3.54.77 \(\int \frac {275+128 x+15 x^2+e^{4 x^3} (5+300 x^2+60 x^3)+e^{2 x^3} (-90-20 x-1200 x^2-540 x^3-60 x^4)}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} (-200-90 x-10 x^2)} \, dx\)

Optimal. Leaf size=25 \[ \log \left ((5+x) \left (-\frac {x}{5}+\left (4-e^{2 x^3}+x\right )^2\right )\right ) \]

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Rubi [F]  time = 2.81, 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 {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{400+275 x+64 x^2+5 x^3+e^{4 x^3} (25+5 x)+e^{2 x^3} \left (-200-90 x-10 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(275 + 128*x + 15*x^2 + E^(4*x^3)*(5 + 300*x^2 + 60*x^3) + E^(2*x^3)*(-90 - 20*x - 1200*x^2 - 540*x^3 - 60
*x^4))/(400 + 275*x + 64*x^2 + 5*x^3 + E^(4*x^3)*(25 + 5*x) + E^(2*x^3)*(-200 - 90*x - 10*x^2)),x]

[Out]

4*x^3 + Log[5 + x] + 39*Defer[Int][(80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2)^(-1), x]
- 10*Defer[Int][E^(2*x^3)/(80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2), x] + 10*Defer[Int
][x/(80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2), x] - 960*Defer[Int][x^2/(80 - 40*E^(2*x
^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2), x] + 240*Defer[Int][(E^(2*x^3)*x^2)/(80 - 40*E^(2*x^3) + 5
*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2), x] - 468*Defer[Int][x^3/(80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x -
10*E^(2*x^3)*x + 5*x^2), x] + 60*Defer[Int][(E^(2*x^3)*x^3)/(80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2
*x^3)*x + 5*x^2), x] - 60*Defer[Int][x^4/(80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {275+128 x+15 x^2+e^{4 x^3} \left (5+300 x^2+60 x^3\right )+e^{2 x^3} \left (-90-20 x-1200 x^2-540 x^3-60 x^4\right )}{(5+x) \left (80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2\right )} \, dx\\ &=\int \left (\frac {1+60 x^2+12 x^3}{5+x}+\frac {39-10 e^{2 x^3}+10 x-960 x^2+240 e^{2 x^3} x^2-468 x^3+60 e^{2 x^3} x^3-60 x^4}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}\right ) \, dx\\ &=\int \frac {1+60 x^2+12 x^3}{5+x} \, dx+\int \frac {39-10 e^{2 x^3}+10 x-960 x^2+240 e^{2 x^3} x^2-468 x^3+60 e^{2 x^3} x^3-60 x^4}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx\\ &=\int \left (12 x^2+\frac {1}{5+x}\right ) \, dx+\int \left (\frac {39}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}-\frac {10 e^{2 x^3}}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}+\frac {10 x}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}-\frac {960 x^2}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}+\frac {240 e^{2 x^3} x^2}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}-\frac {468 x^3}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}+\frac {60 e^{2 x^3} x^3}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}-\frac {60 x^4}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2}\right ) \, dx\\ &=4 x^3+\log (5+x)-10 \int \frac {e^{2 x^3}}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx+10 \int \frac {x}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx+39 \int \frac {1}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx+60 \int \frac {e^{2 x^3} x^3}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx-60 \int \frac {x^4}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx+240 \int \frac {e^{2 x^3} x^2}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx-468 \int \frac {x^3}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx-960 \int \frac {x^2}{80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.87, size = 44, normalized size = 1.76 \begin {gather*} \log (5+x)+\log \left (80-40 e^{2 x^3}+5 e^{4 x^3}+39 x-10 e^{2 x^3} x+5 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(275 + 128*x + 15*x^2 + E^(4*x^3)*(5 + 300*x^2 + 60*x^3) + E^(2*x^3)*(-90 - 20*x - 1200*x^2 - 540*x^
3 - 60*x^4))/(400 + 275*x + 64*x^2 + 5*x^3 + E^(4*x^3)*(25 + 5*x) + E^(2*x^3)*(-200 - 90*x - 10*x^2)),x]

[Out]

Log[5 + x] + Log[80 - 40*E^(2*x^3) + 5*E^(4*x^3) + 39*x - 10*E^(2*x^3)*x + 5*x^2]

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fricas [A]  time = 0.83, size = 35, normalized size = 1.40 \begin {gather*} \log \left (5 \, x^{2} - 10 \, {\left (x + 4\right )} e^{\left (2 \, x^{3}\right )} + 39 \, x + 5 \, e^{\left (4 \, x^{3}\right )} + 80\right ) + \log \left (x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90)*exp(x^3)^2+15*x^2+128*x+275)/((25+
5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp(x^3)^2+5*x^3+64*x^2+275*x+400),x, algorithm="fricas")

[Out]

log(5*x^2 - 10*(x + 4)*e^(2*x^3) + 39*x + 5*e^(4*x^3) + 80) + log(x + 5)

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giac [A]  time = 0.20, size = 41, normalized size = 1.64 \begin {gather*} \log \left (5 \, x^{2} - 10 \, x e^{\left (2 \, x^{3}\right )} + 39 \, x + 5 \, e^{\left (4 \, x^{3}\right )} - 40 \, e^{\left (2 \, x^{3}\right )} + 80\right ) + \log \left (x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90)*exp(x^3)^2+15*x^2+128*x+275)/((25+
5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp(x^3)^2+5*x^3+64*x^2+275*x+400),x, algorithm="giac")

[Out]

log(5*x^2 - 10*x*e^(2*x^3) + 39*x + 5*e^(4*x^3) - 40*e^(2*x^3) + 80) + log(x + 5)

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maple [A]  time = 0.12, size = 33, normalized size = 1.32

method	result	size
risch	\(\ln \left (5+x \right )+\ln \left ({\mathrm e}^{4 x^{3}}+\left (-2 x -8\right ) {\mathrm e}^{2 x^{3}}+x^{2}+\frac {39 x}{5}+16\right )\)	\(33\)
norman	\(\ln \left (5+x \right )+\ln \left (5 \,{\mathrm e}^{4 x^{3}}-10 \,{\mathrm e}^{2 x^{3}} x +5 x^{2}-40 \,{\mathrm e}^{2 x^{3}}+39 x +80\right )\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90)*exp(x^3)^2+15*x^2+128*x+275)/((25+5*x)*e
xp(x^3)^4+(-10*x^2-90*x-200)*exp(x^3)^2+5*x^3+64*x^2+275*x+400),x,method=_RETURNVERBOSE)

[Out]

ln(5+x)+ln(exp(4*x^3)+(-2*x-8)*exp(2*x^3)+x^2+39/5*x+16)

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maxima [A]  time = 0.40, size = 31, normalized size = 1.24 \begin {gather*} \log \left (x^{2} - 2 \, {\left (x + 4\right )} e^{\left (2 \, x^{3}\right )} + \frac {39}{5} \, x + e^{\left (4 \, x^{3}\right )} + 16\right ) + \log \left (x + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^3+300*x^2+5)*exp(x^3)^4+(-60*x^4-540*x^3-1200*x^2-20*x-90)*exp(x^3)^2+15*x^2+128*x+275)/((25+
5*x)*exp(x^3)^4+(-10*x^2-90*x-200)*exp(x^3)^2+5*x^3+64*x^2+275*x+400),x, algorithm="maxima")

[Out]

log(x^2 - 2*(x + 4)*e^(2*x^3) + 39/5*x + e^(4*x^3) + 16) + log(x + 5)

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mupad [B]  time = 0.18, size = 37, normalized size = 1.48 \begin {gather*} \ln \left (x+5\right )+\ln \left (\frac {39\,x}{5}-8\,{\mathrm {e}}^{2\,x^3}+{\mathrm {e}}^{4\,x^3}-2\,x\,{\mathrm {e}}^{2\,x^3}+x^2+16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((128*x + exp(4*x^3)*(300*x^2 + 60*x^3 + 5) - exp(2*x^3)*(20*x + 1200*x^2 + 540*x^3 + 60*x^4 + 90) + 15*x^2
+ 275)/(275*x + exp(4*x^3)*(5*x + 25) - exp(2*x^3)*(90*x + 10*x^2 + 200) + 64*x^2 + 5*x^3 + 400),x)

[Out]

log(x + 5) + log((39*x)/5 - 8*exp(2*x^3) + exp(4*x^3) - 2*x*exp(2*x^3) + x^2 + 16)

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sympy [A]  time = 0.24, size = 36, normalized size = 1.44 \begin {gather*} \log {\left (x + 5 \right )} + \log {\left (x^{2} + \frac {39 x}{5} + \left (- 2 x - 8\right ) e^{2 x^{3}} + e^{4 x^{3}} + 16 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x**3+300*x**2+5)*exp(x**3)**4+(-60*x**4-540*x**3-1200*x**2-20*x-90)*exp(x**3)**2+15*x**2+128*x+
275)/((25+5*x)*exp(x**3)**4+(-10*x**2-90*x-200)*exp(x**3)**2+5*x**3+64*x**2+275*x+400),x)

[Out]

log(x + 5) + log(x**2 + 39*x/5 + (-2*x - 8)*exp(2*x**3) + exp(4*x**3) + 16)

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