3.102.4 \(\int \frac {-162-657 x-260 x^2+1379 x^3+1254 x^4+410 x^5+58 x^6+3 x^7+e^{-\frac {1}{-9-25 x-10 x^2-x^3+3 \log (x)}} (6-50 x-40 x^2-6 x^3)+(108+138 x-330 x^2-168 x^3-18 x^4) \log (x)+(-18+27 x) \log ^2(x)}{81 x+450 x^2+805 x^3+518 x^4+150 x^5+20 x^6+x^7+(-54 x-150 x^2-60 x^3-6 x^4) \log (x)+9 x \log ^2(x)} \, dx\)

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

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Rubi [A]  time = 6.76, antiderivative size = 31, normalized size of antiderivative = 1.03, number of steps used = 20, number of rules used = 4, integrand size = 173, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.023, Rules used = {6688, 6742, 6706, 43} \begin {gather*} 2 e^{\frac {1}{x^3+10 x^2+25 x-3 \log (x)+9}}+3 x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-162 - 657*x - 260*x^2 + 1379*x^3 + 1254*x^4 + 410*x^5 + 58*x^6 + 3*x^7 + (6 - 50*x - 40*x^2 - 6*x^3)/E^(
-9 - 25*x - 10*x^2 - x^3 + 3*Log[x])^(-1) + (108 + 138*x - 330*x^2 - 168*x^3 - 18*x^4)*Log[x] + (-18 + 27*x)*L
og[x]^2)/(81*x + 450*x^2 + 805*x^3 + 518*x^4 + 150*x^5 + 20*x^6 + x^7 + (-54*x - 150*x^2 - 60*x^3 - 6*x^4)*Log
[x] + 9*x*Log[x]^2),x]

[Out]

2*E^(9 + 25*x + 10*x^2 + x^3 - 3*Log[x])^(-1) + 3*x - 2*Log[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 6688

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, 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 {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2-2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}} \left (-3+25 x+20 x^2+3 x^3\right )-6 \left (-18-23 x+55 x^2+28 x^3+3 x^4\right ) \log (x)+9 (-2+3 x) \log ^2(x)}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\\ &=\int \left (\frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}} \left (-3+25 x+20 x^2+3 x^3\right )}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {6 (-2+3 x) \left (9+25 x+10 x^2+x^3\right ) \log (x)}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {9 (-2+3 x) \log ^2(x)}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}} \left (-3+25 x+20 x^2+3 x^3\right )}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )-6 \int \frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right ) \log (x)}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+9 \int \frac {(-2+3 x) \log ^2(x)}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+\int \frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\\ &=2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}}-6 \int \left (\frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2}{3 x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )}{3 x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )}\right ) \, dx+9 \int \left (\frac {-2+3 x}{9 x}+\frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2}{9 x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {2 \left (-18-23 x+55 x^2+28 x^3+3 x^4\right )}{9 x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )}\right ) \, dx+\int \left (-\frac {657}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {162}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {260 x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {1379 x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {1254 x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {410 x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {58 x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {3 x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}\right ) \, dx\\ &=2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}}-2 \int \frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+2 \int \frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )} \, dx-2 \int \frac {-18-23 x+55 x^2+28 x^3+3 x^4}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )} \, dx+3 \int \frac {x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+58 \int \frac {x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-162 \int \frac {1}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-260 \int \frac {x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+410 \int \frac {x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-657 \int \frac {1}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+1254 \int \frac {x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+1379 \int \frac {x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+\int \frac {-2+3 x}{x} \, dx+\int \frac {(-2+3 x) \left (9+25 x+10 x^2+x^3\right )^2}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\\ &=2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}}-2 \int \left (-\frac {657}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {162}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {260 x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {1379 x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {1254 x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {410 x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {58 x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {3 x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}\right ) \, dx+3 \int \frac {x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+58 \int \frac {x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-162 \int \frac {1}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-260 \int \frac {x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+410 \int \frac {x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-657 \int \frac {1}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+1254 \int \frac {x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+1379 \int \frac {x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+\int \left (3-\frac {2}{x}\right ) \, dx+\int \left (-\frac {657}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {162}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}-\frac {260 x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {1379 x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {1254 x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {410 x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {58 x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}+\frac {3 x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2}\right ) \, dx\\ &=2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}}+3 x-2 \log (x)+2 \left (3 \int \frac {x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )-6 \int \frac {x^6}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+2 \left (58 \int \frac {x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )-116 \int \frac {x^5}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-2 \left (162 \int \frac {1}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )-2 \left (260 \int \frac {x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )+324 \int \frac {1}{x \left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+2 \left (410 \int \frac {x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )+520 \int \frac {x}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-2 \left (657 \int \frac {1}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )-820 \int \frac {x^4}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+2 \left (1254 \int \frac {x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )+1314 \int \frac {1}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx+2 \left (1379 \int \frac {x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\right )-2508 \int \frac {x^3}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx-2758 \int \frac {x^2}{\left (9+25 x+10 x^2+x^3-3 \log (x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 31, normalized size = 1.03 \begin {gather*} 2 e^{\frac {1}{9+25 x+10 x^2+x^3-3 \log (x)}}+3 x-2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-162 - 657*x - 260*x^2 + 1379*x^3 + 1254*x^4 + 410*x^5 + 58*x^6 + 3*x^7 + (6 - 50*x - 40*x^2 - 6*x^
3)/E^(-9 - 25*x - 10*x^2 - x^3 + 3*Log[x])^(-1) + (108 + 138*x - 330*x^2 - 168*x^3 - 18*x^4)*Log[x] + (-18 + 2
7*x)*Log[x]^2)/(81*x + 450*x^2 + 805*x^3 + 518*x^4 + 150*x^5 + 20*x^6 + x^7 + (-54*x - 150*x^2 - 60*x^3 - 6*x^
4)*Log[x] + 9*x*Log[x]^2),x]

[Out]

2*E^(9 + 25*x + 10*x^2 + x^3 - 3*Log[x])^(-1) + 3*x - 2*Log[x]

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fricas [A]  time = 0.58, size = 30, normalized size = 1.00 \begin {gather*} 3 \, x + 2 \, e^{\left (\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \relax (x) + 9}\right )} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27*x-18)*log(x)^2+(-18*x^4-168*x^3-330
*x^2+138*x+108)*log(x)+3*x^7+58*x^6+410*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-
150*x^2-54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x, algorithm="fricas")

[Out]

3*x + 2*e^(1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) - 2*log(x)

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giac [A]  time = 0.24, size = 30, normalized size = 1.00 \begin {gather*} 3 \, x + 2 \, e^{\left (\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \relax (x) + 9}\right )} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27*x-18)*log(x)^2+(-18*x^4-168*x^3-330
*x^2+138*x+108)*log(x)+3*x^7+58*x^6+410*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-
150*x^2-54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x, algorithm="giac")

[Out]

3*x + 2*e^(1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) - 2*log(x)

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maple [A]  time = 0.05, size = 35, normalized size = 1.17

method	result	size
risch	\(3 x -2 \ln \relax (x )+2 \,{\mathrm e}^{-\frac {1}{3 \ln \relax (x )-x^{3}-10 x^{2}-25 x -9}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*ln(x)-x^3-10*x^2-25*x-9))+(27*x-18)*ln(x)^2+(-18*x^4-168*x^3-330*x^2+138
*x+108)*ln(x)+3*x^7+58*x^6+410*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*ln(x)^2+(-6*x^4-60*x^3-150*x^2-54
*x)*ln(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x,method=_RETURNVERBOSE)

[Out]

3*x-2*ln(x)+2*exp(-1/(3*ln(x)-x^3-10*x^2-25*x-9))

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maxima [B]  time = 0.38, size = 53, normalized size = 1.77 \begin {gather*} {\left (3 \, x e^{\left (-\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \relax (x) + 9}\right )} + 2\right )} e^{\left (\frac {1}{x^{3} + 10 \, x^{2} + 25 \, x - 3 \, \log \relax (x) + 9}\right )} - 2 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x^3-40*x^2-50*x+6)*exp(-1/(3*log(x)-x^3-10*x^2-25*x-9))+(27*x-18)*log(x)^2+(-18*x^4-168*x^3-330
*x^2+138*x+108)*log(x)+3*x^7+58*x^6+410*x^5+1254*x^4+1379*x^3-260*x^2-657*x-162)/(9*x*log(x)^2+(-6*x^4-60*x^3-
150*x^2-54*x)*log(x)+x^7+20*x^6+150*x^5+518*x^4+805*x^3+450*x^2+81*x),x, algorithm="maxima")

[Out]

(3*x*e^(-1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) + 2)*e^(1/(x^3 + 10*x^2 + 25*x - 3*log(x) + 9)) - 2*log(x)

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mupad [B]  time = 6.68, size = 30, normalized size = 1.00 \begin {gather*} 3\,x+2\,{\mathrm {e}}^{\frac {1}{25\,x-3\,\ln \relax (x)+10\,x^2+x^3+9}}-2\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1379*x^3 - log(x)*(330*x^2 - 138*x + 168*x^3 + 18*x^4 - 108) - exp(1/(25*x - 3*log(x) + 10*x^2 + x^3 + 9)
)*(50*x + 40*x^2 + 6*x^3 - 6) - 260*x^2 - 657*x + 1254*x^4 + 410*x^5 + 58*x^6 + 3*x^7 + log(x)^2*(27*x - 18) -
162)/(81*x + 9*x*log(x)^2 - log(x)*(54*x + 150*x^2 + 60*x^3 + 6*x^4) + 450*x^2 + 805*x^3 + 518*x^4 + 150*x^5
+ 20*x^6 + x^7),x)

[Out]

3*x + 2*exp(1/(25*x - 3*log(x) + 10*x^2 + x^3 + 9)) - 2*log(x)

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sympy [A]  time = 0.86, size = 31, normalized size = 1.03 \begin {gather*} 3 x - 2 \log {\relax (x )} + 2 e^{- \frac {1}{- x^{3} - 10 x^{2} - 25 x + 3 \log {\relax (x )} - 9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-6*x**3-40*x**2-50*x+6)*exp(-1/(3*ln(x)-x**3-10*x**2-25*x-9))+(27*x-18)*ln(x)**2+(-18*x**4-168*x**
3-330*x**2+138*x+108)*ln(x)+3*x**7+58*x**6+410*x**5+1254*x**4+1379*x**3-260*x**2-657*x-162)/(9*x*ln(x)**2+(-6*
x**4-60*x**3-150*x**2-54*x)*ln(x)+x**7+20*x**6+150*x**5+518*x**4+805*x**3+450*x**2+81*x),x)

[Out]

3*x - 2*log(x) + 2*exp(-1/(-x**3 - 10*x**2 - 25*x + 3*log(x) - 9))

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