3.7.26 \(\int \frac {1+2 x^2+2125764 x^3+393660 x^4+5340654 x^5+709344 x^6+4283612 x^7+394200 x^8+1506602 x^9+84636 x^{10}+237168 x^{11}+5832 x^{12}+13122 x^{13}}{x+531441 x^4+78732 x^5+712962 x^6+78840 x^7+357211 x^8+26280 x^9+79218 x^{10}+2916 x^{11}+6561 x^{12}} \, dx\)

Optimal. Leaf size=22 \[ x^2+\log \left (x+x^4 \left (x+9 \left (3+x^2\right )\right )^4\right ) \]

________________________________________________________________________________________

Rubi [B]  time = 0.61, antiderivative size = 54, normalized size of antiderivative = 2.45, number of steps used = 3, number of rules used = 2, integrand size = 112, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2074, 1587} \begin {gather*} x^2+\log \left (6561 x^{11}+2916 x^{10}+79218 x^9+26280 x^8+357211 x^7+78840 x^6+712962 x^5+78732 x^4+531441 x^3+1\right )+\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + 2*x^2 + 2125764*x^3 + 393660*x^4 + 5340654*x^5 + 709344*x^6 + 4283612*x^7 + 394200*x^8 + 1506602*x^9
+ 84636*x^10 + 237168*x^11 + 5832*x^12 + 13122*x^13)/(x + 531441*x^4 + 78732*x^5 + 712962*x^6 + 78840*x^7 + 35
7211*x^8 + 26280*x^9 + 79218*x^10 + 2916*x^11 + 6561*x^12),x]

[Out]

x^2 + Log[x] + Log[1 + 531441*x^3 + 78732*x^4 + 712962*x^5 + 78840*x^6 + 357211*x^7 + 26280*x^8 + 79218*x^9 +
2916*x^10 + 6561*x^11]

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 (\frac {1}{x}+2 x+\frac {x^2 \left (27+x+9 x^2\right )^3 \left (81+7 x+99 x^2\right )}{1+531441 x^3+78732 x^4+712962 x^5+78840 x^6+357211 x^7+26280 x^8+79218 x^9+2916 x^{10}+6561 x^{11}}\right ) \, dx\\ &=x^2+\log (x)+\int \frac {x^2 \left (27+x+9 x^2\right )^3 \left (81+7 x+99 x^2\right )}{1+531441 x^3+78732 x^4+712962 x^5+78840 x^6+357211 x^7+26280 x^8+79218 x^9+2916 x^{10}+6561 x^{11}} \, dx\\ &=x^2+\log (x)+\log \left (1+531441 x^3+78732 x^4+712962 x^5+78840 x^6+357211 x^7+26280 x^8+79218 x^9+2916 x^{10}+6561 x^{11}\right )\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [B]  time = 0.03, size = 54, normalized size = 2.45 \begin {gather*} x^2+\log (x)+\log \left (1+531441 x^3+78732 x^4+712962 x^5+78840 x^6+357211 x^7+26280 x^8+79218 x^9+2916 x^{10}+6561 x^{11}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + 2*x^2 + 2125764*x^3 + 393660*x^4 + 5340654*x^5 + 709344*x^6 + 4283612*x^7 + 394200*x^8 + 150660
2*x^9 + 84636*x^10 + 237168*x^11 + 5832*x^12 + 13122*x^13)/(x + 531441*x^4 + 78732*x^5 + 712962*x^6 + 78840*x^
7 + 357211*x^8 + 26280*x^9 + 79218*x^10 + 2916*x^11 + 6561*x^12),x]

[Out]

x^2 + Log[x] + Log[1 + 531441*x^3 + 78732*x^4 + 712962*x^5 + 78840*x^6 + 357211*x^7 + 26280*x^8 + 79218*x^9 +
2916*x^10 + 6561*x^11]

________________________________________________________________________________________

fricas [B]  time = 0.69, size = 52, normalized size = 2.36 \begin {gather*} x^{2} + \log \left (6561 \, x^{12} + 2916 \, x^{11} + 79218 \, x^{10} + 26280 \, x^{9} + 357211 \, x^{8} + 78840 \, x^{7} + 712962 \, x^{6} + 78732 \, x^{5} + 531441 \, x^{4} + x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13122*x^13+5832*x^12+237168*x^11+84636*x^10+1506602*x^9+394200*x^8+4283612*x^7+709344*x^6+5340654*x
^5+393660*x^4+2125764*x^3+2*x^2+1)/(6561*x^12+2916*x^11+79218*x^10+26280*x^9+357211*x^8+78840*x^7+712962*x^6+7
8732*x^5+531441*x^4+x),x, algorithm="fricas")

[Out]

x^2 + log(6561*x^12 + 2916*x^11 + 79218*x^10 + 26280*x^9 + 357211*x^8 + 78840*x^7 + 712962*x^6 + 78732*x^5 + 5
31441*x^4 + x)

________________________________________________________________________________________

giac [B]  time = 0.35, size = 56, normalized size = 2.55 \begin {gather*} x^{2} + \log \left ({\left | 6561 \, x^{11} + 2916 \, x^{10} + 79218 \, x^{9} + 26280 \, x^{8} + 357211 \, x^{7} + 78840 \, x^{6} + 712962 \, x^{5} + 78732 \, x^{4} + 531441 \, x^{3} + 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13122*x^13+5832*x^12+237168*x^11+84636*x^10+1506602*x^9+394200*x^8+4283612*x^7+709344*x^6+5340654*x
^5+393660*x^4+2125764*x^3+2*x^2+1)/(6561*x^12+2916*x^11+79218*x^10+26280*x^9+357211*x^8+78840*x^7+712962*x^6+7
8732*x^5+531441*x^4+x),x, algorithm="giac")

[Out]

x^2 + log(abs(6561*x^11 + 2916*x^10 + 79218*x^9 + 26280*x^8 + 357211*x^7 + 78840*x^6 + 712962*x^5 + 78732*x^4
+ 531441*x^3 + 1)) + log(abs(x))

________________________________________________________________________________________

maple [B]  time = 0.06, size = 53, normalized size = 2.41

method	result	size
risch	\(x^{2}+\ln \left (6561 x^{12}+2916 x^{11}+79218 x^{10}+26280 x^{9}+357211 x^{8}+78840 x^{7}+712962 x^{6}+78732 x^{5}+531441 x^{4}+x \right )\)	\(53\)
default	\(x^{2}+\ln \relax (x )+\ln \left (6561 x^{11}+2916 x^{10}+79218 x^{9}+26280 x^{8}+357211 x^{7}+78840 x^{6}+712962 x^{5}+78732 x^{4}+531441 x^{3}+1\right )\)	\(55\)
norman	\(x^{2}+\ln \relax (x )+\ln \left (6561 x^{11}+2916 x^{10}+79218 x^{9}+26280 x^{8}+357211 x^{7}+78840 x^{6}+712962 x^{5}+78732 x^{4}+531441 x^{3}+1\right )\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13122*x^13+5832*x^12+237168*x^11+84636*x^10+1506602*x^9+394200*x^8+4283612*x^7+709344*x^6+5340654*x^5+393
660*x^4+2125764*x^3+2*x^2+1)/(6561*x^12+2916*x^11+79218*x^10+26280*x^9+357211*x^8+78840*x^7+712962*x^6+78732*x
^5+531441*x^4+x),x,method=_RETURNVERBOSE)

[Out]

x^2+ln(6561*x^12+2916*x^11+79218*x^10+26280*x^9+357211*x^8+78840*x^7+712962*x^6+78732*x^5+531441*x^4+x)

________________________________________________________________________________________

maxima [B]  time = 0.38, size = 54, normalized size = 2.45 \begin {gather*} x^{2} + \log \left (6561 \, x^{11} + 2916 \, x^{10} + 79218 \, x^{9} + 26280 \, x^{8} + 357211 \, x^{7} + 78840 \, x^{6} + 712962 \, x^{5} + 78732 \, x^{4} + 531441 \, x^{3} + 1\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13122*x^13+5832*x^12+237168*x^11+84636*x^10+1506602*x^9+394200*x^8+4283612*x^7+709344*x^6+5340654*x
^5+393660*x^4+2125764*x^3+2*x^2+1)/(6561*x^12+2916*x^11+79218*x^10+26280*x^9+357211*x^8+78840*x^7+712962*x^6+7
8732*x^5+531441*x^4+x),x, algorithm="maxima")

[Out]

x^2 + log(6561*x^11 + 2916*x^10 + 79218*x^9 + 26280*x^8 + 357211*x^7 + 78840*x^6 + 712962*x^5 + 78732*x^4 + 53
1441*x^3 + 1) + log(x)

________________________________________________________________________________________

mupad [B]  time = 0.70, size = 52, normalized size = 2.36 \begin {gather*} \ln \left (6561\,x^{12}+2916\,x^{11}+79218\,x^{10}+26280\,x^9+357211\,x^8+78840\,x^7+712962\,x^6+78732\,x^5+531441\,x^4+x\right )+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^2 + 2125764*x^3 + 393660*x^4 + 5340654*x^5 + 709344*x^6 + 4283612*x^7 + 394200*x^8 + 1506602*x^9 + 84
636*x^10 + 237168*x^11 + 5832*x^12 + 13122*x^13 + 1)/(x + 531441*x^4 + 78732*x^5 + 712962*x^6 + 78840*x^7 + 35
7211*x^8 + 26280*x^9 + 79218*x^10 + 2916*x^11 + 6561*x^12),x)

[Out]

log(x + 531441*x^4 + 78732*x^5 + 712962*x^6 + 78840*x^7 + 357211*x^8 + 26280*x^9 + 79218*x^10 + 2916*x^11 + 65
61*x^12) + x^2

________________________________________________________________________________________

sympy [B]  time = 0.17, size = 51, normalized size = 2.32 \begin {gather*} x^{2} + \log {\left (6561 x^{12} + 2916 x^{11} + 79218 x^{10} + 26280 x^{9} + 357211 x^{8} + 78840 x^{7} + 712962 x^{6} + 78732 x^{5} + 531441 x^{4} + x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((13122*x**13+5832*x**12+237168*x**11+84636*x**10+1506602*x**9+394200*x**8+4283612*x**7+709344*x**6+5
340654*x**5+393660*x**4+2125764*x**3+2*x**2+1)/(6561*x**12+2916*x**11+79218*x**10+26280*x**9+357211*x**8+78840
*x**7+712962*x**6+78732*x**5+531441*x**4+x),x)

[Out]

x**2 + log(6561*x**12 + 2916*x**11 + 79218*x**10 + 26280*x**9 + 357211*x**8 + 78840*x**7 + 712962*x**6 + 78732
*x**5 + 531441*x**4 + x)

________________________________________________________________________________________