3.88.26 \(\int \frac {-97071-17091 x+9528 x^2+2220 x^3-162 x^4-60 x^5+(-10110-4338 x-168 x^2+168 x^3+24 x^4) \log (5-x)}{-25+5 x} \, dx\)

Optimal. Leaf size=30 \[ 3 \left (3+x+\left (5+(4+x)^2\right )^2\right ) \left (5+\frac {2}{5} (-2 x+\log (5-x))\right ) \]

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Rubi [B]  time = 0.25, antiderivative size = 98, normalized size of antiderivative = 3.27, number of steps used = 17, number of rules used = 7, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.117, Rules used = {6742, 1850, 2417, 2389, 2295, 2395, 43} \begin {gather*} -\frac {12 x^5}{5}-\frac {117 x^4}{5}+\frac {6}{5} x^4 \log (5-x)-\frac {72 x^3}{5}+\frac {96}{5} x^3 \log (5-x)+\frac {3906 x^2}{5}+\frac {636}{5} x^2 \log (5-x)+\frac {19947 x}{5}-\frac {2022}{5} (5-x) \log (5-x)+\frac {12774}{5} \log (5-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-97071 - 17091*x + 9528*x^2 + 2220*x^3 - 162*x^4 - 60*x^5 + (-10110 - 4338*x - 168*x^2 + 168*x^3 + 24*x^4
)*Log[5 - x])/(-25 + 5*x),x]

[Out]

(19947*x)/5 + (3906*x^2)/5 - (72*x^3)/5 - (117*x^4)/5 - (12*x^5)/5 + (12774*Log[5 - x])/5 - (2022*(5 - x)*Log[
5 - x])/5 + (636*x^2*Log[5 - x])/5 + (96*x^3*Log[5 - x])/5 + (6*x^4*Log[5 - x])/5

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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2295

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

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

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 2417

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(Polyx_), x_Symbol] :> Int[ExpandIntegrand[Poly
x*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolynomialQ[Polyx, x]

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 {3 \left (32357+5697 x-3176 x^2-740 x^3+54 x^4+20 x^5\right )}{5 (-5+x)}+\frac {6}{5} \left (337+212 x+48 x^2+4 x^3\right ) \log (5-x)\right ) \, dx\\ &=-\left (\frac {3}{5} \int \frac {32357+5697 x-3176 x^2-740 x^3+54 x^4+20 x^5}{-5+x} \, dx\right )+\frac {6}{5} \int \left (337+212 x+48 x^2+4 x^3\right ) \log (5-x) \, dx\\ &=-\left (\frac {3}{5} \int \left (-9433-\frac {14808}{-5+x}-3026 x+30 x^2+154 x^3+20 x^4\right ) \, dx\right )+\frac {6}{5} \int \left (337 \log (5-x)+212 x \log (5-x)+48 x^2 \log (5-x)+4 x^3 \log (5-x)\right ) \, dx\\ &=\frac {28299 x}{5}+\frac {4539 x^2}{5}-6 x^3-\frac {231 x^4}{10}-\frac {12 x^5}{5}+\frac {44424}{5} \log (5-x)+\frac {24}{5} \int x^3 \log (5-x) \, dx+\frac {288}{5} \int x^2 \log (5-x) \, dx+\frac {1272}{5} \int x \log (5-x) \, dx+\frac {2022}{5} \int \log (5-x) \, dx\\ &=\frac {28299 x}{5}+\frac {4539 x^2}{5}-6 x^3-\frac {231 x^4}{10}-\frac {12 x^5}{5}+\frac {44424}{5} \log (5-x)+\frac {636}{5} x^2 \log (5-x)+\frac {96}{5} x^3 \log (5-x)+\frac {6}{5} x^4 \log (5-x)+\frac {6}{5} \int \frac {x^4}{5-x} \, dx+\frac {96}{5} \int \frac {x^3}{5-x} \, dx+\frac {636}{5} \int \frac {x^2}{5-x} \, dx-\frac {2022}{5} \operatorname {Subst}(\int \log (x) \, dx,x,5-x)\\ &=\frac {26277 x}{5}+\frac {4539 x^2}{5}-6 x^3-\frac {231 x^4}{10}-\frac {12 x^5}{5}+\frac {44424}{5} \log (5-x)-\frac {2022}{5} (5-x) \log (5-x)+\frac {636}{5} x^2 \log (5-x)+\frac {96}{5} x^3 \log (5-x)+\frac {6}{5} x^4 \log (5-x)+\frac {6}{5} \int \left (-125-\frac {625}{-5+x}-25 x-5 x^2-x^3\right ) \, dx+\frac {96}{5} \int \left (-25-\frac {125}{-5+x}-5 x-x^2\right ) \, dx+\frac {636}{5} \int \left (-5-\frac {25}{-5+x}-x\right ) \, dx\\ &=\frac {19947 x}{5}+\frac {3906 x^2}{5}-\frac {72 x^3}{5}-\frac {117 x^4}{5}-\frac {12 x^5}{5}+\frac {12774}{5} \log (5-x)-\frac {2022}{5} (5-x) \log (5-x)+\frac {636}{5} x^2 \log (5-x)+\frac {96}{5} x^3 \log (5-x)+\frac {6}{5} x^4 \log (5-x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.09, size = 78, normalized size = 2.60 \begin {gather*} -\frac {3}{5} \left (-6649 x-1302 x^2+24 x^3+39 x^4+4 x^5-888 \log (5-x)-674 x \log (5-x)-212 x^2 \log (5-x)-32 x^3 \log (5-x)-2 x^4 \log (5-x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-97071 - 17091*x + 9528*x^2 + 2220*x^3 - 162*x^4 - 60*x^5 + (-10110 - 4338*x - 168*x^2 + 168*x^3 +
24*x^4)*Log[5 - x])/(-25 + 5*x),x]

[Out]

(-3*(-6649*x - 1302*x^2 + 24*x^3 + 39*x^4 + 4*x^5 - 888*Log[5 - x] - 674*x*Log[5 - x] - 212*x^2*Log[5 - x] - 3
2*x^3*Log[5 - x] - 2*x^4*Log[5 - x]))/5

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fricas [A]  time = 0.48, size = 50, normalized size = 1.67 \begin {gather*} -\frac {12}{5} \, x^{5} - \frac {117}{5} \, x^{4} - \frac {72}{5} \, x^{3} + \frac {3906}{5} \, x^{2} + \frac {6}{5} \, {\left (x^{4} + 16 \, x^{3} + 106 \, x^{2} + 337 \, x + 444\right )} \log \left (-x + 5\right ) + \frac {19947}{5} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^4+168*x^3-168*x^2-4338*x-10110)*log(5-x)-60*x^5-162*x^4+2220*x^3+9528*x^2-17091*x-97071)/(5*x
-25),x, algorithm="fricas")

[Out]

-12/5*x^5 - 117/5*x^4 - 72/5*x^3 + 3906/5*x^2 + 6/5*(x^4 + 16*x^3 + 106*x^2 + 337*x + 444)*log(-x + 5) + 19947
/5*x

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giac [B]  time = 0.16, size = 55, normalized size = 1.83 \begin {gather*} -\frac {12}{5} \, x^{5} - \frac {117}{5} \, x^{4} - \frac {72}{5} \, x^{3} + \frac {3906}{5} \, x^{2} + \frac {6}{5} \, {\left (x^{4} + 16 \, x^{3} + 106 \, x^{2} + 337 \, x\right )} \log \left (-x + 5\right ) + \frac {19947}{5} \, x + \frac {2664}{5} \, \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^4+168*x^3-168*x^2-4338*x-10110)*log(5-x)-60*x^5-162*x^4+2220*x^3+9528*x^2-17091*x-97071)/(5*x
-25),x, algorithm="giac")

[Out]

-12/5*x^5 - 117/5*x^4 - 72/5*x^3 + 3906/5*x^2 + 6/5*(x^4 + 16*x^3 + 106*x^2 + 337*x)*log(-x + 5) + 19947/5*x +
2664/5*log(x - 5)

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maple [B]  time = 0.42, size = 57, normalized size = 1.90

method	result	size
risch	\(\left (\frac {6}{5} x^{4}+\frac {96}{5} x^{3}+\frac {636}{5} x^{2}+\frac {2022}{5} x \right ) \ln \left (5-x \right )-\frac {12 x^{5}}{5}-\frac {117 x^{4}}{5}-\frac {72 x^{3}}{5}+\frac {3906 x^{2}}{5}+\frac {19947 x}{5}+\frac {2664 \ln \left (x -5\right )}{5}\)	\(57\)
norman	\(\frac {2664 \ln \left (5-x \right )}{5}+\frac {19947 x}{5}+\frac {3906 x^{2}}{5}-\frac {72 x^{3}}{5}-\frac {117 x^{4}}{5}-\frac {12 x^{5}}{5}+\frac {2022 \ln \left (5-x \right ) x}{5}+\frac {636 \ln \left (5-x \right ) x^{2}}{5}+\frac {96 \ln \left (5-x \right ) x^{3}}{5}+\frac {6 \ln \left (5-x \right ) x^{4}}{5}\)	\(75\)
derivativedivides	\(\frac {6 \ln \left (5-x \right ) \left (5-x \right )^{4}}{5}-\frac {417 \left (5-x \right )^{4}}{5}+\frac {12 \left (5-x \right )^{5}}{5}-\frac {216 \ln \left (5-x \right ) \left (5-x \right )^{3}}{5}+\frac {5412 \left (5-x \right )^{3}}{5}+\frac {2976 \ln \left (5-x \right ) \left (5-x \right )^{2}}{5}-\frac {29724 \left (5-x \right )^{2}}{5}-\frac {18582 \left (5-x \right ) \ln \left (5-x \right )}{5}+42393-\frac {42393 x}{5}+\frac {44424 \ln \left (5-x \right )}{5}\)	\(108\)
default	\(\frac {6 \ln \left (5-x \right ) \left (5-x \right )^{4}}{5}-\frac {417 \left (5-x \right )^{4}}{5}+\frac {12 \left (5-x \right )^{5}}{5}-\frac {216 \ln \left (5-x \right ) \left (5-x \right )^{3}}{5}+\frac {5412 \left (5-x \right )^{3}}{5}+\frac {2976 \ln \left (5-x \right ) \left (5-x \right )^{2}}{5}-\frac {29724 \left (5-x \right )^{2}}{5}-\frac {18582 \left (5-x \right ) \ln \left (5-x \right )}{5}+42393-\frac {42393 x}{5}+\frac {44424 \ln \left (5-x \right )}{5}\)	\(108\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((24*x^4+168*x^3-168*x^2-4338*x-10110)*ln(5-x)-60*x^5-162*x^4+2220*x^3+9528*x^2-17091*x-97071)/(5*x-25),x,
method=_RETURNVERBOSE)

[Out]

(6/5*x^4+96/5*x^3+636/5*x^2+2022/5*x)*ln(5-x)-12/5*x^5-117/5*x^4-72/5*x^3+3906/5*x^2+19947/5*x+2664/5*ln(x-5)

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maxima [B]  time = 0.35, size = 146, normalized size = 4.87 \begin {gather*} -\frac {12}{5} \, x^{5} - \frac {117}{5} \, x^{4} - \frac {72}{5} \, x^{3} + \frac {3906}{5} \, x^{2} - 1011 \, \log \left (x - 5\right )^{2} + \frac {2}{5} \, {\left (3 \, x^{4} + 20 \, x^{3} + 150 \, x^{2} + 1500 \, x + 7500 \, \log \left (x - 5\right )\right )} \log \left (-x + 5\right ) + \frac {28}{5} \, {\left (2 \, x^{3} + 15 \, x^{2} + 150 \, x + 750 \, \log \left (x - 5\right )\right )} \log \left (-x + 5\right ) - \frac {84}{5} \, {\left (x^{2} + 10 \, x + 50 \, \log \left (x - 5\right )\right )} \log \left (-x + 5\right ) - \frac {4338}{5} \, {\left (x + 5 \, \log \left (x - 5\right )\right )} \log \left (-x + 5\right ) - 1011 \, \log \left (-x + 5\right )^{2} + \frac {19947}{5} \, x + \frac {2664}{5} \, \log \left (x - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x^4+168*x^3-168*x^2-4338*x-10110)*log(5-x)-60*x^5-162*x^4+2220*x^3+9528*x^2-17091*x-97071)/(5*x
-25),x, algorithm="maxima")

[Out]

-12/5*x^5 - 117/5*x^4 - 72/5*x^3 + 3906/5*x^2 - 1011*log(x - 5)^2 + 2/5*(3*x^4 + 20*x^3 + 150*x^2 + 1500*x + 7
500*log(x - 5))*log(-x + 5) + 28/5*(2*x^3 + 15*x^2 + 150*x + 750*log(x - 5))*log(-x + 5) - 84/5*(x^2 + 10*x +
50*log(x - 5))*log(-x + 5) - 4338/5*(x + 5*log(x - 5))*log(-x + 5) - 1011*log(-x + 5)^2 + 19947/5*x + 2664/5*l
og(x - 5)

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mupad [B]  time = 5.51, size = 66, normalized size = 2.20 \begin {gather*} \frac {2664\,\ln \left (x-5\right )}{5}+x\,\left (\frac {2022\,\ln \left (5-x\right )}{5}+\frac {19947}{5}\right )+x^4\,\left (\frac {6\,\ln \left (5-x\right )}{5}-\frac {117}{5}\right )+x^3\,\left (\frac {96\,\ln \left (5-x\right )}{5}-\frac {72}{5}\right )+x^2\,\left (\frac {636\,\ln \left (5-x\right )}{5}+\frac {3906}{5}\right )-\frac {12\,x^5}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(17091*x - 9528*x^2 - 2220*x^3 + 162*x^4 + 60*x^5 + log(5 - x)*(4338*x + 168*x^2 - 168*x^3 - 24*x^4 + 101
10) + 97071)/(5*x - 25),x)

[Out]

(2664*log(x - 5))/5 + x*((2022*log(5 - x))/5 + 19947/5) + x^4*((6*log(5 - x))/5 - 117/5) + x^3*((96*log(5 - x)
)/5 - 72/5) + x^2*((636*log(5 - x))/5 + 3906/5) - (12*x^5)/5

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sympy [B]  time = 0.17, size = 70, normalized size = 2.33 \begin {gather*} - \frac {12 x^{5}}{5} - \frac {117 x^{4}}{5} - \frac {72 x^{3}}{5} + \frac {3906 x^{2}}{5} + \frac {19947 x}{5} + \left (\frac {6 x^{4}}{5} + \frac {96 x^{3}}{5} + \frac {636 x^{2}}{5} + \frac {2022 x}{5}\right ) \log {\left (5 - x \right )} + \frac {2664 \log {\left (x - 5 \right )}}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((24*x**4+168*x**3-168*x**2-4338*x-10110)*ln(5-x)-60*x**5-162*x**4+2220*x**3+9528*x**2-17091*x-97071
)/(5*x-25),x)

[Out]

-12*x**5/5 - 117*x**4/5 - 72*x**3/5 + 3906*x**2/5 + 19947*x/5 + (6*x**4/5 + 96*x**3/5 + 636*x**2/5 + 2022*x/5)
*log(5 - x) + 2664*log(x - 5)/5

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