3.82.75 \(\int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} (-625 x^2-50 x^3-x^4+(1875 x^3+150 x^4+3 x^5) \log (3))}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} (625 x^2+50 x^3+x^4)+3^{3 x} (250 x+6210 x^2+498 x^3+10 x^4)} \, dx\)

Optimal. Leaf size=25 \[ \frac {x}{-5-3^{3 x}+\frac {-5+x}{x (25+x)}} \]

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Rubi [F]  time = 3.37, 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 {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-250*x - 3115*x^2 - 248*x^3 - 5*x^4 + 3^(3*x)*(-625*x^2 - 50*x^3 - x^4 + (1875*x^3 + 150*x^4 + 3*x^5)*Log
[3]))/(25 + 1240*x + 15426*x^2 + 1240*x^3 + 25*x^4 + 3^(6*x)*(625*x^2 + 50*x^3 + x^4) + 3^(3*x)*(250*x + 6210*
x^2 + 498*x^3 + 10*x^4)),x]

[Out]

-125*Defer[Int][x/(5 + 124*x + 25*27^x*x + 5*x^2 + 27^x*x^2)^2, x] - 5*(2 + 75*Log[3])*Defer[Int][x^2/(5 + 124
*x + 25*27^x*x + 5*x^2 + 27^x*x^2)^2, x] + (1 - 9315*Log[3])*Defer[Int][x^3/(5 + 124*x + 25*27^x*x + 5*x^2 + 2
7^x*x^2)^2, x] - 747*Log[3]*Defer[Int][x^4/(5 + 124*x + 25*27^x*x + 5*x^2 + 27^x*x^2)^2, x] - 15*Log[3]*Defer[
Int][x^5/(5 + 124*x + 25*27^x*x + 5*x^2 + 27^x*x^2)^2, x] - 25*Defer[Int][x/(5 + 124*x + 25*27^x*x + 5*x^2 + 2
7^x*x^2), x] - (1 - 25*Log[27])*Defer[Int][x^2/(5 + 124*x + 25*27^x*x + 5*x^2 + 27^x*x^2), x] + Log[27]*Defer[
Int][x^3/(5 + 124*x + 25*27^x*x + 5*x^2 + 27^x*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (-250-5 \left (623+125\ 27^x\right ) x+3^{1+3 x} x^4 \log (3)+x^3 \left (-5-27^x+50\ 3^{1+3 x} \log (3)\right )+x^2 \left (-248-50\ 27^x+625\ 3^{1+3 x} \log (3)\right )\right )}{\left (5+\left (124+25\ 27^x\right ) x+\left (5+27^x\right ) x^2\right )^2} \, dx\\ &=\int \left (\frac {x \left (-125+x^2 (1-9315 \log (3))-747 x^3 \log (3)-15 x^4 \log (3)-5 x (2+75 \log (3))\right )}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2}+\frac {x (25+x) (-1+x \log (27))}{5+124 x+25\ 27^x x+5 x^2+27^x x^2}\right ) \, dx\\ &=\int \frac {x \left (-125+x^2 (1-9315 \log (3))-747 x^3 \log (3)-15 x^4 \log (3)-5 x (2+75 \log (3))\right )}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2} \, dx+\int \frac {x (25+x) (-1+x \log (27))}{5+124 x+25\ 27^x x+5 x^2+27^x x^2} \, dx\\ &=\int \left (-\frac {125 x}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2}-\frac {747 x^4 \log (3)}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2}-\frac {15 x^5 \log (3)}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2}-\frac {5 x^2 (2+75 \log (3))}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2}-\frac {x^3 (-1+9315 \log (3))}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2}\right ) \, dx+\int \left (-\frac {25 x}{5+124 x+25\ 27^x x+5 x^2+27^x x^2}+\frac {x^3 \log (27)}{5+124 x+25\ 27^x x+5 x^2+27^x x^2}+\frac {x^2 (-1+25 \log (27))}{5+124 x+25\ 27^x x+5 x^2+27^x x^2}\right ) \, dx\\ &=-\left (25 \int \frac {x}{5+124 x+25\ 27^x x+5 x^2+27^x x^2} \, dx\right )-125 \int \frac {x}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2} \, dx+(1-9315 \log (3)) \int \frac {x^3}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2} \, dx-(15 \log (3)) \int \frac {x^5}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2} \, dx-(747 \log (3)) \int \frac {x^4}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2} \, dx-(5 (2+75 \log (3))) \int \frac {x^2}{\left (5+124 x+25\ 27^x x+5 x^2+27^x x^2\right )^2} \, dx+\log (27) \int \frac {x^3}{5+124 x+25\ 27^x x+5 x^2+27^x x^2} \, dx+(-1+25 \log (27)) \int \frac {x^2}{5+124 x+25\ 27^x x+5 x^2+27^x x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-250 x-3115 x^2-248 x^3-5 x^4+3^{3 x} \left (-625 x^2-50 x^3-x^4+\left (1875 x^3+150 x^4+3 x^5\right ) \log (3)\right )}{25+1240 x+15426 x^2+1240 x^3+25 x^4+3^{6 x} \left (625 x^2+50 x^3+x^4\right )+3^{3 x} \left (250 x+6210 x^2+498 x^3+10 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-250*x - 3115*x^2 - 248*x^3 - 5*x^4 + 3^(3*x)*(-625*x^2 - 50*x^3 - x^4 + (1875*x^3 + 150*x^4 + 3*x^
5)*Log[3]))/(25 + 1240*x + 15426*x^2 + 1240*x^3 + 25*x^4 + 3^(6*x)*(625*x^2 + 50*x^3 + x^4) + 3^(3*x)*(250*x +
6210*x^2 + 498*x^3 + 10*x^4)),x]

[Out]

Integrate[(-250*x - 3115*x^2 - 248*x^3 - 5*x^4 + 3^(3*x)*(-625*x^2 - 50*x^3 - x^4 + (1875*x^3 + 150*x^4 + 3*x^
5)*Log[3]))/(25 + 1240*x + 15426*x^2 + 1240*x^3 + 25*x^4 + 3^(6*x)*(625*x^2 + 50*x^3 + x^4) + 3^(3*x)*(250*x +
6210*x^2 + 498*x^3 + 10*x^4)), x]

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fricas [A]  time = 0.89, size = 36, normalized size = 1.44 \begin {gather*} -\frac {x^{3} + 25 \, x^{2}}{{\left (x^{2} + 25 \, x\right )} 3^{3 \, x} + 5 \, x^{2} + 124 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^5+150*x^4+1875*x^3)*log(3)-x^4-50*x^3-625*x^2)*exp(3*x*log(3))-5*x^4-248*x^3-3115*x^2-250*x)/
((x^4+50*x^3+625*x^2)*exp(3*x*log(3))^2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*log(3))+25*x^4+1240*x^3+15426*
x^2+1240*x+25),x, algorithm="fricas")

[Out]

-(x^3 + 25*x^2)/((x^2 + 25*x)*3^(3*x) + 5*x^2 + 124*x + 5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {5 \, x^{4} + 248 \, x^{3} + {\left (x^{4} + 50 \, x^{3} + 625 \, x^{2} - 3 \, {\left (x^{5} + 50 \, x^{4} + 625 \, x^{3}\right )} \log \relax (3)\right )} 3^{3 \, x} + 3115 \, x^{2} + 250 \, x}{25 \, x^{4} + 1240 \, x^{3} + {\left (x^{4} + 50 \, x^{3} + 625 \, x^{2}\right )} 3^{6 \, x} + 2 \, {\left (5 \, x^{4} + 249 \, x^{3} + 3105 \, x^{2} + 125 \, x\right )} 3^{3 \, x} + 15426 \, x^{2} + 1240 \, x + 25}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^5+150*x^4+1875*x^3)*log(3)-x^4-50*x^3-625*x^2)*exp(3*x*log(3))-5*x^4-248*x^3-3115*x^2-250*x)/
((x^4+50*x^3+625*x^2)*exp(3*x*log(3))^2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*log(3))+25*x^4+1240*x^3+15426*
x^2+1240*x+25),x, algorithm="giac")

[Out]

integrate(-(5*x^4 + 248*x^3 + (x^4 + 50*x^3 + 625*x^2 - 3*(x^5 + 50*x^4 + 625*x^3)*log(3))*3^(3*x) + 3115*x^2
+ 250*x)/(25*x^4 + 1240*x^3 + (x^4 + 50*x^3 + 625*x^2)*3^(6*x) + 2*(5*x^4 + 249*x^3 + 3105*x^2 + 125*x)*3^(3*x
) + 15426*x^2 + 1240*x + 25), x)

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maple [A]  time = 0.58, size = 34, normalized size = 1.36

method	result	size
risch	\(-\frac {x^{2} \left (x +25\right )}{27^{x} x^{2}+25 \,27^{x} x +5 x^{2}+124 x +5}\)	\(34\)
norman	\(\frac {-x^{3}-25 x^{2}}{{\mathrm e}^{3 x \ln \relax (3)} x^{2}+25 \,{\mathrm e}^{3 x \ln \relax (3)} x +5 x^{2}+124 x +5}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3*x^5+150*x^4+1875*x^3)*ln(3)-x^4-50*x^3-625*x^2)*exp(3*x*ln(3))-5*x^4-248*x^3-3115*x^2-250*x)/((x^4+50
*x^3+625*x^2)*exp(3*x*ln(3))^2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*ln(3))+25*x^4+1240*x^3+15426*x^2+1240*x
+25),x,method=_RETURNVERBOSE)

[Out]

-x^2*(x+25)/(27^x*x^2+25*27^x*x+5*x^2+124*x+5)

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maxima [A]  time = 0.52, size = 36, normalized size = 1.44 \begin {gather*} -\frac {x^{3} + 25 \, x^{2}}{{\left (x^{2} + 25 \, x\right )} 3^{3 \, x} + 5 \, x^{2} + 124 \, x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x^5+150*x^4+1875*x^3)*log(3)-x^4-50*x^3-625*x^2)*exp(3*x*log(3))-5*x^4-248*x^3-3115*x^2-250*x)/
((x^4+50*x^3+625*x^2)*exp(3*x*log(3))^2+(10*x^4+498*x^3+6210*x^2+250*x)*exp(3*x*log(3))+25*x^4+1240*x^3+15426*
x^2+1240*x+25),x, algorithm="maxima")

[Out]

-(x^3 + 25*x^2)/((x^2 + 25*x)*3^(3*x) + 5*x^2 + 124*x + 5)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {250\,x+{\mathrm {e}}^{3\,x\,\ln \relax (3)}\,\left (625\,x^2-\ln \relax (3)\,\left (3\,x^5+150\,x^4+1875\,x^3\right )+50\,x^3+x^4\right )+3115\,x^2+248\,x^3+5\,x^4}{1240\,x+{\mathrm {e}}^{6\,x\,\ln \relax (3)}\,\left (x^4+50\,x^3+625\,x^2\right )+15426\,x^2+1240\,x^3+25\,x^4+{\mathrm {e}}^{3\,x\,\ln \relax (3)}\,\left (10\,x^4+498\,x^3+6210\,x^2+250\,x\right )+25} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(250*x + exp(3*x*log(3))*(625*x^2 - log(3)*(1875*x^3 + 150*x^4 + 3*x^5) + 50*x^3 + x^4) + 3115*x^2 + 248*
x^3 + 5*x^4)/(1240*x + exp(6*x*log(3))*(625*x^2 + 50*x^3 + x^4) + 15426*x^2 + 1240*x^3 + 25*x^4 + exp(3*x*log(
3))*(250*x + 6210*x^2 + 498*x^3 + 10*x^4) + 25),x)

[Out]

int(-(250*x + exp(3*x*log(3))*(625*x^2 - log(3)*(1875*x^3 + 150*x^4 + 3*x^5) + 50*x^3 + x^4) + 3115*x^2 + 248*
x^3 + 5*x^4)/(1240*x + exp(6*x*log(3))*(625*x^2 + 50*x^3 + x^4) + 15426*x^2 + 1240*x^3 + 25*x^4 + exp(3*x*log(
3))*(250*x + 6210*x^2 + 498*x^3 + 10*x^4) + 25), x)

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sympy [A]  time = 0.31, size = 34, normalized size = 1.36 \begin {gather*} \frac {- x^{3} - 25 x^{2}}{5 x^{2} + 124 x + \left (x^{2} + 25 x\right ) e^{3 x \log {\relax (3 )}} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3*x**5+150*x**4+1875*x**3)*ln(3)-x**4-50*x**3-625*x**2)*exp(3*x*ln(3))-5*x**4-248*x**3-3115*x**2-
250*x)/((x**4+50*x**3+625*x**2)*exp(3*x*ln(3))**2+(10*x**4+498*x**3+6210*x**2+250*x)*exp(3*x*ln(3))+25*x**4+12
40*x**3+15426*x**2+1240*x+25),x)

[Out]

(-x**3 - 25*x**2)/(5*x**2 + 124*x + (x**2 + 25*x)*exp(3*x*log(3)) + 5)

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