\(\int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+(-190+73 x-7 x^2) \log (\frac {-38+7 x}{-5+x})}{190-73 x+7 x^2} \, dx\) [10031]

Optimal result

Integrand size = 56, antiderivative size = 29 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=-4-x+(9-x)^2 x-x \log \left (7+\frac {3}{5-x}\right ) \]

[Out]

(9-x)^2*x-x-4-ln(7+3/(5-x))*x

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.90, number of steps used = 24, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6860, 630, 31, 646, 717, 715, 2535} \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^3-18 x^2+80 x-\frac {38}{7} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {38}{7} \log (5-x) \]

[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 - (38*Log[38 - 7*x])/7 + ((38 - 7*x)*Log[(38 - 7*x)/(5 - x)])/7 + (38*Log[5 - x])/7

Rule 31

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

Rule 630

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 646

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 715

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 717

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 2535

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.), x_Symbol] :> Simp[(a +
b*x)*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^p/b), x] - Dist[B*n*p*((b*c - a*d)/b), Int[(A + B*Log[e*((a + b*
x)/(c + d*x))^n])^(p - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && NeQ[b*c - a*d, 0] && IGtQ
[p, 0]

Rule 6860

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{align*} \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}+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {3}{7} \int \frac {1}{-5+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 {375554}{21} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {337514}{21} \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 {932558}{49} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )-\frac {106712}{7} \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-\frac {38}{7} \log (38-7 x)+\frac {1}{7} (38-7 x) \log \left (\frac {38-7 x}{5-x}\right )+\frac {38}{7} \log (5-x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.83 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=80 x-18 x^2+x^3-\frac {38}{7} \log (38-7 x)+\frac {38}{7} \log (5-x)-\frac {1}{7} (-38+7 x) \log \left (\frac {-38+7 x}{-5+x}\right ) \]

[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

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97

method	result	size
norman	\(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x\)	\(28\)
risch	\(x^{3}+80 x -18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x\)	\(28\)
parallelrisch	\(x^{3}+\frac {36186}{7}-18 x^{2}-\ln \left (\frac {7 x -38}{-5+x}\right ) x +80 x\)	\(29\)
derivativedivides	\(-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}+\left (-5+x \right )^{3}-3 \left (-5+x \right )^{2}+125-25 x -\frac {38 \ln \left (7-\frac {3}{-5+x}\right )}{7}\)	\(54\)
default	\(-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}+\left (-5+x \right )^{3}-3 \left (-5+x \right )^{2}+125-25 x -\frac {38 \ln \left (7-\frac {3}{-5+x}\right )}{7}\)	\(54\)
parts	\(x^{3}-18 x^{2}+80 x +5 \ln \left (-5+x \right )-\frac {38 \ln \left (7 x -38\right )}{7}-\frac {3 \ln \left (-\frac {3}{-5+x}\right )}{7}-\frac {\ln \left (7-\frac {3}{-5+x}\right ) \left (7-\frac {3}{-5+x}\right ) \left (-5+x \right )}{7}\)	\(61\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 \, x^{2} - x \log \left (\frac {7 \, x - 38}{x - 5}\right ) + 80 \, x \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 x^{2} - x \log {\left (\frac {7 x - 38}{x - 5} \right )} + 80 x \]

[In]

integrate(((-7*x**2+73*x-190)*ln((7*x-38)/(-5+x))+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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (25) = 50\).

Time = 0.24 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.97 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=x^{3} - 18 \, x^{2} - \frac {1}{21} \, {\left (21 \, x + 1330 \, \log \left (x - 5\right ) - 114\right )} \log \left (7 \, x - 38\right ) + \frac {190}{3} \, \log \left (7 \, x - 38\right )^{2} + {\left (x - 5\right )} \log \left (x - 5\right ) - \frac {190}{3} \, \log \left (7 \, x - 38\right ) \log \left (x - 5\right ) + \frac {190}{3} \, \log \left (x - 5\right )^{2} - \frac {190}{3} \, {\left (\log \left (7 \, x - 38\right ) - \log \left (x - 5\right )\right )} \log \left (\frac {7 \, x}{x - 5} - \frac {38}{x - 5}\right ) + 80 \, x - \frac {38}{7} \, \log \left (7 \, x - 38\right ) + 5 \, \log \left (x - 5\right ) \]

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (25) = 50\).

Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 4.03 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=\frac {3 \, {\left (\frac {25 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} - \frac {359 \, {\left (7 \, x - 38\right )}}{x - 5} + 1279\right )}}{\frac {{\left (7 \, x - 38\right )}^{3}}{{\left (x - 5\right )}^{3}} - \frac {21 \, {\left (7 \, x - 38\right )}^{2}}{{\left (x - 5\right )}^{2}} + \frac {147 \, {\left (7 \, x - 38\right )}}{x - 5} - 343} + \frac {3 \, \log \left (\frac {7 \, x - 38}{x - 5}\right )}{\frac {7 \, x - 38}{x - 5} - 7} - 5 \, \log \left (\frac {7 \, x - 38}{x - 5}\right ) \]

[In]

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

Mupad [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {15200-12683 x+3758 x^2-471 x^3+21 x^4+\left (-190+73 x-7 x^2\right ) \log \left (\frac {-38+7 x}{-5+x}\right )}{190-73 x+7 x^2} \, dx=80\,x-x\,\ln \left (\frac {7\,x-38}{x-5}\right )-18\,x^2+x^3 \]

[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