3.10.13 \(\int \frac {12 x^3+45 x^6+225 e^x x^6+(-48 x^3+30 x^5+e^x (-120 x^3+150 x^5)) \log (x)+(-12 x^2+5 x^4+e^x (16-40 x^2+25 x^4)) \log ^2(x)}{225 x^6+(-120 x^3+150 x^5) \log (x)+(16-40 x^2+25 x^4) \log ^2(x)} \, dx\)

Optimal. Leaf size=28 \[ e^x+\frac {x}{5-\frac {4}{x \left (x+\frac {3 x^2}{\log (x)}\right )}} \]

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Rubi [F]  time = 8.77, 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 {12 x^3+45 x^6+225 e^x x^6+\left (-48 x^3+30 x^5+e^x \left (-120 x^3+150 x^5\right )\right ) \log (x)+\left (-12 x^2+5 x^4+e^x \left (16-40 x^2+25 x^4\right )\right ) \log ^2(x)}{225 x^6+\left (-120 x^3+150 x^5\right ) \log (x)+\left (16-40 x^2+25 x^4\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(12*x^3 + 45*x^6 + 225*E^x*x^6 + (-48*x^3 + 30*x^5 + E^x*(-120*x^3 + 150*x^5))*Log[x] + (-12*x^2 + 5*x^4 +
E^x*(16 - 40*x^2 + 25*x^4))*Log[x]^2)/(225*x^6 + (-120*x^3 + 150*x^5)*Log[x] + (16 - 40*x^2 + 25*x^4)*Log[x]^
2),x]

[Out]

E^x + (3*x)/10 - (6*x)/(5*(4 - 5*x^2)) + x^3/(2*(4 - 5*x^2)) - (1728*Defer[Int][(15*x^3 - 4*Log[x] + 5*x^2*Log
[x])^(-2), x])/25 - (144*Defer[Int][x^2/(15*x^3 - 4*Log[x] + 5*x^2*Log[x])^2, x])/5 + 12*Defer[Int][x^3/(15*x^
3 - 4*Log[x] + 5*x^2*Log[x])^2, x] + 36*Defer[Int][x^4/(15*x^3 - 4*Log[x] + 5*x^2*Log[x])^2, x] + (576*Defer[I
nt][1/((2 - Sqrt[5]*x)*(15*x^3 - 4*Log[x] + 5*x^2*Log[x])^2), x])/5 + (576*Defer[Int][1/((2 + Sqrt[5]*x)*(15*x
^3 - 4*Log[x] + 5*x^2*Log[x])^2), x])/5 - (18432*Defer[Int][1/((-4 + 5*x^2)^2*(15*x^3 - 4*Log[x] + 5*x^2*Log[x
])^2), x])/25 - (96*Defer[Int][1/((2 - Sqrt[5]*x)*(15*x^3 - 4*Log[x] + 5*x^2*Log[x])), x])/(5*Sqrt[5]) + (96*D
efer[Int][1/((2 + Sqrt[5]*x)*(15*x^3 - 4*Log[x] + 5*x^2*Log[x])), x])/(5*Sqrt[5]) + (768*Defer[Int][x/((-4 + 5
*x^2)^2*(15*x^3 - 4*Log[x] + 5*x^2*Log[x])), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {12 x^3+45 x^6+225 e^x x^6+\left (-48 x^3+30 x^5+e^x \left (-120 x^3+150 x^5\right )\right ) \log (x)+\left (-12 x^2+5 x^4+e^x \left (16-40 x^2+25 x^4\right )\right ) \log ^2(x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx\\ &=\int \left (e^x+\frac {12 x^3}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}+\frac {45 x^6}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}-\frac {48 x^3 \log (x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}+\frac {30 x^5 \log (x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}-\frac {12 x^2 \log ^2(x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}+\frac {5 x^4 \log ^2(x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}\right ) \, dx\\ &=5 \int \frac {x^4 \log ^2(x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+12 \int \frac {x^3}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx-12 \int \frac {x^2 \log ^2(x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+30 \int \frac {x^5 \log (x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+45 \int \frac {x^6}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx-48 \int \frac {x^3 \log (x)}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+\int e^x \, dx\\ &=e^x+5 \int \left (\frac {x^4}{\left (-4+5 x^2\right )^2}+\frac {225 x^{10}}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}-\frac {30 x^7}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )}\right ) \, dx+12 \int \frac {x^3}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx-12 \int \left (\frac {x^2}{\left (-4+5 x^2\right )^2}+\frac {225 x^8}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}-\frac {30 x^5}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )}\right ) \, dx+30 \int \left (-\frac {15 x^8}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}+\frac {x^5}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )}\right ) \, dx+45 \int \frac {x^6}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx-48 \int \left (-\frac {15 x^6}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2}+\frac {x^3}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )}\right ) \, dx\\ &=e^x+5 \int \frac {x^4}{\left (-4+5 x^2\right )^2} \, dx-12 \int \frac {x^2}{\left (-4+5 x^2\right )^2} \, dx+12 \int \frac {x^3}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+30 \int \frac {x^5}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )} \, dx+45 \int \frac {x^6}{\left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx-48 \int \frac {x^3}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )} \, dx-150 \int \frac {x^7}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )} \, dx+360 \int \frac {x^5}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )} \, dx-450 \int \frac {x^8}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+720 \int \frac {x^6}{\left (-4+5 x^2\right ) \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx+1125 \int \frac {x^{10}}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx-2700 \int \frac {x^8}{\left (-4+5 x^2\right )^2 \left (15 x^3-4 \log (x)+5 x^2 \log (x)\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 50, normalized size = 1.79 \begin {gather*} \frac {3 x^3 \left (5 e^x+x\right )+\left (x^3+e^x \left (-4+5 x^2\right )\right ) \log (x)}{15 x^3+\left (-4+5 x^2\right ) \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(12*x^3 + 45*x^6 + 225*E^x*x^6 + (-48*x^3 + 30*x^5 + E^x*(-120*x^3 + 150*x^5))*Log[x] + (-12*x^2 + 5
*x^4 + E^x*(16 - 40*x^2 + 25*x^4))*Log[x]^2)/(225*x^6 + (-120*x^3 + 150*x^5)*Log[x] + (16 - 40*x^2 + 25*x^4)*L
og[x]^2),x]

[Out]

(3*x^3*(5*E^x + x) + (x^3 + E^x*(-4 + 5*x^2))*Log[x])/(15*x^3 + (-4 + 5*x^2)*Log[x])

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fricas [A]  time = 0.58, size = 49, normalized size = 1.75 \begin {gather*} \frac {3 \, x^{4} + 15 \, x^{3} e^{x} + {\left (x^{3} + {\left (5 \, x^{2} - 4\right )} e^{x}\right )} \log \relax (x)}{15 \, x^{3} + {\left (5 \, x^{2} - 4\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((25*x^4-40*x^2+16)*exp(x)+5*x^4-12*x^2)*log(x)^2+((150*x^5-120*x^3)*exp(x)+30*x^5-48*x^3)*log(x)+2
25*x^6*exp(x)+45*x^6+12*x^3)/((25*x^4-40*x^2+16)*log(x)^2+(150*x^5-120*x^3)*log(x)+225*x^6),x, algorithm="fric
as")

[Out]

(3*x^4 + 15*x^3*e^x + (x^3 + (5*x^2 - 4)*e^x)*log(x))/(15*x^3 + (5*x^2 - 4)*log(x))

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giac [A]  time = 0.58, size = 54, normalized size = 1.93 \begin {gather*} \frac {3 \, x^{4} + 15 \, x^{3} e^{x} + x^{3} \log \relax (x) + 5 \, x^{2} e^{x} \log \relax (x) - 4 \, e^{x} \log \relax (x)}{15 \, x^{3} + 5 \, x^{2} \log \relax (x) - 4 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((25*x^4-40*x^2+16)*exp(x)+5*x^4-12*x^2)*log(x)^2+((150*x^5-120*x^3)*exp(x)+30*x^5-48*x^3)*log(x)+2
25*x^6*exp(x)+45*x^6+12*x^3)/((25*x^4-40*x^2+16)*log(x)^2+(150*x^5-120*x^3)*log(x)+225*x^6),x, algorithm="giac
")

[Out]

(3*x^4 + 15*x^3*e^x + x^3*log(x) + 5*x^2*e^x*log(x) - 4*e^x*log(x))/(15*x^3 + 5*x^2*log(x) - 4*log(x))

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maple [A]  time = 0.05, size = 36, normalized size = 1.29

method	result	size
default	\(\frac {x^{3} \ln \relax (x )+3 x^{4}}{5 x^{2} \ln \relax (x )+15 x^{3}-4 \ln \relax (x )}+{\mathrm e}^{x}\)	\(36\)
risch	\(\frac {x^{3}+5 \,{\mathrm e}^{x} x^{2}-4 \,{\mathrm e}^{x}}{5 x^{2}-4}-\frac {12 x^{4}}{\left (5 x^{2}-4\right ) \left (5 x^{2} \ln \relax (x )+15 x^{3}-4 \ln \relax (x )\right )}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((25*x^4-40*x^2+16)*exp(x)+5*x^4-12*x^2)*ln(x)^2+((150*x^5-120*x^3)*exp(x)+30*x^5-48*x^3)*ln(x)+225*x^6*e
xp(x)+45*x^6+12*x^3)/((25*x^4-40*x^2+16)*ln(x)^2+(150*x^5-120*x^3)*ln(x)+225*x^6),x,method=_RETURNVERBOSE)

[Out]

(x^3*ln(x)+3*x^4)/(5*x^2*ln(x)+15*x^3-4*ln(x))+exp(x)

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maxima [A]  time = 0.59, size = 50, normalized size = 1.79 \begin {gather*} \frac {3 \, x^{4} + x^{3} \log \relax (x) + {\left (15 \, x^{3} + {\left (5 \, x^{2} - 4\right )} \log \relax (x)\right )} e^{x}}{15 \, x^{3} + {\left (5 \, x^{2} - 4\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((25*x^4-40*x^2+16)*exp(x)+5*x^4-12*x^2)*log(x)^2+((150*x^5-120*x^3)*exp(x)+30*x^5-48*x^3)*log(x)+2
25*x^6*exp(x)+45*x^6+12*x^3)/((25*x^4-40*x^2+16)*log(x)^2+(150*x^5-120*x^3)*log(x)+225*x^6),x, algorithm="maxi
ma")

[Out]

(3*x^4 + x^3*log(x) + (15*x^3 + (5*x^2 - 4)*log(x))*e^x)/(15*x^3 + (5*x^2 - 4)*log(x))

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mupad [B]  time = 0.87, size = 99, normalized size = 3.54 \begin {gather*} \frac {x}{5}+{\mathrm {e}}^x+\frac {4\,x}{25\,\left (x^2-\frac {4}{5}\right )}-\frac {12\,\left (75\,x^{10}+25\,x^9-180\,x^8-40\,x^7+16\,x^5\right )}{\left (15\,x^3+\ln \relax (x)\,\left (5\,x^2-4\right )\right )\,\left (5\,x^2-4\right )\,\left (75\,x^6+25\,x^5-180\,x^4-40\,x^3+16\,x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((225*x^6*exp(x) - log(x)*(exp(x)*(120*x^3 - 150*x^5) + 48*x^3 - 30*x^5) + log(x)^2*(exp(x)*(25*x^4 - 40*x^
2 + 16) - 12*x^2 + 5*x^4) + 12*x^3 + 45*x^6)/(log(x)^2*(25*x^4 - 40*x^2 + 16) - log(x)*(120*x^3 - 150*x^5) + 2
25*x^6),x)

[Out]

x/5 + exp(x) + (4*x)/(25*(x^2 - 4/5)) - (12*(16*x^5 - 40*x^7 - 180*x^8 + 25*x^9 + 75*x^10))/((15*x^3 + log(x)*
(5*x^2 - 4))*(5*x^2 - 4)*(16*x - 40*x^3 - 180*x^4 + 25*x^5 + 75*x^6))

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sympy [B]  time = 0.47, size = 46, normalized size = 1.64 \begin {gather*} - \frac {12 x^{4}}{75 x^{5} - 60 x^{3} + \left (25 x^{4} - 40 x^{2} + 16\right ) \log {\relax (x )}} + \frac {x}{5} + \frac {4 x}{25 x^{2} - 20} + e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((25*x**4-40*x**2+16)*exp(x)+5*x**4-12*x**2)*ln(x)**2+((150*x**5-120*x**3)*exp(x)+30*x**5-48*x**3)*
ln(x)+225*x**6*exp(x)+45*x**6+12*x**3)/((25*x**4-40*x**2+16)*ln(x)**2+(150*x**5-120*x**3)*ln(x)+225*x**6),x)

[Out]

-12*x**4/(75*x**5 - 60*x**3 + (25*x**4 - 40*x**2 + 16)*log(x)) + x/5 + 4*x/(25*x**2 - 20) + exp(x)

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