3.91.56 \(\int \frac {130 x^3+e^{2 x} (78 x+52 x^2)+e^x (208 x^2+52 x^3)+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} (72 x^7+12 x^4 \log (2))+e^x (48 x^8+24 x^5 \log (2))} \, dx\)

Optimal. Leaf size=24 \[ 5-\frac {13}{3 x^2 \left (2 x \left (e^x+x\right )^2+\log (2)\right )} \]

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Rubi [F]  time = 3.94, 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 {130 x^3+e^{2 x} \left (78 x+52 x^2\right )+e^x \left (208 x^2+52 x^3\right )+26 \log (2)}{12 e^{4 x} x^5+48 e^{3 x} x^6+12 x^9+12 x^6 \log (2)+3 x^3 \log ^2(2)+e^{2 x} \left (72 x^7+12 x^4 \log (2)\right )+e^x \left (48 x^8+24 x^5 \log (2)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(130*x^3 + E^(2*x)*(78*x + 52*x^2) + E^x*(208*x^2 + 52*x^3) + 26*Log[2])/(12*E^(4*x)*x^5 + 48*E^(3*x)*x^6
+ 12*x^9 + 12*x^6*Log[2] + 3*x^3*Log[2]^2 + E^(2*x)*(72*x^7 + 12*x^4*Log[2]) + E^x*(48*x^8 + 24*x^5*Log[2])),x
]

[Out]

(52*Defer[Int][(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3 + Log[2])^(-2), x])/3 - (52*Defer[Int][E^x/(2*E^(2*x)*x + 4*E^
x*x^2 + 2*x^3 + Log[2])^2, x])/3 - (13*Log[2]*Defer[Int][1/(x^3*(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3 + Log[2])^2),
x])/3 - (13*Log[4]*Defer[Int][1/(x^2*(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3 + Log[2])^2), x])/3 + (52*Defer[Int][E^
x/(x*(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3 + Log[2])^2), x])/3 - (52*Defer[Int][x/(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3
+ Log[2])^2, x])/3 + 26*Defer[Int][1/(x^3*(4*E^(2*x)*x + 8*E^x*x^2 + 4*x^3 + Log[4])), x] + (52*Defer[Int][1/(
x^2*(4*E^(2*x)*x + 8*E^x*x^2 + 4*x^3 + Log[4])), x])/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {26 \left (5 x^3+2 e^x x^2 (4+x)+e^{2 x} x (3+2 x)+\log (2)\right )}{3 x^3 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx\\ &=\frac {26}{3} \int \frac {5 x^3+2 e^x x^2 (4+x)+e^{2 x} x (3+2 x)+\log (2)}{x^3 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx\\ &=\frac {26}{3} \int \left (\frac {3+2 x}{x^3 \left (4 e^{2 x} x+8 e^x x^2+4 x^3+\log (4)\right )}-\frac {-4 e^x x^2-4 x^3+4 e^x x^3+4 x^4+\log (2)+x \log (4)}{2 x^3 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}\right ) \, dx\\ &=-\left (\frac {13}{3} \int \frac {-4 e^x x^2-4 x^3+4 e^x x^3+4 x^4+\log (2)+x \log (4)}{x^3 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx\right )+\frac {26}{3} \int \frac {3+2 x}{x^3 \left (4 e^{2 x} x+8 e^x x^2+4 x^3+\log (4)\right )} \, dx\\ &=-\left (\frac {13}{3} \int \left (-\frac {4}{\left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}+\frac {4 e^x}{\left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}-\frac {4 e^x}{x \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}+\frac {4 x}{\left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}+\frac {\log (2)}{x^3 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}+\frac {\log (4)}{x^2 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2}\right ) \, dx\right )+\frac {26}{3} \int \left (\frac {3}{x^3 \left (4 e^{2 x} x+8 e^x x^2+4 x^3+\log (4)\right )}+\frac {2}{x^2 \left (4 e^{2 x} x+8 e^x x^2+4 x^3+\log (4)\right )}\right ) \, dx\\ &=\frac {52}{3} \int \frac {1}{\left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx-\frac {52}{3} \int \frac {e^x}{\left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx+\frac {52}{3} \int \frac {e^x}{x \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx-\frac {52}{3} \int \frac {x}{\left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx+\frac {52}{3} \int \frac {1}{x^2 \left (4 e^{2 x} x+8 e^x x^2+4 x^3+\log (4)\right )} \, dx+26 \int \frac {1}{x^3 \left (4 e^{2 x} x+8 e^x x^2+4 x^3+\log (4)\right )} \, dx-\frac {1}{3} (13 \log (2)) \int \frac {1}{x^3 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx-\frac {1}{3} (13 \log (4)) \int \frac {1}{x^2 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.73, size = 33, normalized size = 1.38 \begin {gather*} -\frac {13}{3 x^2 \left (2 e^{2 x} x+4 e^x x^2+2 x^3+\log (2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(130*x^3 + E^(2*x)*(78*x + 52*x^2) + E^x*(208*x^2 + 52*x^3) + 26*Log[2])/(12*E^(4*x)*x^5 + 48*E^(3*x
)*x^6 + 12*x^9 + 12*x^6*Log[2] + 3*x^3*Log[2]^2 + E^(2*x)*(72*x^7 + 12*x^4*Log[2]) + E^x*(48*x^8 + 24*x^5*Log[
2])),x]

[Out]

-13/(3*x^2*(2*E^(2*x)*x + 4*E^x*x^2 + 2*x^3 + Log[2]))

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fricas [A]  time = 0.50, size = 32, normalized size = 1.33 \begin {gather*} -\frac {13}{3 \, {\left (2 \, x^{5} + 4 \, x^{4} e^{x} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+
(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^5*log(2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x, algorith
m="fricas")

[Out]

-13/3/(2*x^5 + 4*x^4*e^x + 2*x^3*e^(2*x) + x^2*log(2))

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giac [A]  time = 1.98, size = 32, normalized size = 1.33 \begin {gather*} -\frac {26}{3 \, {\left (2 \, x^{5} + 4 \, x^{4} e^{x} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+
(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^5*log(2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x, algorith
m="giac")

[Out]

-26/3/(2*x^5 + 4*x^4*e^x + 2*x^3*e^(2*x) + x^2*log(2))

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maple [A]  time = 0.07, size = 30, normalized size = 1.25

method	result	size
risch	\(-\frac {13}{3 x^{2} \left (2 x \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} x^{2}+2 x^{3}+\ln \relax (2)\right )}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*ln(2)+130*x^3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+(12*x^4
*ln(2)+72*x^7)*exp(x)^2+(24*x^5*ln(2)+48*x^8)*exp(x)+3*x^3*ln(2)^2+12*x^6*ln(2)+12*x^9),x,method=_RETURNVERBOS
E)

[Out]

-13/3/x^2/(2*x*exp(2*x)+4*exp(x)*x^2+2*x^3+ln(2))

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maxima [A]  time = 0.51, size = 32, normalized size = 1.33 \begin {gather*} -\frac {13}{3 \, {\left (2 \, x^{5} + 4 \, x^{4} e^{x} + 2 \, x^{3} e^{\left (2 \, x\right )} + x^{2} \log \relax (2)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((52*x^2+78*x)*exp(x)^2+(52*x^3+208*x^2)*exp(x)+26*log(2)+130*x^3)/(12*x^5*exp(x)^4+48*x^6*exp(x)^3+
(12*x^4*log(2)+72*x^7)*exp(x)^2+(24*x^5*log(2)+48*x^8)*exp(x)+3*x^3*log(2)^2+12*x^6*log(2)+12*x^9),x, algorith
m="maxima")

[Out]

-13/3/(2*x^5 + 4*x^4*e^x + 2*x^3*e^(2*x) + x^2*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {26\,\ln \relax (2)+{\mathrm {e}}^{2\,x}\,\left (52\,x^2+78\,x\right )+{\mathrm {e}}^x\,\left (52\,x^3+208\,x^2\right )+130\,x^3}{3\,x^3\,{\ln \relax (2)}^2+12\,x^5\,{\mathrm {e}}^{4\,x}+48\,x^6\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^x\,\left (48\,x^8+24\,\ln \relax (2)\,x^5\right )+12\,x^6\,\ln \relax (2)+{\mathrm {e}}^{2\,x}\,\left (72\,x^7+12\,\ln \relax (2)\,x^4\right )+12\,x^9} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((26*log(2) + exp(2*x)*(78*x + 52*x^2) + exp(x)*(208*x^2 + 52*x^3) + 130*x^3)/(3*x^3*log(2)^2 + 12*x^5*exp(
4*x) + 48*x^6*exp(3*x) + exp(x)*(24*x^5*log(2) + 48*x^8) + 12*x^6*log(2) + exp(2*x)*(12*x^4*log(2) + 72*x^7) +
12*x^9),x)

[Out]

int((26*log(2) + exp(2*x)*(78*x + 52*x^2) + exp(x)*(208*x^2 + 52*x^3) + 130*x^3)/(3*x^3*log(2)^2 + 12*x^5*exp(
4*x) + 48*x^6*exp(3*x) + exp(x)*(24*x^5*log(2) + 48*x^8) + 12*x^6*log(2) + exp(2*x)*(12*x^4*log(2) + 72*x^7) +
12*x^9), x)

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sympy [A]  time = 0.24, size = 34, normalized size = 1.42 \begin {gather*} - \frac {13}{6 x^{5} + 12 x^{4} e^{x} + 6 x^{3} e^{2 x} + 3 x^{2} \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((52*x**2+78*x)*exp(x)**2+(52*x**3+208*x**2)*exp(x)+26*ln(2)+130*x**3)/(12*x**5*exp(x)**4+48*x**6*ex
p(x)**3+(12*x**4*ln(2)+72*x**7)*exp(x)**2+(24*x**5*ln(2)+48*x**8)*exp(x)+3*x**3*ln(2)**2+12*x**6*ln(2)+12*x**9
),x)

[Out]

-13/(6*x**5 + 12*x**4*exp(x) + 6*x**3*exp(2*x) + 3*x**2*log(2))

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