3.96.21 \(\int \frac {-640 e^5 x-80 x^3}{9+30 x^4+6400 e^{10} x^4+25 x^8+e^5 (480 x^2+800 x^6)+(6+160 e^5 x^2+10 x^4) \log (2)+\log ^2(2)} \, dx\)

Optimal. Leaf size=23 \[ \frac {4}{3+5 x^3 \left (\frac {16 e^5}{x}+x\right )+\log (2)} \]

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Rubi [A]  time = 0.20, antiderivative size = 21, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 6, integrand size = 72, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6, 1593, 6688, 12, 1247, 629} \begin {gather*} \frac {4}{5 x^4+80 e^5 x^2+3+\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-640*E^5*x - 80*x^3)/(9 + 30*x^4 + 6400*E^10*x^4 + 25*x^8 + E^5*(480*x^2 + 800*x^6) + (6 + 160*E^5*x^2 +
10*x^4)*Log[2] + Log[2]^2),x]

[Out]

4/(3 + 80*E^5*x^2 + 5*x^4 + Log[2])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-640 e^5 x-80 x^3}{9+\left (30+6400 e^{10}\right ) x^4+25 x^8+e^5 \left (480 x^2+800 x^6\right )+\left (6+160 e^5 x^2+10 x^4\right ) \log (2)+\log ^2(2)} \, dx\\ &=\int \frac {x \left (-640 e^5-80 x^2\right )}{9+\left (30+6400 e^{10}\right ) x^4+25 x^8+e^5 \left (480 x^2+800 x^6\right )+\left (6+160 e^5 x^2+10 x^4\right ) \log (2)+\log ^2(2)} \, dx\\ &=\int \frac {80 x \left (-8 e^5-x^2\right )}{\left (3+80 e^5 x^2+5 x^4+\log (2)\right )^2} \, dx\\ &=80 \int \frac {x \left (-8 e^5-x^2\right )}{\left (3+80 e^5 x^2+5 x^4+\log (2)\right )^2} \, dx\\ &=40 \operatorname {Subst}\left (\int \frac {-8 e^5-x}{\left (3+80 e^5 x+5 x^2+\log (2)\right )^2} \, dx,x,x^2\right )\\ &=\frac {4}{3+80 e^5 x^2+5 x^4+\log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.91 \begin {gather*} \frac {4}{3+80 e^5 x^2+5 x^4+\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-640*E^5*x - 80*x^3)/(9 + 30*x^4 + 6400*E^10*x^4 + 25*x^8 + E^5*(480*x^2 + 800*x^6) + (6 + 160*E^5*
x^2 + 10*x^4)*Log[2] + Log[2]^2),x]

[Out]

4/(3 + 80*E^5*x^2 + 5*x^4 + Log[2])

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fricas [A]  time = 0.53, size = 20, normalized size = 0.87 \begin {gather*} \frac {4}{5 \, x^{4} + 80 \, x^{2} e^{5} + \log \relax (2) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-640*x*exp(5)-80*x^3)/(log(2)^2+(160*x^2*exp(5)+10*x^4+6)*log(2)+6400*x^4*exp(5)^2+(800*x^6+480*x^2
)*exp(5)+25*x^8+30*x^4+9),x, algorithm="fricas")

[Out]

4/(5*x^4 + 80*x^2*e^5 + log(2) + 3)

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giac [A]  time = 6.70, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-640*x*exp(5)-80*x^3)/(log(2)^2+(160*x^2*exp(5)+10*x^4+6)*log(2)+6400*x^4*exp(5)^2+(800*x^6+480*x^2
)*exp(5)+25*x^8+30*x^4+9),x, algorithm="giac")

[Out]

0

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maple [A]  time = 0.17, size = 21, normalized size = 0.91

method	result	size
gosper	\(\frac {4}{5 x^{4}+80 x^{2} {\mathrm e}^{5}+\ln \relax (2)+3}\)	\(21\)
norman	\(\frac {4}{5 x^{4}+80 x^{2} {\mathrm e}^{5}+\ln \relax (2)+3}\)	\(21\)
risch	\(\frac {1}{\frac {5 x^{4}}{4}+20 x^{2} {\mathrm e}^{5}+\frac {\ln \relax (2)}{4}+\frac {3}{4}}\)	\(22\)
default	\(2 \left (\munderset {\textit {\_R} =\RootOf \left (25 \textit {\_Z}^{4}+800 \textit {\_Z}^{3} {\mathrm e}^{5}+\left (6400 \,{\mathrm e}^{10}+10 \ln \relax (2)+30\right ) \textit {\_Z}^{2}+\left (160 \,{\mathrm e}^{5} \ln \relax (2)+480 \,{\mathrm e}^{5}\right ) \textit {\_Z} +9+\ln \relax (2)^{2}+6 \ln \relax (2)\right )}{\sum }\frac {\left (-\textit {\_R} -8 \,{\mathrm e}^{5}\right ) \ln \left (x^{2}-\textit {\_R} \right )}{5 \textit {\_R}^{3}+120 \textit {\_R}^{2} {\mathrm e}^{5}+640 \textit {\_R} \,{\mathrm e}^{10}+\textit {\_R} \ln \relax (2)+8 \,{\mathrm e}^{5} \ln \relax (2)+3 \textit {\_R} +24 \,{\mathrm e}^{5}}\right )\)	\(110\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-640*x*exp(5)-80*x^3)/(ln(2)^2+(160*x^2*exp(5)+10*x^4+6)*ln(2)+6400*x^4*exp(5)^2+(800*x^6+480*x^2)*exp(5)
+25*x^8+30*x^4+9),x,method=_RETURNVERBOSE)

[Out]

4/(5*x^4+80*x^2*exp(5)+ln(2)+3)

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maxima [A]  time = 0.35, size = 20, normalized size = 0.87 \begin {gather*} \frac {4}{5 \, x^{4} + 80 \, x^{2} e^{5} + \log \relax (2) + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-640*x*exp(5)-80*x^3)/(log(2)^2+(160*x^2*exp(5)+10*x^4+6)*log(2)+6400*x^4*exp(5)^2+(800*x^6+480*x^2
)*exp(5)+25*x^8+30*x^4+9),x, algorithm="maxima")

[Out]

4/(5*x^4 + 80*x^2*e^5 + log(2) + 3)

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mupad [B]  time = 11.27, size = 1, normalized size = 0.04 \begin {gather*} 0 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(640*x*exp(5) + 80*x^3)/(exp(5)*(480*x^2 + 800*x^6) + 6400*x^4*exp(10) + log(2)^2 + 30*x^4 + 25*x^8 + log
(2)*(160*x^2*exp(5) + 10*x^4 + 6) + 9),x)

[Out]

0

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sympy [A]  time = 1.09, size = 19, normalized size = 0.83 \begin {gather*} \frac {4}{5 x^{4} + 80 x^{2} e^{5} + \log {\relax (2 )} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-640*x*exp(5)-80*x**3)/(ln(2)**2+(160*x**2*exp(5)+10*x**4+6)*ln(2)+6400*x**4*exp(5)**2+(800*x**6+48
0*x**2)*exp(5)+25*x**8+30*x**4+9),x)

[Out]

4/(5*x**4 + 80*x**2*exp(5) + log(2) + 3)

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