3.56.30 \(\int \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+(35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5) \log (7+e^4+x+x^2-x^3-2 x^4+x^5)}{7 x^2+e^4 x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx\)

Optimal. Leaf size=31 \[ 5 \left (x-\frac {\log \left (7+e^4+x \left (-x+\left (1+x-x^2\right )^2\right )\right )}{x}\right ) \]

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Rubi [A]  time = 3.54, antiderivative size = 32, normalized size of antiderivative = 1.03, number of steps used = 17, number of rules used = 7, integrand size = 131, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {6, 6688, 12, 14, 6742, 2100, 2525} \begin {gather*} 5 x-\frac {5 \log \left (x^5-2 x^4-x^3+x^2+x+e^4+7\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5*x + 25*x^2 + 5*E^4*x^2 + 20*x^3 + 45*x^4 - 30*x^5 - 10*x^6 + 5*x^7 + (35 + 5*E^4 + 5*x + 5*x^2 - 5*x^3
- 10*x^4 + 5*x^5)*Log[7 + E^4 + x + x^2 - x^3 - 2*x^4 + x^5])/(7*x^2 + E^4*x^2 + x^3 + x^4 - x^5 - 2*x^6 + x^
7),x]

[Out]

5*x - (5*Log[7 + E^4 + x + x^2 - x^3 - 2*x^4 + x^5])/x

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2100

Int[(Pm_)/(Qn_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x]}, Simp[(Coeff[Pm, x, m]*Log[Qn])/(n*Coe
ff[Qn, x, n]), x] + Dist[1/(n*Coeff[Qn, x, n]), Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x, m]*D[Qn, x
], x]/Qn, x], x] /; EqQ[m, n - 1]] /; PolyQ[Pm, x] && PolyQ[Qn, x]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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 \frac {-5 x+25 x^2+5 e^4 x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{\left (7+e^4\right ) x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx\\ &=\int \frac {-5 x+\left (25+5 e^4\right ) x^2+20 x^3+45 x^4-30 x^5-10 x^6+5 x^7+\left (35+5 e^4+5 x+5 x^2-5 x^3-10 x^4+5 x^5\right ) \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{\left (7+e^4\right ) x^2+x^3+x^4-x^5-2 x^6+x^7} \, dx\\ &=\int \frac {5 \left (\frac {x \left (-1+\left (5+e^4\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6\right )}{7+e^4+x+x^2-x^3-2 x^4+x^5}+\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )\right )}{x^2} \, dx\\ &=5 \int \frac {\frac {x \left (-1+\left (5+e^4\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6\right )}{7+e^4+x+x^2-x^3-2 x^4+x^5}+\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x^2} \, dx\\ &=5 \int \left (\frac {-1+5 \left (1+\frac {e^4}{5}\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6}{x \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}+\frac {\log \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}{x^2}\right ) \, dx\\ &=5 \int \frac {-1+5 \left (1+\frac {e^4}{5}\right ) x+4 x^2+9 x^3-6 x^4-2 x^5+x^6}{x \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )} \, dx+5 \int \frac {\log \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}{x^2} \, dx\\ &=-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+5 \int \frac {1+2 x-3 x^2-8 x^3+5 x^4}{x \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )} \, dx+5 \int \left (1+\frac {1}{\left (-7-e^4\right ) x}+\frac {-13-2 e^4+\left (22+3 e^4\right ) x+\left (55+8 e^4\right ) x^2-\left (37+5 e^4\right ) x^3+x^4}{\left (7+e^4\right ) \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}\right ) \, dx\\ &=5 x-\frac {5 \log (x)}{7+e^4}-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+5 \int \left (\frac {1}{\left (7+e^4\right ) x}+\frac {13+2 e^4-\left (22+3 e^4\right ) x-\left (55+8 e^4\right ) x^2+\left (37+5 e^4\right ) x^3-x^4}{\left (7+e^4\right ) \left (7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5\right )}\right ) \, dx+\frac {5 \int \frac {-13-2 e^4+\left (22+3 e^4\right ) x+\left (55+8 e^4\right ) x^2-\left (37+5 e^4\right ) x^3+x^4}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}\\ &=5 x+\frac {\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{7+e^4}-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+\frac {\int \frac {-2 \left (33+5 e^4\right )+3 \left (36+5 e^4\right ) x+2 \left (139+20 e^4\right ) x^2-\left (177+25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {5 \int \frac {13+2 e^4-\left (22+3 e^4\right ) x-\left (55+8 e^4\right ) x^2+\left (37+5 e^4\right ) x^3-x^4}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}\\ &=5 x-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+\frac {\int \frac {2 \left (33+5 e^4\right )-3 \left (36+5 e^4\right ) x-2 \left (139+20 e^4\right ) x^2+\left (177+25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {\int \left (\frac {2 \left (-33-5 e^4\right )}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {3 \left (36+5 e^4\right ) x}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {2 \left (139+20 e^4\right ) x^2}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {\left (-177-25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}\right ) \, dx}{7+e^4}\\ &=5 x-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}+\frac {\int \left (\frac {2 \left (33+5 e^4\right )}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {3 \left (-36-5 e^4\right ) x}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {2 \left (-139-20 e^4\right ) x^2}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}+\frac {\left (177+25 e^4\right ) x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5}\right ) \, dx}{7+e^4}-\frac {\left (2 \left (33+5 e^4\right )\right ) \int \frac {1}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {\left (3 \left (36+5 e^4\right )\right ) \int \frac {x}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}+\frac {\left (2 \left (139+20 e^4\right )\right ) \int \frac {x^2}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}-\frac {\left (177+25 e^4\right ) \int \frac {x^3}{7 \left (1+\frac {e^4}{7}\right )+x+x^2-x^3-2 x^4+x^5} \, dx}{7+e^4}\\ &=5 x-\frac {5 \log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 32, normalized size = 1.03 \begin {gather*} 5 \left (x-\frac {\log \left (7+e^4+x+x^2-x^3-2 x^4+x^5\right )}{x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5*x + 25*x^2 + 5*E^4*x^2 + 20*x^3 + 45*x^4 - 30*x^5 - 10*x^6 + 5*x^7 + (35 + 5*E^4 + 5*x + 5*x^2 -
5*x^3 - 10*x^4 + 5*x^5)*Log[7 + E^4 + x + x^2 - x^3 - 2*x^4 + x^5])/(7*x^2 + E^4*x^2 + x^3 + x^4 - x^5 - 2*x^
6 + x^7),x]

[Out]

5*(x - Log[7 + E^4 + x + x^2 - x^3 - 2*x^4 + x^5]/x)

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fricas [A]  time = 0.53, size = 33, normalized size = 1.06 \begin {gather*} \frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*
x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/(x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x, algorithm="fricas")

[Out]

5*(x^2 - log(x^5 - 2*x^4 - x^3 + x^2 + x + e^4 + 7))/x

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giac [A]  time = 0.30, size = 33, normalized size = 1.06 \begin {gather*} \frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*
x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/(x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x, algorithm="giac")

[Out]

5*(x^2 - log(x^5 - 2*x^4 - x^3 + x^2 + x + e^4 + 7))/x

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maple [A]  time = 0.26, size = 32, normalized size = 1.03

method	result	size
risch	\(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+5 x\)	\(32\)
norman	\(\frac {5 x^{2}-5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}\)	\(35\)
default	\(-\frac {5 \ln \left ({\mathrm e}^{4}+x^{5}-2 x^{4}-x^{3}+x^{2}+x +7\right )}{x}+\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+\left (5 \,{\mathrm e}^{4}+37\right ) \textit {\_R}^{3}+\left (-8 \,{\mathrm e}^{4}-55\right ) \textit {\_R}^{2}+\left (-3 \,{\mathrm e}^{4}-22\right ) \textit {\_R} +2 \,{\mathrm e}^{4}+13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}+5 x +\frac {5 \left (\munderset {\textit {\_R} =\RootOf \left ({\mathrm e}^{4}+\textit {\_Z}^{5}-2 \textit {\_Z}^{4}-\textit {\_Z}^{3}+\textit {\_Z}^{2}+\textit {\_Z} +7\right )}{\sum }\frac {\left (\textit {\_R}^{4}+\left (-5 \,{\mathrm e}^{4}-37\right ) \textit {\_R}^{3}+\left (8 \,{\mathrm e}^{4}+55\right ) \textit {\_R}^{2}+\left (3 \,{\mathrm e}^{4}+22\right ) \textit {\_R} -2 \,{\mathrm e}^{4}-13\right ) \ln \left (x -\textit {\_R} \right )}{5 \textit {\_R}^{4}-8 \textit {\_R}^{3}-3 \textit {\_R}^{2}+2 \textit {\_R} +1}\right )}{{\mathrm e}^{4}+7}\)	\(232\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*ln(exp(4)+x^5-2*x^4-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*x^6-30*
x^5+45*x^4+20*x^3+25*x^2-5*x)/(x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.38, size = 33, normalized size = 1.06 \begin {gather*} \frac {5 \, {\left (x^{2} - \log \left (x^{5} - 2 \, x^{4} - x^{3} + x^{2} + x + e^{4} + 7\right )\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(4)+5*x^5-10*x^4-5*x^3+5*x^2+5*x+35)*log(exp(4)+x^5-2*x^4-x^3+x^2+x+7)+5*x^2*exp(4)+5*x^7-10*
x^6-30*x^5+45*x^4+20*x^3+25*x^2-5*x)/(x^2*exp(4)+x^7-2*x^6-x^5+x^4+x^3+7*x^2),x, algorithm="maxima")

[Out]

5*(x^2 - log(x^5 - 2*x^4 - x^3 + x^2 + x + e^4 + 7))/x

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mupad [B]  time = 4.15, size = 33, normalized size = 1.06 \begin {gather*} -\frac {5\,\left (\ln \left (x^5-2\,x^4-x^3+x^2+x+{\mathrm {e}}^4+7\right )-x^2\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^2*exp(4) - 5*x + 25*x^2 + 20*x^3 + 45*x^4 - 30*x^5 - 10*x^6 + 5*x^7 + log(x + exp(4) + x^2 - x^3 - 2*
x^4 + x^5 + 7)*(5*x + 5*exp(4) + 5*x^2 - 5*x^3 - 10*x^4 + 5*x^5 + 35))/(x^2*exp(4) + 7*x^2 + x^3 + x^4 - x^5 -
2*x^6 + x^7),x)

[Out]

-(5*(log(x + exp(4) + x^2 - x^3 - 2*x^4 + x^5 + 7) - x^2))/x

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sympy [A]  time = 0.26, size = 29, normalized size = 0.94 \begin {gather*} 5 x - \frac {5 \log {\left (x^{5} - 2 x^{4} - x^{3} + x^{2} + x + 7 + e^{4} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*exp(4)+5*x**5-10*x**4-5*x**3+5*x**2+5*x+35)*ln(exp(4)+x**5-2*x**4-x**3+x**2+x+7)+5*x**2*exp(4)+5
*x**7-10*x**6-30*x**5+45*x**4+20*x**3+25*x**2-5*x)/(x**2*exp(4)+x**7-2*x**6-x**5+x**4+x**3+7*x**2),x)

[Out]

5*x - 5*log(x**5 - 2*x**4 - x**3 + x**2 + x + 7 + exp(4))/x

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