3.54.80 \(\int \frac {4 x^3-5996 x^4-6959 x^5-1812 x^6+84 x^7+(-300 x^2-388 x^3-109 x^4+5 x^5) \log (25-x)}{-25+x} \, dx\)

Optimal. Leaf size=26 \[ 2+\left (2 x+x^2\right )^2 \left (12 x^3+x \log (25-x)\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.56, antiderivative size = 48, normalized size of antiderivative = 1.85, number of steps used = 16, number of rules used = 6, integrand size = 60, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6741, 6742, 1620, 2417, 2395, 43} \begin {gather*} 12 x^7+48 x^6+48 x^5+x^5 \log (25-x)+4 x^4 \log (25-x)+4 x^3 \log (25-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^3 - 5996*x^4 - 6959*x^5 - 1812*x^6 + 84*x^7 + (-300*x^2 - 388*x^3 - 109*x^4 + 5*x^5)*Log[25 - x])/(-2
5 + x),x]

[Out]

48*x^5 + 48*x^6 + 12*x^7 + 4*x^3*Log[25 - x] + 4*x^4*Log[25 - x] + x^5*Log[25 - x]

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 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]

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 6741

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

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 \frac {x^2 (2+x) \left (-2 x+2999 x^2+1980 x^3-84 x^4+150 \log (25-x)+119 x \log (25-x)-5 x^2 \log (25-x)\right )}{25-x} \, dx\\ &=\int \left (\frac {x^3 \left (4-5996 x-6959 x^2-1812 x^3+84 x^4\right )}{-25+x}+x^2 \left (12+16 x+5 x^2\right ) \log (25-x)\right ) \, dx\\ &=\int \frac {x^3 \left (4-5996 x-6959 x^2-1812 x^3+84 x^4\right )}{-25+x} \, dx+\int x^2 \left (12+16 x+5 x^2\right ) \log (25-x) \, dx\\ &=\int \left (455625+\frac {11390625}{-25+x}+18225 x+729 x^2+29 x^3+241 x^4+288 x^5+84 x^6\right ) \, dx+\int \left (12 x^2 \log (25-x)+16 x^3 \log (25-x)+5 x^4 \log (25-x)\right ) \, dx\\ &=455625 x+\frac {18225 x^2}{2}+243 x^3+\frac {29 x^4}{4}+\frac {241 x^5}{5}+48 x^6+12 x^7+11390625 \log (25-x)+5 \int x^4 \log (25-x) \, dx+12 \int x^2 \log (25-x) \, dx+16 \int x^3 \log (25-x) \, dx\\ &=455625 x+\frac {18225 x^2}{2}+243 x^3+\frac {29 x^4}{4}+\frac {241 x^5}{5}+48 x^6+12 x^7+11390625 \log (25-x)+4 x^3 \log (25-x)+4 x^4 \log (25-x)+x^5 \log (25-x)+4 \int \frac {x^3}{25-x} \, dx+4 \int \frac {x^4}{25-x} \, dx+\int \frac {x^5}{25-x} \, dx\\ &=455625 x+\frac {18225 x^2}{2}+243 x^3+\frac {29 x^4}{4}+\frac {241 x^5}{5}+48 x^6+12 x^7+11390625 \log (25-x)+4 x^3 \log (25-x)+4 x^4 \log (25-x)+x^5 \log (25-x)+4 \int \left (-625-\frac {15625}{-25+x}-25 x-x^2\right ) \, dx+4 \int \left (-15625-\frac {390625}{-25+x}-625 x-25 x^2-x^3\right ) \, dx+\int \left (-390625-\frac {9765625}{-25+x}-15625 x-625 x^2-25 x^3-x^4\right ) \, dx\\ &=48 x^5+48 x^6+12 x^7+4 x^3 \log (25-x)+4 x^4 \log (25-x)+x^5 \log (25-x)\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 48, normalized size = 1.85 \begin {gather*} 48 x^5+48 x^6+12 x^7+4 x^3 \log (25-x)+4 x^4 \log (25-x)+x^5 \log (25-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^3 - 5996*x^4 - 6959*x^5 - 1812*x^6 + 84*x^7 + (-300*x^2 - 388*x^3 - 109*x^4 + 5*x^5)*Log[25 - x
])/(-25 + x),x]

[Out]

48*x^5 + 48*x^6 + 12*x^7 + 4*x^3*Log[25 - x] + 4*x^4*Log[25 - x] + x^5*Log[25 - x]

________________________________________________________________________________________

fricas [A]  time = 0.62, size = 37, normalized size = 1.42 \begin {gather*} 12 \, x^{7} + 48 \, x^{6} + 48 \, x^{5} + {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} \log \left (-x + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5-109*x^4-388*x^3-300*x^2)*log(-x+25)+84*x^7-1812*x^6-6959*x^5-5996*x^4+4*x^3)/(x-25),x, algor
ithm="fricas")

[Out]

12*x^7 + 48*x^6 + 48*x^5 + (x^5 + 4*x^4 + 4*x^3)*log(-x + 25)

________________________________________________________________________________________

giac [A]  time = 0.15, size = 37, normalized size = 1.42 \begin {gather*} 12 \, x^{7} + 48 \, x^{6} + 48 \, x^{5} + {\left (x^{5} + 4 \, x^{4} + 4 \, x^{3}\right )} \log \left (-x + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5-109*x^4-388*x^3-300*x^2)*log(-x+25)+84*x^7-1812*x^6-6959*x^5-5996*x^4+4*x^3)/(x-25),x, algor
ithm="giac")

[Out]

12*x^7 + 48*x^6 + 48*x^5 + (x^5 + 4*x^4 + 4*x^3)*log(-x + 25)

________________________________________________________________________________________

maple [A]  time = 0.21, size = 38, normalized size = 1.46

method	result	size
risch	\(\left (x^{5}+4 x^{4}+4 x^{3}\right ) \ln \left (-x +25\right )+12 x^{7}+48 x^{6}+48 x^{5}\)	\(38\)
norman	\(x^{5} \ln \left (-x +25\right )+48 x^{5}+48 x^{6}+12 x^{7}+4 \ln \left (-x +25\right ) x^{3}+4 \ln \left (-x +25\right ) x^{4}\)	\(49\)
derivativedivides	\(-12 \left (-x +25\right )^{7}-\ln \left (-x +25\right ) \left (-x +25\right )^{5}-164748 \left (-x +25\right )^{5}+2148 \left (-x +25\right )^{6}+129 \ln \left (-x +25\right ) \left (-x +25\right )^{4}+7018500 \left (-x +25\right )^{4}-6654 \ln \left (-x +25\right ) \left (-x +25\right )^{3}-179362500 \left (-x +25\right )^{3}+171550 \ln \left (-x +25\right ) \left (-x +25\right )^{2}[phone] \left (-x +25\right )^{2}-2210625 \ln \left (-x +25\right ) \left (-x +25\right )[phone] x -585351562500+11390625 \ln \left (-x +25\right )\)	\(141\)
default	\(-12 \left (-x +25\right )^{7}-\ln \left (-x +25\right ) \left (-x +25\right )^{5}-164748 \left (-x +25\right )^{5}+2148 \left (-x +25\right )^{6}+129 \ln \left (-x +25\right ) \left (-x +25\right )^{4}+7018500 \left (-x +25\right )^{4}-6654 \ln \left (-x +25\right ) \left (-x +25\right )^{3}-179362500 \left (-x +25\right )^{3}+171550 \ln \left (-x +25\right ) \left (-x +25\right )^{2}[phone] \left (-x +25\right )^{2}-2210625 \ln \left (-x +25\right ) \left (-x +25\right )[phone] x -585351562500+11390625 \ln \left (-x +25\right )\)	\(141\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^5-109*x^4-388*x^3-300*x^2)*ln(-x+25)+84*x^7-1812*x^6-6959*x^5-5996*x^4+4*x^3)/(x-25),x,method=_RETUR
NVERBOSE)

[Out]

(x^5+4*x^4+4*x^3)*ln(-x+25)+12*x^7+48*x^6+48*x^5

________________________________________________________________________________________

maxima [B]  time = 0.42, size = 136, normalized size = 5.23 \begin {gather*} 12 \, x^{7} + 48 \, x^{6} + 48 \, x^{5} + \frac {1}{12} \, {\left (12 \, x^{5} + 375 \, x^{4} + 12500 \, x^{3} + 468750 \, x^{2} + 23437500 \, x + 585937500 \, \log \left (x - 25\right )\right )} \log \left (-x + 25\right ) - \frac {109}{12} \, {\left (3 \, x^{4} + 100 \, x^{3} + 3750 \, x^{2} + 187500 \, x + 4687500 \, \log \left (x - 25\right )\right )} \log \left (-x + 25\right ) - \frac {194}{3} \, {\left (2 \, x^{3} + 75 \, x^{2} + 3750 \, x + 93750 \, \log \left (x - 25\right )\right )} \log \left (-x + 25\right ) - 150 \, {\left (x^{2} + 50 \, x + 1250 \, \log \left (x - 25\right )\right )} \log \left (-x + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^5-109*x^4-388*x^3-300*x^2)*log(-x+25)+84*x^7-1812*x^6-6959*x^5-5996*x^4+4*x^3)/(x-25),x, algor
ithm="maxima")

[Out]

12*x^7 + 48*x^6 + 48*x^5 + 1/12*(12*x^5 + 375*x^4 + 12500*x^3 + 468750*x^2 + 23437500*x + 585937500*log(x - 25
))*log(-x + 25) - 109/12*(3*x^4 + 100*x^3 + 3750*x^2 + 187500*x + 4687500*log(x - 25))*log(-x + 25) - 194/3*(2
*x^3 + 75*x^2 + 3750*x + 93750*log(x - 25))*log(-x + 25) - 150*(x^2 + 50*x + 1250*log(x - 25))*log(-x + 25)

________________________________________________________________________________________

mupad [B]  time = 0.14, size = 21, normalized size = 0.81 \begin {gather*} x^3\,\left (\ln \left (25-x\right )+12\,x^2\right )\,{\left (x+2\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(25 - x)*(300*x^2 + 388*x^3 + 109*x^4 - 5*x^5) - 4*x^3 + 5996*x^4 + 6959*x^5 + 1812*x^6 - 84*x^7)/(x
- 25),x)

[Out]

x^3*(log(25 - x) + 12*x^2)*(x + 2)^2

________________________________________________________________________________________

sympy [A]  time = 0.13, size = 32, normalized size = 1.23 \begin {gather*} 12 x^{7} + 48 x^{6} + 48 x^{5} + \left (x^{5} + 4 x^{4} + 4 x^{3}\right ) \log {\left (25 - x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**5-109*x**4-388*x**3-300*x**2)*ln(-x+25)+84*x**7-1812*x**6-6959*x**5-5996*x**4+4*x**3)/(x-25),
x)

[Out]

12*x**7 + 48*x**6 + 48*x**5 + (x**5 + 4*x**4 + 4*x**3)*log(25 - x)

________________________________________________________________________________________