3.4.51 \(\int \frac {625+9225 x+46950 x^2+78550 x^3+753 x^4+3798 x^5+6266 x^6+15 x^7+75 x^8+125 x^9+e^8 (x^4+15 x^5+75 x^6+125 x^7)+e^4 (-50 x^2-750 x^3-3783 x^4-6267 x^5-30 x^6-150 x^7-250 x^8)}{625+9375 x+46950 x^2+78550 x^3+750 x^4+3753 x^5+6266 x^6+15 x^7+75 x^8+125 x^9+e^8 (x^4+15 x^5+75 x^6+125 x^7)+e^4 (-50 x^2-750 x^3-3753 x^4-6267 x^5-30 x^6-150 x^7-250 x^8)} \, dx\)

Optimal. Leaf size=30 \[ x-\log \left (1+\frac {3}{\left (-e^4+\frac {25}{x^2}+x\right ) (1+5 x)^2}\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.71, antiderivative size = 71, normalized size of antiderivative = 2.37, number of steps used = 4, number of rules used = 2, integrand size = 219, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.009, Rules used = {2074, 1587} \begin {gather*} \log \left (x^3-e^4 x^2+25\right )-\log \left (25 x^5+5 \left (2-5 e^4\right ) x^4+\left (1-10 e^4\right ) x^3+\left (628-e^4\right ) x^2+250 x+25\right )+x+2 \log (5 x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(625 + 9225*x + 46950*x^2 + 78550*x^3 + 753*x^4 + 3798*x^5 + 6266*x^6 + 15*x^7 + 75*x^8 + 125*x^9 + E^8*(x
^4 + 15*x^5 + 75*x^6 + 125*x^7) + E^4*(-50*x^2 - 750*x^3 - 3783*x^4 - 6267*x^5 - 30*x^6 - 150*x^7 - 250*x^8))/
(625 + 9375*x + 46950*x^2 + 78550*x^3 + 750*x^4 + 3753*x^5 + 6266*x^6 + 15*x^7 + 75*x^8 + 125*x^9 + E^8*(x^4 +
15*x^5 + 75*x^6 + 125*x^7) + E^4*(-50*x^2 - 750*x^3 - 3753*x^4 - 6267*x^5 - 30*x^6 - 150*x^7 - 250*x^8)),x]

[Out]

x + 2*Log[1 + 5*x] + Log[25 - E^4*x^2 + x^3] - Log[25 + 250*x + (628 - E^4)*x^2 + (1 - 10*E^4)*x^3 + 5*(2 - 5*
E^4)*x^4 + 25*x^5]

Rule 1587

Int[(Pp_)/(Qq_), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*Log[RemoveConte
nt[Qq, x]])/(q*Coeff[Qq, x, q]), x] /; EqQ[p, q - 1] && EqQ[Pp, Simplify[(Coeff[Pp, x, p]*D[Qq, x])/(q*Coeff[Q
q, x, q])]]] /; PolyQ[Pp, x] && PolyQ[Qq, x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {10}{1+5 x}+\frac {x \left (-2 e^4+3 x\right )}{25-e^4 x^2+x^3}+\frac {-250-2 \left (628-e^4\right ) x-3 \left (1-10 e^4\right ) x^2-20 \left (2-5 e^4\right ) x^3-125 x^4}{25+250 x+628 \left (1-\frac {e^4}{628}\right ) x^2+\left (1-10 e^4\right ) x^3+10 \left (1-\frac {5 e^4}{2}\right ) x^4+25 x^5}\right ) \, dx\\ &=x+2 \log (1+5 x)+\int \frac {x \left (-2 e^4+3 x\right )}{25-e^4 x^2+x^3} \, dx+\int \frac {-250-2 \left (628-e^4\right ) x-3 \left (1-10 e^4\right ) x^2-20 \left (2-5 e^4\right ) x^3-125 x^4}{25+250 x+628 \left (1-\frac {e^4}{628}\right ) x^2+\left (1-10 e^4\right ) x^3+10 \left (1-\frac {5 e^4}{2}\right ) x^4+25 x^5} \, dx\\ &=x+2 \log (1+5 x)+\log \left (25-e^4 x^2+x^3\right )-\log \left (25+250 x+\left (628-e^4\right ) x^2+\left (1-10 e^4\right ) x^3+5 \left (2-5 e^4\right ) x^4+25 x^5\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.43, size = 145, normalized size = 4.83 \begin {gather*} x+2 \log (1+5 x)+\log \left (3124-5 e^4+3 (1+5 x)+10 e^4 (1+5 x)-3 (1+5 x)^2-5 e^4 (1+5 x)^2+(1+5 x)^3\right )-\log \left (15-30 (1+5 x)+3139 (1+5 x)^2-5 e^4 (1+5 x)^2+3 (1+5 x)^3+10 e^4 (1+5 x)^3-3 (1+5 x)^4-5 e^4 (1+5 x)^4+(1+5 x)^5\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(625 + 9225*x + 46950*x^2 + 78550*x^3 + 753*x^4 + 3798*x^5 + 6266*x^6 + 15*x^7 + 75*x^8 + 125*x^9 +
E^8*(x^4 + 15*x^5 + 75*x^6 + 125*x^7) + E^4*(-50*x^2 - 750*x^3 - 3783*x^4 - 6267*x^5 - 30*x^6 - 150*x^7 - 250*
x^8))/(625 + 9375*x + 46950*x^2 + 78550*x^3 + 750*x^4 + 3753*x^5 + 6266*x^6 + 15*x^7 + 75*x^8 + 125*x^9 + E^8*
(x^4 + 15*x^5 + 75*x^6 + 125*x^7) + E^4*(-50*x^2 - 750*x^3 - 3753*x^4 - 6267*x^5 - 30*x^6 - 150*x^7 - 250*x^8)
),x]

[Out]

x + 2*Log[1 + 5*x] + Log[3124 - 5*E^4 + 3*(1 + 5*x) + 10*E^4*(1 + 5*x) - 3*(1 + 5*x)^2 - 5*E^4*(1 + 5*x)^2 + (
1 + 5*x)^3] - Log[15 - 30*(1 + 5*x) + 3139*(1 + 5*x)^2 - 5*E^4*(1 + 5*x)^2 + 3*(1 + 5*x)^3 + 10*E^4*(1 + 5*x)^
3 - 3*(1 + 5*x)^4 - 5*E^4*(1 + 5*x)^4 + (1 + 5*x)^5]

________________________________________________________________________________________

fricas [B]  time = 0.89, size = 67, normalized size = 2.23 \begin {gather*} x - \log \left (25 \, x^{5} + 10 \, x^{4} + x^{3} + 628 \, x^{2} - {\left (25 \, x^{4} + 10 \, x^{3} + x^{2}\right )} e^{4} + 250 \, x + 25\right ) + \log \left (x^{3} - x^{2} e^{4} + 25\right ) + 2 \, \log \left (5 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*x^7+75*x^6+15*x^5+x^4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3783*x^4-750*x^3-50*x^2)*exp
(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3798*x^5+753*x^4+78550*x^3+46950*x^2+9225*x+625)/((125*x^7+75*x^6+15*x^5+x^
4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3753*x^4-750*x^3-50*x^2)*exp(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3
753*x^5+750*x^4+78550*x^3+46950*x^2+9375*x+625),x, algorithm="fricas")

[Out]

x - log(25*x^5 + 10*x^4 + x^3 + 628*x^2 - (25*x^4 + 10*x^3 + x^2)*e^4 + 250*x + 25) + log(x^3 - x^2*e^4 + 25)
+ 2*log(5*x + 1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {125 \, x^{9} + 75 \, x^{8} + 15 \, x^{7} + 6266 \, x^{6} + 3798 \, x^{5} + 753 \, x^{4} + 78550 \, x^{3} + 46950 \, x^{2} + {\left (125 \, x^{7} + 75 \, x^{6} + 15 \, x^{5} + x^{4}\right )} e^{8} - {\left (250 \, x^{8} + 150 \, x^{7} + 30 \, x^{6} + 6267 \, x^{5} + 3783 \, x^{4} + 750 \, x^{3} + 50 \, x^{2}\right )} e^{4} + 9225 \, x + 625}{125 \, x^{9} + 75 \, x^{8} + 15 \, x^{7} + 6266 \, x^{6} + 3753 \, x^{5} + 750 \, x^{4} + 78550 \, x^{3} + 46950 \, x^{2} + {\left (125 \, x^{7} + 75 \, x^{6} + 15 \, x^{5} + x^{4}\right )} e^{8} - {\left (250 \, x^{8} + 150 \, x^{7} + 30 \, x^{6} + 6267 \, x^{5} + 3753 \, x^{4} + 750 \, x^{3} + 50 \, x^{2}\right )} e^{4} + 9375 \, x + 625}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*x^7+75*x^6+15*x^5+x^4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3783*x^4-750*x^3-50*x^2)*exp
(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3798*x^5+753*x^4+78550*x^3+46950*x^2+9225*x+625)/((125*x^7+75*x^6+15*x^5+x^
4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3753*x^4-750*x^3-50*x^2)*exp(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3
753*x^5+750*x^4+78550*x^3+46950*x^2+9375*x+625),x, algorithm="giac")

[Out]

integrate((125*x^9 + 75*x^8 + 15*x^7 + 6266*x^6 + 3798*x^5 + 753*x^4 + 78550*x^3 + 46950*x^2 + (125*x^7 + 75*x
^6 + 15*x^5 + x^4)*e^8 - (250*x^8 + 150*x^7 + 30*x^6 + 6267*x^5 + 3783*x^4 + 750*x^3 + 50*x^2)*e^4 + 9225*x +
625)/(125*x^9 + 75*x^8 + 15*x^7 + 6266*x^6 + 3753*x^5 + 750*x^4 + 78550*x^3 + 46950*x^2 + (125*x^7 + 75*x^6 +
15*x^5 + x^4)*e^8 - (250*x^8 + 150*x^7 + 30*x^6 + 6267*x^5 + 3753*x^4 + 750*x^3 + 50*x^2)*e^4 + 9375*x + 625),
x)

________________________________________________________________________________________

maple [B]  time = 0.61, size = 67, normalized size = 2.23

method	result	size
risch	\(x +2 \ln \left (1+5 x \right )+\ln \left (-x^{2} {\mathrm e}^{4}+x^{3}+25\right )-\ln \left (25 x^{5}+\left (-25 \,{\mathrm e}^{4}+10\right ) x^{4}+\left (-10 \,{\mathrm e}^{4}+1\right ) x^{3}+\left (-{\mathrm e}^{4}+628\right ) x^{2}+250 x +25\right )\)	\(67\)
norman	\(x +2 \ln \left (1+5 x \right )-\ln \left (25 x^{4} {\mathrm e}^{4}-25 x^{5}+10 x^{3} {\mathrm e}^{4}-10 x^{4}+x^{2} {\mathrm e}^{4}-x^{3}-628 x^{2}-250 x -25\right )+\ln \left (x^{2} {\mathrm e}^{4}-x^{3}-25\right )\)	\(73\)
default	\(x -\left (\munderset {\textit {\_R} =\RootOf \left (625+25 \textit {\_Z}^{8}+\left (-50 \,{\mathrm e}^{4}+10\right ) \textit {\_Z}^{7}+\left (-20 \,{\mathrm e}^{4}+25 \,{\mathrm e}^{8}+1\right ) \textit {\_Z}^{6}+\left (-2 \,{\mathrm e}^{4}+10 \,{\mathrm e}^{8}+1253\right ) \textit {\_Z}^{5}+\left (-1253 \,{\mathrm e}^{4}+{\mathrm e}^{8}+500\right ) \textit {\_Z}^{4}+\left (-500 \,{\mathrm e}^{4}+50\right ) \textit {\_Z}^{3}+\left (-50 \,{\mathrm e}^{4}+15700\right ) \textit {\_Z}^{2}+6250 \textit {\_Z} \right )}{\sum }\frac {\left (-6250-50 \textit {\_R}^{7}+10 \left (10 \,{\mathrm e}^{4}-1\right ) \textit {\_R}^{6}+10 \left (2 \,{\mathrm e}^{4}-5 \,{\mathrm e}^{8}\right ) \textit {\_R}^{5}+\left (-10 \,{\mathrm e}^{8}-2497\right ) \textit {\_R}^{4}+500 \left (5 \,{\mathrm e}^{4}-1\right ) \textit {\_R}^{3}+500 \textit {\_R}^{2} {\mathrm e}^{4}-31400 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{350 \textit {\_R}^{6} {\mathrm e}^{4}-200 \textit {\_R}^{7}+120 \textit {\_R}^{5} {\mathrm e}^{4}-150 \textit {\_R}^{5} {\mathrm e}^{8}-70 \textit {\_R}^{6}+10 \textit {\_R}^{4} {\mathrm e}^{4}-50 \textit {\_R}^{4} {\mathrm e}^{8}-6 \textit {\_R}^{5}+5012 \textit {\_R}^{3} {\mathrm e}^{4}-4 \textit {\_R}^{3} {\mathrm e}^{8}-6265 \textit {\_R}^{4}+1500 \textit {\_R}^{2} {\mathrm e}^{4}-2000 \textit {\_R}^{3}+100 \textit {\_R} \,{\mathrm e}^{4}-150 \textit {\_R}^{2}-31400 \textit {\_R} -6250}\right )+2 \ln \left (1+5 x \right )\)	\(265\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((125*x^7+75*x^6+15*x^5+x^4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3783*x^4-750*x^3-50*x^2)*exp(4)+12
5*x^9+75*x^8+15*x^7+6266*x^6+3798*x^5+753*x^4+78550*x^3+46950*x^2+9225*x+625)/((125*x^7+75*x^6+15*x^5+x^4)*exp
(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3753*x^4-750*x^3-50*x^2)*exp(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3753*x^
5+750*x^4+78550*x^3+46950*x^2+9375*x+625),x,method=_RETURNVERBOSE)

[Out]

x+2*ln(1+5*x)+ln(-x^2*exp(4)+x^3+25)-ln(25*x^5+(-25*exp(4)+10)*x^4+(-10*exp(4)+1)*x^3+(-exp(4)+628)*x^2+250*x+
25)

________________________________________________________________________________________

maxima [B]  time = 0.42, size = 67, normalized size = 2.23 \begin {gather*} x - \log \left (25 \, x^{5} - 5 \, x^{4} {\left (5 \, e^{4} - 2\right )} - x^{3} {\left (10 \, e^{4} - 1\right )} - x^{2} {\left (e^{4} - 628\right )} + 250 \, x + 25\right ) + \log \left (x^{3} - x^{2} e^{4} + 25\right ) + 2 \, \log \left (5 \, x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*x^7+75*x^6+15*x^5+x^4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3783*x^4-750*x^3-50*x^2)*exp
(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3798*x^5+753*x^4+78550*x^3+46950*x^2+9225*x+625)/((125*x^7+75*x^6+15*x^5+x^
4)*exp(4)^2+(-250*x^8-150*x^7-30*x^6-6267*x^5-3753*x^4-750*x^3-50*x^2)*exp(4)+125*x^9+75*x^8+15*x^7+6266*x^6+3
753*x^5+750*x^4+78550*x^3+46950*x^2+9375*x+625),x, algorithm="maxima")

[Out]

x - log(25*x^5 - 5*x^4*(5*e^4 - 2) - x^3*(10*e^4 - 1) - x^2*(e^4 - 628) + 250*x + 25) + log(x^3 - x^2*e^4 + 25
) + 2*log(5*x + 1)

________________________________________________________________________________________

mupad [B]  time = 1.08, size = 68, normalized size = 2.27 \begin {gather*} x+\ln \left (x^3-{\mathrm {e}}^4\,x^2+25\right )+2\,\ln \left (x+\frac {1}{5}\right )-\ln \left (10\,x-\frac {x^2\,{\mathrm {e}}^4}{25}-\frac {2\,x^3\,{\mathrm {e}}^4}{5}-x^4\,{\mathrm {e}}^4+\frac {628\,x^2}{25}+\frac {x^3}{25}+\frac {2\,x^4}{5}+x^5+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9225*x + exp(8)*(x^4 + 15*x^5 + 75*x^6 + 125*x^7) - exp(4)*(50*x^2 + 750*x^3 + 3783*x^4 + 6267*x^5 + 30*x
^6 + 150*x^7 + 250*x^8) + 46950*x^2 + 78550*x^3 + 753*x^4 + 3798*x^5 + 6266*x^6 + 15*x^7 + 75*x^8 + 125*x^9 +
625)/(9375*x + exp(8)*(x^4 + 15*x^5 + 75*x^6 + 125*x^7) - exp(4)*(50*x^2 + 750*x^3 + 3753*x^4 + 6267*x^5 + 30*
x^6 + 150*x^7 + 250*x^8) + 46950*x^2 + 78550*x^3 + 750*x^4 + 3753*x^5 + 6266*x^6 + 15*x^7 + 75*x^8 + 125*x^9 +
625),x)

[Out]

x + log(x^3 - x^2*exp(4) + 25) + 2*log(x + 1/5) - log(10*x - (x^2*exp(4))/25 - (2*x^3*exp(4))/5 - x^4*exp(4) +
(628*x^2)/25 + x^3/25 + (2*x^4)/5 + x^5 + 1)

________________________________________________________________________________________

sympy [B]  time = 19.38, size = 68, normalized size = 2.27 \begin {gather*} x + 2 \log {\left (5 x + 1 \right )} + \log {\left (x^{3} - x^{2} e^{4} + 25 \right )} - \log {\left (x^{5} + x^{4} \left (\frac {2}{5} - e^{4}\right ) + x^{3} \left (\frac {1}{25} - \frac {2 e^{4}}{5}\right ) + x^{2} \left (\frac {628}{25} - \frac {e^{4}}{25}\right ) + 10 x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((125*x**7+75*x**6+15*x**5+x**4)*exp(4)**2+(-250*x**8-150*x**7-30*x**6-6267*x**5-3783*x**4-750*x**3-
50*x**2)*exp(4)+125*x**9+75*x**8+15*x**7+6266*x**6+3798*x**5+753*x**4+78550*x**3+46950*x**2+9225*x+625)/((125*
x**7+75*x**6+15*x**5+x**4)*exp(4)**2+(-250*x**8-150*x**7-30*x**6-6267*x**5-3753*x**4-750*x**3-50*x**2)*exp(4)+
125*x**9+75*x**8+15*x**7+6266*x**6+3753*x**5+750*x**4+78550*x**3+46950*x**2+9375*x+625),x)

[Out]

x + 2*log(5*x + 1) + log(x**3 - x**2*exp(4) + 25) - log(x**5 + x**4*(2/5 - exp(4)) + x**3*(1/25 - 2*exp(4)/5)
+ x**2*(628/25 - exp(4)/25) + 10*x + 1)

________________________________________________________________________________________