 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] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 703

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

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {15200}{190-73 x+7 x^2}-\frac {12683 x}{190-73 x+7 x^2}+\frac {3758 x^2}{190-73 x+7 x^2}-\frac {471 x^3}{190-73 x+7 x^2}+\frac {21 x^4}{190-73 x+7 x^2}-\log \left (\frac {-38+7 x}{-5+x}\right )\right ) \, dx\\ &=21 \int \frac {x^4}{190-73 x+7 x^2} \, dx-471 \int \frac {x^3}{190-73 x+7 x^2} \, dx+3758 \int \frac {x^2}{190-73 x+7 x^2} \, dx-12683 \int \frac {x}{190-73 x+7 x^2} \, dx+15200 \int \frac {1}{190-73 x+7 x^2} \, dx-\int \log \left (\frac {-38+7 x}{-5+x}\right ) \, dx\\ &=\frac {3758 x}{7}+(5-x) \log \left (\frac {38-7 x}{5-x}\right )+3 \int \frac {1}{-38+7 x} \, dx+21 \int \left (\frac {3999}{343}+\frac {73 x}{49}+\frac {x^2}{7}-\frac {759810-194837 x}{343 \left (190-73 x+7 x^2\right )}\right ) \, dx-471 \int \left (\frac {73}{49}+\frac {x}{7}-\frac {13870-3999 x}{49 \left (190-73 x+7 x^2\right )}\right ) \, dx+\frac {3758}{7} \int \frac {-190+73 x}{190-73 x+7 x^2} \, dx+\frac {106400}{3} \int \frac {1}{-38+7 x} \, dx-\frac {106400}{3} \int \frac {1}{-35+7 x} \, dx+\frac {443905}{3} \int \frac {1}{-35+7 x} \, dx-\frac {481954}{3} \int \frac {1}{-38+7 x} \, dx\\ &=80 x-18 x^2+x^3-\frac {375545}{21} \log (38-7 x)+(5-x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {48215}{3} \log (5-x)-\frac {3}{49} \int \frac {759810-194837 x}{190-73 x+7 x^2} \, dx+\frac {471}{49} \int \frac {13870-3999 x}{190-73 x+7 x^2} \, dx-\frac {657650}{3} \int \frac {1}{-35+7 x} \, dx+\frac {5426552}{21} \int \frac {1}{-38+7 x} \, dx\\ &=80 x-18 x^2+x^3+\frac {932579}{49} \log (38-7 x)+(5-x) \log \left (\frac {38-7 x}{5-x}\right )-15245 \log (5-x)-30625 \int \frac {1}{-35+7 x} \, dx+\frac {2085136}{49} \int \frac {1}{-38+7 x} \, dx+137375 \int \frac {1}{-35+7 x} \, dx-\frac {8614904}{49} \int \frac {1}{-38+7 x} \, dx\\ &=80 x-18 x^2+x^3-5 \log (38-7 x)+(5-x) \log \left (\frac {38-7 x}{5-x}\right )+5 \log (5-x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 53, normalized size = 1.83 \begin {gather*} 80 x-18 x^2+x^3-\frac {38}{7} \log (38-7 x)+\frac {38}{7} \log (5-x)-\frac {1}{7} (-38+7 x) \log \left (\frac {-38+7 x}{-5+x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15200 - 12683*x + 3758*x^2 - 471*x^3 + 21*x^4 + (-190 + 73*x - 7*x^2)*Log[(-38 + 7*x)/(-5 + x)])/(1
90 - 73*x + 7*x^2),x]

[Out]

80*x - 18*x^2 + x^3 - (38*Log[38 - 7*x])/7 + (38*Log[5 - x])/7 - ((-38 + 7*x)*Log[(-38 + 7*x)/(-5 + x)])/7

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fricas [A]  time = 1.50, size = 27, normalized size = 0.93 \begin {gather*} x^{3} - 18 \, x^{2} - x \log \left (\frac {7 \, x - 38}{x - 5}\right ) + 80 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^2+73*x-190)*log((7*x-38)/(x-5))+21*x^4-471*x^3+3758*x^2-12683*x+15200)/(7*x^2-73*x+190),x, al
gorithm="fricas")

[Out]

x^3 - 18*x^2 - x*log((7*x - 38)/(x - 5)) + 80*x

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giac [B]  time = 0.23, size = 117, normalized size = 4.03 \begin {gather*} \frac {3 \, {\left (\frac {25 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} - \frac {359 \, {\left (7 \, x - 38\right )}}{x - 5} + 1279\right )}}{\frac {{\left (7 \, x - 38\right )}^{3}}{{\left (x - 5\right )}^{3}} - \frac {21 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} + \frac {147 \, {\left (7 \, x - 38\right )}}{x - 5} - 343} + \frac {3 \, \log \left (\frac {7 \, x - 38}{x - 5}\right )}{\frac {7 \, x - 38}{x - 5} - 7} - 5 \, \log \left (\frac {7 \, x - 38}{x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^2+73*x-190)*log((7*x-38)/(x-5))+21*x^4-471*x^3+3758*x^2-12683*x+15200)/(7*x^2-73*x+190),x, al
gorithm="giac")

[Out]

3*(25*(7*x - 38)^2/(x - 5)^2 - 359*(7*x - 38)/(x - 5) + 1279)/((7*x - 38)^3/(x - 5)^3 - 21*(7*x - 38)^2/(x - 5
)^2 + 147*(7*x - 38)/(x - 5) - 343) + 3*log((7*x - 38)/(x - 5))/((7*x - 38)/(x - 5) - 7) - 5*log((7*x - 38)/(x
- 5))

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maple [A]  time = 0.17, size = 28, normalized size = 0.97

method	result	size
norman	\(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{x -5}\right ) x\)	\(28\)
risch	\(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{x -5}\right ) x\)	\(28\)
derivativedivides	\(-\frac {\ln \left (7-\frac {3}{x -5}\right ) \left (7-\frac {3}{x -5}\right ) \left (x -5\right )}{7}-\frac {38 \ln \left (7-\frac {3}{x -5}\right )}{7}+\left (x -5\right )^{3}-3 \left (x -5\right )^{2}-25 x +125\)	\(54\)
default	\(-\frac {\ln \left (7-\frac {3}{x -5}\right ) \left (7-\frac {3}{x -5}\right ) \left (x -5\right )}{7}-\frac {38 \ln \left (7-\frac {3}{x -5}\right )}{7}+\left (x -5\right )^{3}-3 \left (x -5\right )^{2}-25 x +125\)	\(54\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-7*x^2+73*x-190)*ln((7*x-38)/(x-5))+21*x^4-471*x^3+3758*x^2-12683*x+15200)/(7*x^2-73*x+190),x,method=_RE
TURNVERBOSE)

[Out]

x^3+80*x-18*x^2-ln((7*x-38)/(x-5))*x

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maxima [B]  time = 0.41, size = 115, normalized size = 3.97 \begin {gather*} x^{3} - 18 \, x^{2} - \frac {1}{21} \, {\left (21 \, x + 1330 \, \log \left (x - 5\right ) - 114\right )} \log \left (7 \, x - 38\right ) + \frac {190}{3} \, \log \left (7 \, x - 38\right )^{2} + {\left (x - 5\right )} \log \left (x - 5\right ) - \frac {190}{3} \, \log \left (7 \, x - 38\right ) \log \left (x - 5\right ) + \frac {190}{3} \, \log \left (x - 5\right )^{2} - \frac {190}{3} \, {\left (\log \left (7 \, x - 38\right ) - \log \left (x - 5\right )\right )} \log \left (\frac {7 \, x}{x - 5} - \frac {38}{x - 5}\right ) + 80 \, x - \frac {38}{7} \, \log \left (7 \, x - 38\right ) + 5 \, \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x^2+73*x-190)*log((7*x-38)/(x-5))+21*x^4-471*x^3+3758*x^2-12683*x+15200)/(7*x^2-73*x+190),x, al
gorithm="maxima")

[Out]

x^3 - 18*x^2 - 1/21*(21*x + 1330*log(x - 5) - 114)*log(7*x - 38) + 190/3*log(7*x - 38)^2 + (x - 5)*log(x - 5)
- 190/3*log(7*x - 38)*log(x - 5) + 190/3*log(x - 5)^2 - 190/3*(log(7*x - 38) - log(x - 5))*log(7*x/(x - 5) - 3
8/(x - 5)) + 80*x - 38/7*log(7*x - 38) + 5*log(x - 5)

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mupad [B]  time = 0.28, size = 27, normalized size = 0.93 \begin {gather*} 80\,x-x\,\ln \left (\frac {7\,x-38}{x-5}\right )-18\,x^2+x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12683*x + log((7*x - 38)/(x - 5))*(7*x^2 - 73*x + 190) - 3758*x^2 + 471*x^3 - 21*x^4 - 15200)/(7*x^2 - 7
3*x + 190),x)

[Out]

80*x - x*log((7*x - 38)/(x - 5)) - 18*x^2 + x^3

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sympy [A]  time = 0.17, size = 22, normalized size = 0.76 \begin {gather*} x^{3} - 18 x^{2} - x \log {\left (\frac {7 x - 38}{x - 5} \right )} + 80 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-7*x**2+73*x-190)*ln((7*x-38)/(x-5))+21*x**4-471*x**3+3758*x**2-12683*x+15200)/(7*x**2-73*x+190),x
)

[Out]

x**3 - 18*x**2 - x*log((7*x - 38)/(x - 5)) + 80*x

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