3.20.6 \(\int \frac {2 x+18 x^2+54 x^3+6 e^{2 x} x^3+54 x^4+e^x (2 x+12 x^2+24 x^3+18 x^4)+(6 x+42 x^2+90 x^3+54 x^4+e^{2 x} (2 x+2 x^2+2 x^3)+e^x (6 x+24 x^2+26 x^3+12 x^4)) \log (x)+(6 x+30 x^2+42 x^3+18 x^4+e^x (4 x+8 x^2+6 x^3+2 x^4)) \log ^2(x)+(2 x+6 x^2+6 x^3+2 x^4) \log ^3(x)}{1+9 x+27 x^2+27 x^3+(3+21 x+45 x^2+27 x^3) \log (x)+(3+15 x+21 x^2+9 x^3) \log ^2(x)+(1+3 x+3 x^2+x^3) \log ^3(x)} \, dx\)

Optimal. Leaf size=29 \[ \left (x+\frac {e^x}{-\frac {2-3 (1+x)-\log (x)}{x}+\log (x)}\right )^2 \]

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Rubi [F]  time = 14.94, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2 x+18 x^2+54 x^3+6 e^{2 x} x^3+54 x^4+e^x \left (2 x+12 x^2+24 x^3+18 x^4\right )+\left (6 x+42 x^2+90 x^3+54 x^4+e^{2 x} \left (2 x+2 x^2+2 x^3\right )+e^x \left (6 x+24 x^2+26 x^3+12 x^4\right )\right ) \log (x)+\left (6 x+30 x^2+42 x^3+18 x^4+e^x \left (4 x+8 x^2+6 x^3+2 x^4\right )\right ) \log ^2(x)+\left (2 x+6 x^2+6 x^3+2 x^4\right ) \log ^3(x)}{1+9 x+27 x^2+27 x^3+\left (3+21 x+45 x^2+27 x^3\right ) \log (x)+\left (3+15 x+21 x^2+9 x^3\right ) \log ^2(x)+\left (1+3 x+3 x^2+x^3\right ) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2*x + 18*x^2 + 54*x^3 + 6*E^(2*x)*x^3 + 54*x^4 + E^x*(2*x + 12*x^2 + 24*x^3 + 18*x^4) + (6*x + 42*x^2 + 9
0*x^3 + 54*x^4 + E^(2*x)*(2*x + 2*x^2 + 2*x^3) + E^x*(6*x + 24*x^2 + 26*x^3 + 12*x^4))*Log[x] + (6*x + 30*x^2
+ 42*x^3 + 18*x^4 + E^x*(4*x + 8*x^2 + 6*x^3 + 2*x^4))*Log[x]^2 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[x]^3)/(1 +
9*x + 27*x^2 + 27*x^3 + (3 + 21*x + 45*x^2 + 27*x^3)*Log[x] + (3 + 15*x + 21*x^2 + 9*x^3)*Log[x]^2 + (1 + 3*x
+ 3*x^2 + x^3)*Log[x]^3),x]

[Out]

x^2 + 24*Defer[Int][(-1 - 3*x - Log[x] - x*Log[x])^(-1), x] + 4*Defer[Int][E^(2*x)/(1 + 3*x + Log[x] + x*Log[x
])^3, x] - 6*Defer[Int][(E^(2*x)*x)/(1 + 3*x + Log[x] + x*Log[x])^3, x] - 2*Defer[Int][(E^(2*x)*x^2)/(1 + 3*x
+ Log[x] + x*Log[x])^3, x] - 4*Defer[Int][E^(2*x)/((1 + x)*(1 + 3*x + Log[x] + x*Log[x])^3), x] + 4*Defer[Int]
[E^x/(1 + 3*x + Log[x] + x*Log[x])^2, x] + 2*Defer[Int][E^(2*x)/(1 + 3*x + Log[x] + x*Log[x])^2, x] - 6*Defer[
Int][(E^x*x)/(1 + 3*x + Log[x] + x*Log[x])^2, x] - 2*Defer[Int][(E^x*x^2)/(1 + 3*x + Log[x] + x*Log[x])^2, x]
+ 2*Defer[Int][(E^(2*x)*x^2)/(1 + 3*x + Log[x] + x*Log[x])^2, x] - 4*Defer[Int][E^x/((1 + x)*(1 + 3*x + Log[x]
+ x*Log[x])^2), x] - 2*Defer[Int][E^(2*x)/((1 + x)*(1 + 3*x + Log[x] + x*Log[x])^2), x] + 24*Defer[Int][(1 +
3*x + Log[x] + x*Log[x])^(-1), x] + 2*Defer[Int][E^x/(1 + 3*x + Log[x] + x*Log[x]), x] + 2*Defer[Int][(E^x*x)/
(1 + 3*x + Log[x] + x*Log[x]), x] + 2*Defer[Int][(E^x*x^2)/(1 + 3*x + Log[x] + x*Log[x]), x] - 2*Defer[Int][E^
x/((1 + x)*(1 + 3*x + Log[x] + x*Log[x])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 x \left (3 e^{2 x} x^2+(1+3 x)^3+e^x \left (1+6 x+12 x^2+9 x^3\right )+\left (3 (1+x) (1+3 x)^2+e^{2 x} \left (1+x+x^2\right )+e^x \left (3+12 x+13 x^2+6 x^3\right )\right ) \log (x)+(1+x) \left (3+12 x+9 x^2+e^x \left (2+2 x+x^2\right )\right ) \log ^2(x)+(1+x)^3 \log ^3(x)\right )}{(1+3 x+(1+x) \log (x))^3} \, dx\\ &=2 \int \frac {x \left (3 e^{2 x} x^2+(1+3 x)^3+e^x \left (1+6 x+12 x^2+9 x^3\right )+\left (3 (1+x) (1+3 x)^2+e^{2 x} \left (1+x+x^2\right )+e^x \left (3+12 x+13 x^2+6 x^3\right )\right ) \log (x)+(1+x) \left (3+12 x+9 x^2+e^x \left (2+2 x+x^2\right )\right ) \log ^2(x)+(1+x)^3 \log ^3(x)\right )}{(1+3 x+(1+x) \log (x))^3} \, dx\\ &=2 \int \left (\frac {x (1+3 x)^3}{(1+3 x+\log (x)+x \log (x))^3}+\frac {3 x (1+x) (1+3 x)^2 \log (x)}{(1+3 x+\log (x)+x \log (x))^3}+\frac {3 x (1+x) \log ^2(x)}{(1+3 x+\log (x)+x \log (x))^3}+\frac {12 x^2 (1+x) \log ^2(x)}{(1+3 x+\log (x)+x \log (x))^3}+\frac {9 x^3 (1+x) \log ^2(x)}{(1+3 x+\log (x)+x \log (x))^3}+\frac {x (1+x)^3 \log ^3(x)}{(1+3 x+\log (x)+x \log (x))^3}+\frac {e^{2 x} x \left (3 x^2+\log (x)+x \log (x)+x^2 \log (x)\right )}{(1+3 x+\log (x)+x \log (x))^3}+\frac {e^x x \left (1+3 x+3 x^2+2 \log (x)+2 x \log (x)+x^2 \log (x)\right )}{(1+3 x+\log (x)+x \log (x))^2}\right ) \, dx\\ &=2 \int \frac {x (1+3 x)^3}{(1+3 x+\log (x)+x \log (x))^3} \, dx+2 \int \frac {x (1+x)^3 \log ^3(x)}{(1+3 x+\log (x)+x \log (x))^3} \, dx+2 \int \frac {e^{2 x} x \left (3 x^2+\log (x)+x \log (x)+x^2 \log (x)\right )}{(1+3 x+\log (x)+x \log (x))^3} \, dx+2 \int \frac {e^x x \left (1+3 x+3 x^2+2 \log (x)+2 x \log (x)+x^2 \log (x)\right )}{(1+3 x+\log (x)+x \log (x))^2} \, dx+6 \int \frac {x (1+x) (1+3 x)^2 \log (x)}{(1+3 x+\log (x)+x \log (x))^3} \, dx+6 \int \frac {x (1+x) \log ^2(x)}{(1+3 x+\log (x)+x \log (x))^3} \, dx+18 \int \frac {x^3 (1+x) \log ^2(x)}{(1+3 x+\log (x)+x \log (x))^3} \, dx+24 \int \frac {x^2 (1+x) \log ^2(x)}{(1+3 x+\log (x)+x \log (x))^3} \, dx\\ &=2 \int \left (\frac {x}{(1+3 x+\log (x)+x \log (x))^3}+\frac {9 x^2}{(1+3 x+\log (x)+x \log (x))^3}+\frac {27 x^3}{(1+3 x+\log (x)+x \log (x))^3}+\frac {27 x^4}{(1+3 x+\log (x)+x \log (x))^3}\right ) \, dx+2 \int \left (-\frac {e^{2 x} x \left (1+4 x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))^3}+\frac {e^{2 x} x \left (1+x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))^2}\right ) \, dx+2 \int \left (x-\frac {x (1+3 x)^3}{(1+3 x+\log (x)+x \log (x))^3}+\frac {3 x (1+3 x)^2}{(1+3 x+\log (x)+x \log (x))^2}-\frac {3 x (1+3 x)}{1+3 x+\log (x)+x \log (x)}\right ) \, dx+2 \int \left (-\frac {e^x x \left (1+4 x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))^2}+\frac {e^x x \left (2+2 x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))}\right ) \, dx+6 \int \left (-\frac {x (1+3 x)^3}{(1+3 x+\log (x)+x \log (x))^3}+\frac {x (1+3 x)^2}{(1+3 x+\log (x)+x \log (x))^2}\right ) \, dx+6 \int \left (\frac {x (1+3 x)^2}{(1+x) (1+3 x+\log (x)+x \log (x))^3}-\frac {2 x (1+3 x)}{(1+x) (1+3 x+\log (x)+x \log (x))^2}+\frac {x}{(1+x) (1+3 x+\log (x)+x \log (x))}\right ) \, dx+18 \int \left (\frac {x^3 (1+3 x)^2}{(1+x) (1+3 x+\log (x)+x \log (x))^3}-\frac {2 x^3 (1+3 x)}{(1+x) (1+3 x+\log (x)+x \log (x))^2}+\frac {x^3}{(1+x) (1+3 x+\log (x)+x \log (x))}\right ) \, dx+24 \int \left (\frac {x^2 (1+3 x)^2}{(1+x) (1+3 x+\log (x)+x \log (x))^3}-\frac {2 x^2 (1+3 x)}{(1+x) (1+3 x+\log (x)+x \log (x))^2}+\frac {x^2}{(1+x) (1+3 x+\log (x)+x \log (x))}\right ) \, dx\\ &=x^2+2 \int \frac {x}{(1+3 x+\log (x)+x \log (x))^3} \, dx-2 \int \frac {x (1+3 x)^3}{(1+3 x+\log (x)+x \log (x))^3} \, dx-2 \int \frac {e^{2 x} x \left (1+4 x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))^3} \, dx+2 \int \frac {e^{2 x} x \left (1+x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))^2} \, dx-2 \int \frac {e^x x \left (1+4 x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))^2} \, dx+2 \int \frac {e^x x \left (2+2 x+x^2\right )}{(1+x) (1+3 x+\log (x)+x \log (x))} \, dx+6 \int \frac {x (1+3 x)^2}{(1+x) (1+3 x+\log (x)+x \log (x))^3} \, dx-6 \int \frac {x (1+3 x)^3}{(1+3 x+\log (x)+x \log (x))^3} \, dx+2 \left (6 \int \frac {x (1+3 x)^2}{(1+3 x+\log (x)+x \log (x))^2} \, dx\right )+6 \int \frac {x}{(1+x) (1+3 x+\log (x)+x \log (x))} \, dx-6 \int \frac {x (1+3 x)}{1+3 x+\log (x)+x \log (x)} \, dx-12 \int \frac {x (1+3 x)}{(1+x) (1+3 x+\log (x)+x \log (x))^2} \, dx+18 \int \frac {x^2}{(1+3 x+\log (x)+x \log (x))^3} \, dx+18 \int \frac {x^3 (1+3 x)^2}{(1+x) (1+3 x+\log (x)+x \log (x))^3} \, dx+18 \int \frac {x^3}{(1+x) (1+3 x+\log (x)+x \log (x))} \, dx+24 \int \frac {x^2 (1+3 x)^2}{(1+x) (1+3 x+\log (x)+x \log (x))^3} \, dx+24 \int \frac {x^2}{(1+x) (1+3 x+\log (x)+x \log (x))} \, dx-36 \int \frac {x^3 (1+3 x)}{(1+x) (1+3 x+\log (x)+x \log (x))^2} \, dx-48 \int \frac {x^2 (1+3 x)}{(1+x) (1+3 x+\log (x)+x \log (x))^2} \, dx+54 \int \frac {x^3}{(1+3 x+\log (x)+x \log (x))^3} \, dx+54 \int \frac {x^4}{(1+3 x+\log (x)+x \log (x))^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 33, normalized size = 1.14 \begin {gather*} \frac {x^2 \left (1+e^x+3 x+(1+x) \log (x)\right )^2}{(1+3 x+(1+x) \log (x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2*x + 18*x^2 + 54*x^3 + 6*E^(2*x)*x^3 + 54*x^4 + E^x*(2*x + 12*x^2 + 24*x^3 + 18*x^4) + (6*x + 42*x
^2 + 90*x^3 + 54*x^4 + E^(2*x)*(2*x + 2*x^2 + 2*x^3) + E^x*(6*x + 24*x^2 + 26*x^3 + 12*x^4))*Log[x] + (6*x + 3
0*x^2 + 42*x^3 + 18*x^4 + E^x*(4*x + 8*x^2 + 6*x^3 + 2*x^4))*Log[x]^2 + (2*x + 6*x^2 + 6*x^3 + 2*x^4)*Log[x]^3
)/(1 + 9*x + 27*x^2 + 27*x^3 + (3 + 21*x + 45*x^2 + 27*x^3)*Log[x] + (3 + 15*x + 21*x^2 + 9*x^3)*Log[x]^2 + (1
+ 3*x + 3*x^2 + x^3)*Log[x]^3),x]

[Out]

(x^2*(1 + E^x + 3*x + (1 + x)*Log[x])^2)/(1 + 3*x + (1 + x)*Log[x])^2

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fricas [B]  time = 0.78, size = 120, normalized size = 4.14 \begin {gather*} \frac {9 \, x^{4} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (x)^{2} + x^{2} + 2 \, {\left (3 \, x^{3} + x^{2}\right )} e^{x} + 2 \, {\left (3 \, x^{4} + 4 \, x^{3} + x^{2} + {\left (x^{3} + x^{2}\right )} e^{x}\right )} \log \relax (x)}{{\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 9 \, x^{2} + 2 \, {\left (3 \, x^{2} + 4 \, x + 1\right )} \log \relax (x) + 6 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+6*x^3+6*x^2+2*x)*log(x)^3+((2*x^4+6*x^3+8*x^2+4*x)*exp(x)+18*x^4+42*x^3+30*x^2+6*x)*log(x)^2
+((2*x^3+2*x^2+2*x)*exp(x)^2+(12*x^4+26*x^3+24*x^2+6*x)*exp(x)+54*x^4+90*x^3+42*x^2+6*x)*log(x)+6*exp(x)^2*x^3
+(18*x^4+24*x^3+12*x^2+2*x)*exp(x)+54*x^4+54*x^3+18*x^2+2*x)/((x^3+3*x^2+3*x+1)*log(x)^3+(9*x^3+21*x^2+15*x+3)
*log(x)^2+(27*x^3+45*x^2+21*x+3)*log(x)+27*x^3+27*x^2+9*x+1),x, algorithm="fricas")

[Out]

(9*x^4 + 6*x^3 + x^2*e^(2*x) + (x^4 + 2*x^3 + x^2)*log(x)^2 + x^2 + 2*(3*x^3 + x^2)*e^x + 2*(3*x^4 + 4*x^3 + x
^2 + (x^3 + x^2)*e^x)*log(x))/((x^2 + 2*x + 1)*log(x)^2 + 9*x^2 + 2*(3*x^2 + 4*x + 1)*log(x) + 6*x + 1)

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giac [B]  time = 0.41, size = 148, normalized size = 5.10 \begin {gather*} \frac {x^{4} \log \relax (x)^{2} + 6 \, x^{4} \log \relax (x) + 2 \, x^{3} e^{x} \log \relax (x) + 2 \, x^{3} \log \relax (x)^{2} + 9 \, x^{4} + 6 \, x^{3} e^{x} + 8 \, x^{3} \log \relax (x) + 2 \, x^{2} e^{x} \log \relax (x) + x^{2} \log \relax (x)^{2} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + 2 \, x^{2} e^{x} + 2 \, x^{2} \log \relax (x) + x^{2}}{x^{2} \log \relax (x)^{2} + 6 \, x^{2} \log \relax (x) + 2 \, x \log \relax (x)^{2} + 9 \, x^{2} + 8 \, x \log \relax (x) + \log \relax (x)^{2} + 6 \, x + 2 \, \log \relax (x) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+6*x^3+6*x^2+2*x)*log(x)^3+((2*x^4+6*x^3+8*x^2+4*x)*exp(x)+18*x^4+42*x^3+30*x^2+6*x)*log(x)^2
+((2*x^3+2*x^2+2*x)*exp(x)^2+(12*x^4+26*x^3+24*x^2+6*x)*exp(x)+54*x^4+90*x^3+42*x^2+6*x)*log(x)+6*exp(x)^2*x^3
+(18*x^4+24*x^3+12*x^2+2*x)*exp(x)+54*x^4+54*x^3+18*x^2+2*x)/((x^3+3*x^2+3*x+1)*log(x)^3+(9*x^3+21*x^2+15*x+3)
*log(x)^2+(27*x^3+45*x^2+21*x+3)*log(x)+27*x^3+27*x^2+9*x+1),x, algorithm="giac")

[Out]

(x^4*log(x)^2 + 6*x^4*log(x) + 2*x^3*e^x*log(x) + 2*x^3*log(x)^2 + 9*x^4 + 6*x^3*e^x + 8*x^3*log(x) + 2*x^2*e^
x*log(x) + x^2*log(x)^2 + 6*x^3 + x^2*e^(2*x) + 2*x^2*e^x + 2*x^2*log(x) + x^2)/(x^2*log(x)^2 + 6*x^2*log(x) +
2*x*log(x)^2 + 9*x^2 + 8*x*log(x) + log(x)^2 + 6*x + 2*log(x) + 1)

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maple [A]  time = 0.07, size = 40, normalized size = 1.38

method	result	size
risch	\(x^{2}+\frac {\left (2 x \ln \relax (x )+6 x +{\mathrm e}^{x}+2 \ln \relax (x )+2\right ) x^{2} {\mathrm e}^{x}}{\left (x \ln \relax (x )+\ln \relax (x )+3 x +1\right )^{2}}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^4+6*x^3+6*x^2+2*x)*ln(x)^3+((2*x^4+6*x^3+8*x^2+4*x)*exp(x)+18*x^4+42*x^3+30*x^2+6*x)*ln(x)^2+((2*x^3
+2*x^2+2*x)*exp(x)^2+(12*x^4+26*x^3+24*x^2+6*x)*exp(x)+54*x^4+90*x^3+42*x^2+6*x)*ln(x)+6*exp(x)^2*x^3+(18*x^4+
24*x^3+12*x^2+2*x)*exp(x)+54*x^4+54*x^3+18*x^2+2*x)/((x^3+3*x^2+3*x+1)*ln(x)^3+(9*x^3+21*x^2+15*x+3)*ln(x)^2+(
27*x^3+45*x^2+21*x+3)*ln(x)+27*x^3+27*x^2+9*x+1),x,method=_RETURNVERBOSE)

[Out]

x^2+(2*x*ln(x)+6*x+exp(x)+2*ln(x)+2)*x^2*exp(x)/(x*ln(x)+ln(x)+3*x+1)^2

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maxima [B]  time = 0.57, size = 120, normalized size = 4.14 \begin {gather*} \frac {9 \, x^{4} + 6 \, x^{3} + x^{2} e^{\left (2 \, x\right )} + {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} \log \relax (x)^{2} + x^{2} + 2 \, {\left (3 \, x^{3} + x^{2} + {\left (x^{3} + x^{2}\right )} \log \relax (x)\right )} e^{x} + 2 \, {\left (3 \, x^{4} + 4 \, x^{3} + x^{2}\right )} \log \relax (x)}{{\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} + 9 \, x^{2} + 2 \, {\left (3 \, x^{2} + 4 \, x + 1\right )} \log \relax (x) + 6 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^4+6*x^3+6*x^2+2*x)*log(x)^3+((2*x^4+6*x^3+8*x^2+4*x)*exp(x)+18*x^4+42*x^3+30*x^2+6*x)*log(x)^2
+((2*x^3+2*x^2+2*x)*exp(x)^2+(12*x^4+26*x^3+24*x^2+6*x)*exp(x)+54*x^4+90*x^3+42*x^2+6*x)*log(x)+6*exp(x)^2*x^3
+(18*x^4+24*x^3+12*x^2+2*x)*exp(x)+54*x^4+54*x^3+18*x^2+2*x)/((x^3+3*x^2+3*x+1)*log(x)^3+(9*x^3+21*x^2+15*x+3)
*log(x)^2+(27*x^3+45*x^2+21*x+3)*log(x)+27*x^3+27*x^2+9*x+1),x, algorithm="maxima")

[Out]

(9*x^4 + 6*x^3 + x^2*e^(2*x) + (x^4 + 2*x^3 + x^2)*log(x)^2 + x^2 + 2*(3*x^3 + x^2 + (x^3 + x^2)*log(x))*e^x +
2*(3*x^4 + 4*x^3 + x^2)*log(x))/((x^2 + 2*x + 1)*log(x)^2 + 9*x^2 + 2*(3*x^2 + 4*x + 1)*log(x) + 6*x + 1)

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mupad [B]  time = 1.43, size = 32, normalized size = 1.10 \begin {gather*} \frac {x^2\,{\left (3\,x+{\mathrm {e}}^x+\ln \relax (x)+x\,\ln \relax (x)+1\right )}^2}{{\left (3\,x+\ln \relax (x)+x\,\ln \relax (x)+1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + exp(x)*(2*x + 12*x^2 + 24*x^3 + 18*x^4) + 6*x^3*exp(2*x) + log(x)*(6*x + exp(x)*(6*x + 24*x^2 + 26*
x^3 + 12*x^4) + exp(2*x)*(2*x + 2*x^2 + 2*x^3) + 42*x^2 + 90*x^3 + 54*x^4) + log(x)^3*(2*x + 6*x^2 + 6*x^3 + 2
*x^4) + log(x)^2*(6*x + exp(x)*(4*x + 8*x^2 + 6*x^3 + 2*x^4) + 30*x^2 + 42*x^3 + 18*x^4) + 18*x^2 + 54*x^3 + 5
4*x^4)/(9*x + log(x)^2*(15*x + 21*x^2 + 9*x^3 + 3) + 27*x^2 + 27*x^3 + log(x)*(21*x + 45*x^2 + 27*x^3 + 3) + l
og(x)^3*(3*x + 3*x^2 + x^3 + 1) + 1),x)

[Out]

(x^2*(3*x + exp(x) + log(x) + x*log(x) + 1)^2)/(3*x + log(x) + x*log(x) + 1)^2

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sympy [B]  time = 0.67, size = 216, normalized size = 7.45 \begin {gather*} x^{2} + \frac {\left (x^{3} \log {\relax (x )} + 3 x^{3} + x^{2} \log {\relax (x )} + x^{2}\right ) e^{2 x} + \left (2 x^{4} \log {\relax (x )}^{2} + 12 x^{4} \log {\relax (x )} + 18 x^{4} + 4 x^{3} \log {\relax (x )}^{2} + 16 x^{3} \log {\relax (x )} + 12 x^{3} + 2 x^{2} \log {\relax (x )}^{2} + 4 x^{2} \log {\relax (x )} + 2 x^{2}\right ) e^{x}}{x^{3} \log {\relax (x )}^{3} + 9 x^{3} \log {\relax (x )}^{2} + 27 x^{3} \log {\relax (x )} + 27 x^{3} + 3 x^{2} \log {\relax (x )}^{3} + 21 x^{2} \log {\relax (x )}^{2} + 45 x^{2} \log {\relax (x )} + 27 x^{2} + 3 x \log {\relax (x )}^{3} + 15 x \log {\relax (x )}^{2} + 21 x \log {\relax (x )} + 9 x + \log {\relax (x )}^{3} + 3 \log {\relax (x )}^{2} + 3 \log {\relax (x )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**4+6*x**3+6*x**2+2*x)*ln(x)**3+((2*x**4+6*x**3+8*x**2+4*x)*exp(x)+18*x**4+42*x**3+30*x**2+6*x)
*ln(x)**2+((2*x**3+2*x**2+2*x)*exp(x)**2+(12*x**4+26*x**3+24*x**2+6*x)*exp(x)+54*x**4+90*x**3+42*x**2+6*x)*ln(
x)+6*exp(x)**2*x**3+(18*x**4+24*x**3+12*x**2+2*x)*exp(x)+54*x**4+54*x**3+18*x**2+2*x)/((x**3+3*x**2+3*x+1)*ln(
x)**3+(9*x**3+21*x**2+15*x+3)*ln(x)**2+(27*x**3+45*x**2+21*x+3)*ln(x)+27*x**3+27*x**2+9*x+1),x)

[Out]

x**2 + ((x**3*log(x) + 3*x**3 + x**2*log(x) + x**2)*exp(2*x) + (2*x**4*log(x)**2 + 12*x**4*log(x) + 18*x**4 +
4*x**3*log(x)**2 + 16*x**3*log(x) + 12*x**3 + 2*x**2*log(x)**2 + 4*x**2*log(x) + 2*x**2)*exp(x))/(x**3*log(x)*
*3 + 9*x**3*log(x)**2 + 27*x**3*log(x) + 27*x**3 + 3*x**2*log(x)**3 + 21*x**2*log(x)**2 + 45*x**2*log(x) + 27*
x**2 + 3*x*log(x)**3 + 15*x*log(x)**2 + 21*x*log(x) + 9*x + log(x)**3 + 3*log(x)**2 + 3*log(x) + 1)

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