3.81.9 \(\int \frac {e^{4 x} (-392 x-28 x^2+812 x^3+114 x^4+4 x^5)+e^{4 x} (364 x+868 x^2+230 x^3+12 x^4) \log (3 x)+e^{4 x} (56 x+118 x^2+12 x^3) \log ^2(3 x)+e^{4 x} (2 x+4 x^2) \log ^3(3 x)}{x^3+3 x^2 \log (3 x)+3 x \log ^2(3 x)+\log ^3(3 x)} \, dx\)

Optimal. Leaf size=21 \[ e^{4 x} \left (x+\frac {14 x}{x+\log (3 x)}\right )^2 \]

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Rubi [F]  time = 3.13, 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 {e^{4 x} \left (-392 x-28 x^2+812 x^3+114 x^4+4 x^5\right )+e^{4 x} \left (364 x+868 x^2+230 x^3+12 x^4\right ) \log (3 x)+e^{4 x} \left (56 x+118 x^2+12 x^3\right ) \log ^2(3 x)+e^{4 x} \left (2 x+4 x^2\right ) \log ^3(3 x)}{x^3+3 x^2 \log (3 x)+3 x \log ^2(3 x)+\log ^3(3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(4*x)*(-392*x - 28*x^2 + 812*x^3 + 114*x^4 + 4*x^5) + E^(4*x)*(364*x + 868*x^2 + 230*x^3 + 12*x^4)*Log[
3*x] + E^(4*x)*(56*x + 118*x^2 + 12*x^3)*Log[3*x]^2 + E^(4*x)*(2*x + 4*x^2)*Log[3*x]^3)/(x^3 + 3*x^2*Log[3*x]
+ 3*x*Log[3*x]^2 + Log[3*x]^3),x]

[Out]

E^(4*x)*x^2 - 392*Defer[Int][(E^(4*x)*x)/(x + Log[3*x])^3, x] - 392*Defer[Int][(E^(4*x)*x^2)/(x + Log[3*x])^3,
x] + 364*Defer[Int][(E^(4*x)*x)/(x + Log[3*x])^2, x] + 756*Defer[Int][(E^(4*x)*x^2)/(x + Log[3*x])^2, x] + 56
*Defer[Int][(E^(4*x)*x)/(x + Log[3*x]), x] + 112*Defer[Int][(E^(4*x)*x^2)/(x + Log[3*x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{4 x} x \left (-196-14 x+406 x^2+57 x^3+2 x^4+\left (182+434 x+115 x^2+6 x^3\right ) \log (3 x)+\left (28+59 x+6 x^2\right ) \log ^2(3 x)+(1+2 x) \log ^3(3 x)\right )}{(x+\log (3 x))^3} \, dx\\ &=2 \int \frac {e^{4 x} x \left (-196-14 x+406 x^2+57 x^3+2 x^4+\left (182+434 x+115 x^2+6 x^3\right ) \log (3 x)+\left (28+59 x+6 x^2\right ) \log ^2(3 x)+(1+2 x) \log ^3(3 x)\right )}{(x+\log (3 x))^3} \, dx\\ &=2 \int \left (e^{4 x} x (1+2 x)-\frac {196 e^{4 x} x (1+x)}{(x+\log (3 x))^3}+\frac {14 e^{4 x} x (13+27 x)}{(x+\log (3 x))^2}+\frac {28 e^{4 x} x (1+2 x)}{x+\log (3 x)}\right ) \, dx\\ &=2 \int e^{4 x} x (1+2 x) \, dx+28 \int \frac {e^{4 x} x (13+27 x)}{(x+\log (3 x))^2} \, dx+56 \int \frac {e^{4 x} x (1+2 x)}{x+\log (3 x)} \, dx-392 \int \frac {e^{4 x} x (1+x)}{(x+\log (3 x))^3} \, dx\\ &=2 \int \left (e^{4 x} x+2 e^{4 x} x^2\right ) \, dx+28 \int \left (\frac {13 e^{4 x} x}{(x+\log (3 x))^2}+\frac {27 e^{4 x} x^2}{(x+\log (3 x))^2}\right ) \, dx+56 \int \left (\frac {e^{4 x} x}{x+\log (3 x)}+\frac {2 e^{4 x} x^2}{x+\log (3 x)}\right ) \, dx-392 \int \left (\frac {e^{4 x} x}{(x+\log (3 x))^3}+\frac {e^{4 x} x^2}{(x+\log (3 x))^3}\right ) \, dx\\ &=2 \int e^{4 x} x \, dx+4 \int e^{4 x} x^2 \, dx+56 \int \frac {e^{4 x} x}{x+\log (3 x)} \, dx+112 \int \frac {e^{4 x} x^2}{x+\log (3 x)} \, dx+364 \int \frac {e^{4 x} x}{(x+\log (3 x))^2} \, dx-392 \int \frac {e^{4 x} x}{(x+\log (3 x))^3} \, dx-392 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^3} \, dx+756 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^2} \, dx\\ &=\frac {1}{2} e^{4 x} x+e^{4 x} x^2-\frac {1}{2} \int e^{4 x} \, dx-2 \int e^{4 x} x \, dx+56 \int \frac {e^{4 x} x}{x+\log (3 x)} \, dx+112 \int \frac {e^{4 x} x^2}{x+\log (3 x)} \, dx+364 \int \frac {e^{4 x} x}{(x+\log (3 x))^2} \, dx-392 \int \frac {e^{4 x} x}{(x+\log (3 x))^3} \, dx-392 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^3} \, dx+756 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^2} \, dx\\ &=-\frac {e^{4 x}}{8}+e^{4 x} x^2+\frac {1}{2} \int e^{4 x} \, dx+56 \int \frac {e^{4 x} x}{x+\log (3 x)} \, dx+112 \int \frac {e^{4 x} x^2}{x+\log (3 x)} \, dx+364 \int \frac {e^{4 x} x}{(x+\log (3 x))^2} \, dx-392 \int \frac {e^{4 x} x}{(x+\log (3 x))^3} \, dx-392 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^3} \, dx+756 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^2} \, dx\\ &=e^{4 x} x^2+56 \int \frac {e^{4 x} x}{x+\log (3 x)} \, dx+112 \int \frac {e^{4 x} x^2}{x+\log (3 x)} \, dx+364 \int \frac {e^{4 x} x}{(x+\log (3 x))^2} \, dx-392 \int \frac {e^{4 x} x}{(x+\log (3 x))^3} \, dx-392 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^3} \, dx+756 \int \frac {e^{4 x} x^2}{(x+\log (3 x))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 26, normalized size = 1.24 \begin {gather*} \frac {e^{4 x} x^2 (14+x+\log (3 x))^2}{(x+\log (3 x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(4*x)*(-392*x - 28*x^2 + 812*x^3 + 114*x^4 + 4*x^5) + E^(4*x)*(364*x + 868*x^2 + 230*x^3 + 12*x^4
)*Log[3*x] + E^(4*x)*(56*x + 118*x^2 + 12*x^3)*Log[3*x]^2 + E^(4*x)*(2*x + 4*x^2)*Log[3*x]^3)/(x^3 + 3*x^2*Log
[3*x] + 3*x*Log[3*x]^2 + Log[3*x]^3),x]

[Out]

(E^(4*x)*x^2*(14 + x + Log[3*x])^2)/(x + Log[3*x])^2

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fricas [B]  time = 0.77, size = 73, normalized size = 3.48 \begin {gather*} \frac {x^{2} e^{\left (4 \, x\right )} \log \left (3 \, x\right )^{2} + 2 \, {\left (x^{3} + 14 \, x^{2}\right )} e^{\left (4 \, x\right )} \log \left (3 \, x\right ) + {\left (x^{4} + 28 \, x^{3} + 196 \, x^{2}\right )} e^{\left (4 \, x\right )}}{x^{2} + 2 \, x \log \left (3 \, x\right ) + \log \left (3 \, x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+2*x)*exp(2*x)^2*log(3*x)^3+(12*x^3+118*x^2+56*x)*exp(2*x)^2*log(3*x)^2+(12*x^4+230*x^3+868*x
^2+364*x)*exp(2*x)^2*log(3*x)+(4*x^5+114*x^4+812*x^3-28*x^2-392*x)*exp(2*x)^2)/(log(3*x)^3+3*x*log(3*x)^2+3*x^
2*log(3*x)+x^3),x, algorithm="fricas")

[Out]

(x^2*e^(4*x)*log(3*x)^2 + 2*(x^3 + 14*x^2)*e^(4*x)*log(3*x) + (x^4 + 28*x^3 + 196*x^2)*e^(4*x))/(x^2 + 2*x*log
(3*x) + log(3*x)^2)

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giac [B]  time = 0.20, size = 87, normalized size = 4.14 \begin {gather*} \frac {x^{4} e^{\left (4 \, x\right )} + 2 \, x^{3} e^{\left (4 \, x\right )} \log \left (3 \, x\right ) + x^{2} e^{\left (4 \, x\right )} \log \left (3 \, x\right )^{2} + 28 \, x^{3} e^{\left (4 \, x\right )} + 28 \, x^{2} e^{\left (4 \, x\right )} \log \left (3 \, x\right ) + 196 \, x^{2} e^{\left (4 \, x\right )}}{x^{2} + 2 \, x \log \left (3 \, x\right ) + \log \left (3 \, x\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+2*x)*exp(2*x)^2*log(3*x)^3+(12*x^3+118*x^2+56*x)*exp(2*x)^2*log(3*x)^2+(12*x^4+230*x^3+868*x
^2+364*x)*exp(2*x)^2*log(3*x)+(4*x^5+114*x^4+812*x^3-28*x^2-392*x)*exp(2*x)^2)/(log(3*x)^3+3*x*log(3*x)^2+3*x^
2*log(3*x)+x^3),x, algorithm="giac")

[Out]

(x^4*e^(4*x) + 2*x^3*e^(4*x)*log(3*x) + x^2*e^(4*x)*log(3*x)^2 + 28*x^3*e^(4*x) + 28*x^2*e^(4*x)*log(3*x) + 19
6*x^2*e^(4*x))/(x^2 + 2*x*log(3*x) + log(3*x)^2)

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maple [A]  time = 0.04, size = 34, normalized size = 1.62

method	result	size
risch	\(x^{2} {\mathrm e}^{4 x}+\frac {28 \left (x +\ln \left (3 x \right )+7\right ) x^{2} {\mathrm e}^{4 x}}{\left (x +\ln \left (3 x \right )\right )^{2}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^2+2*x)*exp(2*x)^2*ln(3*x)^3+(12*x^3+118*x^2+56*x)*exp(2*x)^2*ln(3*x)^2+(12*x^4+230*x^3+868*x^2+364*x
)*exp(2*x)^2*ln(3*x)+(4*x^5+114*x^4+812*x^3-28*x^2-392*x)*exp(2*x)^2)/(ln(3*x)^3+3*x*ln(3*x)^2+3*x^2*ln(3*x)+x
^3),x,method=_RETURNVERBOSE)

[Out]

x^2*exp(4*x)+28*(x+ln(3*x)+7)*x^2*exp(4*x)/(x+ln(3*x))^2

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maxima [B]  time = 0.51, size = 83, normalized size = 3.95 \begin {gather*} \frac {{\left (x^{4} + 2 \, x^{3} {\left (\log \relax (3) + 14\right )} + x^{2} \log \relax (x)^{2} + {\left (\log \relax (3)^{2} + 28 \, \log \relax (3) + 196\right )} x^{2} + 2 \, {\left (x^{3} + x^{2} {\left (\log \relax (3) + 14\right )}\right )} \log \relax (x)\right )} e^{\left (4 \, x\right )}}{x^{2} + 2 \, x \log \relax (3) + \log \relax (3)^{2} + 2 \, {\left (x + \log \relax (3)\right )} \log \relax (x) + \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^2+2*x)*exp(2*x)^2*log(3*x)^3+(12*x^3+118*x^2+56*x)*exp(2*x)^2*log(3*x)^2+(12*x^4+230*x^3+868*x
^2+364*x)*exp(2*x)^2*log(3*x)+(4*x^5+114*x^4+812*x^3-28*x^2-392*x)*exp(2*x)^2)/(log(3*x)^3+3*x*log(3*x)^2+3*x^
2*log(3*x)+x^3),x, algorithm="maxima")

[Out]

(x^4 + 2*x^3*(log(3) + 14) + x^2*log(x)^2 + (log(3)^2 + 28*log(3) + 196)*x^2 + 2*(x^3 + x^2*(log(3) + 14))*log
(x))*e^(4*x)/(x^2 + 2*x*log(3) + log(3)^2 + 2*(x + log(3))*log(x) + log(x)^2)

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mupad [B]  time = 5.57, size = 434, normalized size = 20.67 \begin {gather*} \frac {{\mathrm {e}}^{4\,x}\,\left (224\,x^6+2129\,x^5+3727\,x^4+2439\,x^3+449\,x^2\right )}{x^3+3\,x^2+3\,x+1}-\frac {\frac {28\,x\,{\ln \left (3\,x\right )}^2\,\left (x\,{\mathrm {e}}^{4\,x}+2\,x^2\,{\mathrm {e}}^{4\,x}\right )}{x+1}-\frac {14\,x\,\left (14\,x\,{\mathrm {e}}^{4\,x}+x^2\,{\mathrm {e}}^{4\,x}-29\,x^3\,{\mathrm {e}}^{4\,x}-4\,x^4\,{\mathrm {e}}^{4\,x}\right )}{x+1}+\frac {14\,x\,\ln \left (3\,x\right )\,\left (13\,x\,{\mathrm {e}}^{4\,x}+31\,x^2\,{\mathrm {e}}^{4\,x}+8\,x^3\,{\mathrm {e}}^{4\,x}\right )}{x+1}}{x^2+2\,x\,\ln \left (3\,x\right )+{\ln \left (3\,x\right )}^2}-\frac {\frac {14\,x\,\left (93\,x^3\,{\mathrm {e}}^{4\,x}-29\,x^2\,{\mathrm {e}}^{4\,x}-15\,x\,{\mathrm {e}}^{4\,x}+227\,x^4\,{\mathrm {e}}^{4\,x}+148\,x^5\,{\mathrm {e}}^{4\,x}+16\,x^6\,{\mathrm {e}}^{4\,x}\right )}{{\left (x+1\right )}^3}+\frac {28\,x\,{\ln \left (3\,x\right )}^2\,\left (2\,x\,{\mathrm {e}}^{4\,x}+11\,x^2\,{\mathrm {e}}^{4\,x}+16\,x^3\,{\mathrm {e}}^{4\,x}+8\,x^4\,{\mathrm {e}}^{4\,x}\right )}{{\left (x+1\right )}^3}+\frac {28\,x\,\ln \left (3\,x\right )\,\left (15\,x\,{\mathrm {e}}^{4\,x}+85\,x^2\,{\mathrm {e}}^{4\,x}+139\,x^3\,{\mathrm {e}}^{4\,x}+90\,x^4\,{\mathrm {e}}^{4\,x}+16\,x^5\,{\mathrm {e}}^{4\,x}\right )}{{\left (x+1\right )}^3}}{x+\ln \left (3\,x\right )}+\ln \left (3\,x\right )\,{\mathrm {e}}^{4\,x}\,\left (\frac {224\,x^5+448\,x^4+308\,x^3+476\,x^2+700\,x+308}{x^3+3\,x^2+3\,x+1}-\frac {420\,x^2+700\,x+308}{x^3+3\,x^2+3\,x+1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x)*(812*x^3 - 28*x^2 - 392*x + 114*x^4 + 4*x^5) + log(3*x)^3*exp(4*x)*(2*x + 4*x^2) + log(3*x)*exp(
4*x)*(364*x + 868*x^2 + 230*x^3 + 12*x^4) + log(3*x)^2*exp(4*x)*(56*x + 118*x^2 + 12*x^3))/(3*x*log(3*x)^2 + 3
*x^2*log(3*x) + log(3*x)^3 + x^3),x)

[Out]

(exp(4*x)*(449*x^2 + 2439*x^3 + 3727*x^4 + 2129*x^5 + 224*x^6))/(3*x + 3*x^2 + x^3 + 1) - ((28*x*log(3*x)^2*(x
*exp(4*x) + 2*x^2*exp(4*x)))/(x + 1) - (14*x*(14*x*exp(4*x) + x^2*exp(4*x) - 29*x^3*exp(4*x) - 4*x^4*exp(4*x))
)/(x + 1) + (14*x*log(3*x)*(13*x*exp(4*x) + 31*x^2*exp(4*x) + 8*x^3*exp(4*x)))/(x + 1))/(2*x*log(3*x) + log(3*
x)^2 + x^2) - ((14*x*(93*x^3*exp(4*x) - 29*x^2*exp(4*x) - 15*x*exp(4*x) + 227*x^4*exp(4*x) + 148*x^5*exp(4*x)
+ 16*x^6*exp(4*x)))/(x + 1)^3 + (28*x*log(3*x)^2*(2*x*exp(4*x) + 11*x^2*exp(4*x) + 16*x^3*exp(4*x) + 8*x^4*exp
(4*x)))/(x + 1)^3 + (28*x*log(3*x)*(15*x*exp(4*x) + 85*x^2*exp(4*x) + 139*x^3*exp(4*x) + 90*x^4*exp(4*x) + 16*
x^5*exp(4*x)))/(x + 1)^3)/(x + log(3*x)) + log(3*x)*exp(4*x)*((700*x + 476*x^2 + 308*x^3 + 448*x^4 + 224*x^5 +
308)/(3*x + 3*x^2 + x^3 + 1) - (700*x + 420*x^2 + 308)/(3*x + 3*x^2 + x^3 + 1))

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sympy [B]  time = 0.43, size = 66, normalized size = 3.14 \begin {gather*} \frac {\left (x^{4} + 2 x^{3} \log {\left (3 x \right )} + 28 x^{3} + x^{2} \log {\left (3 x \right )}^{2} + 28 x^{2} \log {\left (3 x \right )} + 196 x^{2}\right ) e^{4 x}}{x^{2} + 2 x \log {\left (3 x \right )} + \log {\left (3 x \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**2+2*x)*exp(2*x)**2*ln(3*x)**3+(12*x**3+118*x**2+56*x)*exp(2*x)**2*ln(3*x)**2+(12*x**4+230*x**
3+868*x**2+364*x)*exp(2*x)**2*ln(3*x)+(4*x**5+114*x**4+812*x**3-28*x**2-392*x)*exp(2*x)**2)/(ln(3*x)**3+3*x*ln
(3*x)**2+3*x**2*ln(3*x)+x**3),x)

[Out]

(x**4 + 2*x**3*log(3*x) + 28*x**3 + x**2*log(3*x)**2 + 28*x**2*log(3*x) + 196*x**2)*exp(4*x)/(x**2 + 2*x*log(3
*x) + log(3*x)**2)

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