3.91.41 \(\int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 (160 x^6+208 x^7)+e^3 (192 x^5+480 x^6+208 x^7)+e^6 (96 x^5+480 x^6+312 x^7)}{256+1024 x+1440 x^2+832 x^3+169 x^4+169 e^{12} x^4+e^9 (832 x^3+676 x^4)+e^3 (1024 x+2880 x^2+2496 x^3+676 x^4)+e^6 (1440 x^2+2496 x^3+1014 x^4)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^4}{-3+\left (4+\frac {4}{\left (1+e^3\right ) x}\right )^2} \]

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Rubi [A]  time = 0.28, antiderivative size = 35, normalized size of antiderivative = 1.46, number of steps used = 5, number of rules used = 4, integrand size = 168, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {6, 1680, 12, 1590} \begin {gather*} \frac {\left (1+e^3\right )^2 x^6}{13 \left (1+e^3\right )^2 x^2+32 \left (1+e^3\right ) x+16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(96*x^5 + 160*x^6 + 52*x^7 + 52*E^12*x^7 + E^9*(160*x^6 + 208*x^7) + E^3*(192*x^5 + 480*x^6 + 208*x^7) + E
^6*(96*x^5 + 480*x^6 + 312*x^7))/(256 + 1024*x + 1440*x^2 + 832*x^3 + 169*x^4 + 169*E^12*x^4 + E^9*(832*x^3 +
676*x^4) + E^3*(1024*x + 2880*x^2 + 2496*x^3 + 676*x^4) + E^6*(1440*x^2 + 2496*x^3 + 1014*x^4)),x]

[Out]

((1 + E^3)^2*x^6)/(16 + 32*(1 + E^3)*x + 13*(1 + E^3)^2*x^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 1590

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[(Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*Rr^(n + 1))/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x
, r]), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {96 x^5+160 x^6+52 x^7+52 e^{12} x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+\left (169+169 e^{12}\right ) x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx\\ &=\int \frac {96 x^5+160 x^6+\left (52+52 e^{12}\right ) x^7+e^9 \left (160 x^6+208 x^7\right )+e^3 \left (192 x^5+480 x^6+208 x^7\right )+e^6 \left (96 x^5+480 x^6+312 x^7\right )}{256+1024 x+1440 x^2+832 x^3+\left (169+169 e^{12}\right ) x^4+e^9 \left (832 x^3+676 x^4\right )+e^3 \left (1024 x+2880 x^2+2496 x^3+676 x^4\right )+e^6 \left (1440 x^2+2496 x^3+1014 x^4\right )} \, dx\\ &=\operatorname {Subst}\left (\int \frac {4 \left (16-13 \left (1+e^3\right ) x\right )^5 \left (72-104 \left (1+e^3\right ) x-169 \left (1+e^3\right )^2 x^2\right )}{28561 \left (1+e^3\right )^3 \left (48-169 \left (1+e^3\right )^2 x^2\right )^2} \, dx,x,\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}+x\right )\\ &=\frac {4 \operatorname {Subst}\left (\int \frac {\left (16-13 \left (1+e^3\right ) x\right )^5 \left (72-104 \left (1+e^3\right ) x-169 \left (1+e^3\right )^2 x^2\right )}{\left (48-169 \left (1+e^3\right )^2 x^2\right )^2} \, dx,x,\frac {832+2496 e^3+2496 e^6+832 e^9}{4 \left (169+676 e^3+1014 e^6+676 e^9+169 e^{12}\right )}+x\right )}{28561 \left (1+e^3\right )^3}\\ &=\frac {\left (1+e^3\right )^2 x^6}{16+32 \left (1+e^3\right ) x+13 \left (1+e^3\right )^2 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.05, size = 69, normalized size = 2.88 \begin {gather*} -\frac {7245824+14491648 \left (1+e^3\right ) x+5887232 \left (1+e^3\right )^2 x^2-371293 \left (1+e^3\right )^6 x^6}{371293 \left (1+e^3\right )^4 \left (16+32 \left (1+e^3\right ) x+13 \left (1+e^3\right )^2 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(96*x^5 + 160*x^6 + 52*x^7 + 52*E^12*x^7 + E^9*(160*x^6 + 208*x^7) + E^3*(192*x^5 + 480*x^6 + 208*x^
7) + E^6*(96*x^5 + 480*x^6 + 312*x^7))/(256 + 1024*x + 1440*x^2 + 832*x^3 + 169*x^4 + 169*E^12*x^4 + E^9*(832*
x^3 + 676*x^4) + E^3*(1024*x + 2880*x^2 + 2496*x^3 + 676*x^4) + E^6*(1440*x^2 + 2496*x^3 + 1014*x^4)),x]

[Out]

-1/371293*(7245824 + 14491648*(1 + E^3)*x + 5887232*(1 + E^3)^2*x^2 - 371293*(1 + E^3)^6*x^6)/((1 + E^3)^4*(16
+ 32*(1 + E^3)*x + 13*(1 + E^3)^2*x^2))

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fricas [B]  time = 0.62, size = 164, normalized size = 6.83 \begin {gather*} \frac {371293 \, x^{6} e^{18} + 2227758 \, x^{6} e^{15} + 5569395 \, x^{6} e^{12} + 7425860 \, x^{6} e^{9} + 371293 \, x^{6} - 5887232 \, x^{2} + 13 \, {\left (428415 \, x^{6} - 452864 \, x^{2}\right )} e^{6} + 2 \, {\left (1113879 \, x^{6} - 5887232 \, x^{2} - 7245824 \, x\right )} e^{3} - 14491648 \, x - 7245824}{371293 \, {\left (13 \, x^{2} e^{18} + 13 \, x^{2} + 2 \, {\left (39 \, x^{2} + 16 \, x\right )} e^{15} + {\left (195 \, x^{2} + 160 \, x + 16\right )} e^{12} + 4 \, {\left (65 \, x^{2} + 80 \, x + 16\right )} e^{9} + {\left (195 \, x^{2} + 320 \, x + 96\right )} e^{6} + 2 \, {\left (39 \, x^{2} + 80 \, x + 32\right )} e^{3} + 32 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96*x^5)*exp(3)^2+(208*x^7+480*x^6+192*x
^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169*x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*ex
p(3)^2+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+1024*x+256),x, algorithm="fricas")

[Out]

1/371293*(371293*x^6*e^18 + 2227758*x^6*e^15 + 5569395*x^6*e^12 + 7425860*x^6*e^9 + 371293*x^6 - 5887232*x^2 +
13*(428415*x^6 - 452864*x^2)*e^6 + 2*(1113879*x^6 - 5887232*x^2 - 7245824*x)*e^3 - 14491648*x - 7245824)/(13*
x^2*e^18 + 13*x^2 + 2*(39*x^2 + 16*x)*e^15 + (195*x^2 + 160*x + 16)*e^12 + 4*(65*x^2 + 80*x + 16)*e^9 + (195*x
^2 + 320*x + 96)*e^6 + 2*(39*x^2 + 80*x + 32)*e^3 + 32*x + 16)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96*x^5)*exp(3)^2+(208*x^7+480*x^6+192*x
^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169*x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*ex
p(3)^2+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+1024*x+256),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.43, size = 48, normalized size = 2.00

method	result	size
gosper	\(\frac {x^{6} \left ({\mathrm e}^{6}+2 \,{\mathrm e}^{3}+1\right )}{13 x^{2} {\mathrm e}^{6}+26 x^{2} {\mathrm e}^{3}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16}\)	\(48\)
norman	\(\frac {x^{6} \left ({\mathrm e}^{6}+2 \,{\mathrm e}^{3}+1\right )}{13 x^{2} {\mathrm e}^{6}+26 x^{2} {\mathrm e}^{3}+32 x \,{\mathrm e}^{3}+13 x^{2}+32 x +16}\)	\(48\)
risch	\(\frac {6591 \,{\mathrm e}^{6} x^{4}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {2197 x^{4} {\mathrm e}^{9}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {6591 \,{\mathrm e}^{3} x^{4}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {5408 x^{3} {\mathrm e}^{6}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {10816 \,{\mathrm e}^{3} x^{3}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {2197 x^{4}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {10608 x^{2} {\mathrm e}^{3}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {5408 x^{3}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {10608 x^{2}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}-\frac {19456 x}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2}}+\frac {-\frac {10444800 x}{169}-\frac {7245824}{169 \left ({\mathrm e}^{3}+1\right )}}{\left (169 \,{\mathrm e}^{3}+169\right ) \left (13 \,{\mathrm e}^{3}+13\right )^{2} \left (x^{2} {\mathrm e}^{6}+2 x^{2} {\mathrm e}^{3}+\frac {32 x \,{\mathrm e}^{3}}{13}+x^{2}+\frac {32 x}{13}+\frac {16}{13}\right )}\)	\(279\)
default	\(\text {Expression too large to display}\)	\(2908\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96*x^5)*exp(3)^2+(208*x^7+480*x^6+192*x^5)*ex
p(3)+52*x^7+160*x^6+96*x^5)/(169*x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*exp(3)^2
+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+1024*x+256),x,method=_RETURNVERBOSE)

[Out]

x^6*(exp(3)^2+2*exp(3)+1)/(13*x^2*exp(3)^2+26*x^2*exp(3)+32*x*exp(3)+13*x^2+32*x+16)

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maxima [B]  time = 0.35, size = 143, normalized size = 5.96 \begin {gather*} -\frac {4096 \, {\left (2550 \, x {\left (e^{3} + 1\right )} + 1769\right )}}{371293 \, {\left (13 \, x^{2} {\left (e^{18} + 6 \, e^{15} + 15 \, e^{12} + 20 \, e^{9} + 15 \, e^{6} + 6 \, e^{3} + 1\right )} + 32 \, x {\left (e^{15} + 5 \, e^{12} + 10 \, e^{9} + 10 \, e^{6} + 5 \, e^{3} + 1\right )} + 16 \, e^{12} + 64 \, e^{9} + 96 \, e^{6} + 64 \, e^{3} + 16\right )}} + \frac {2197 \, x^{4} {\left (e^{9} + 3 \, e^{6} + 3 \, e^{3} + 1\right )} - 5408 \, x^{3} {\left (e^{6} + 2 \, e^{3} + 1\right )} + 10608 \, x^{2} {\left (e^{3} + 1\right )} - 19456 \, x}{28561 \, {\left (e^{9} + 3 \, e^{6} + 3 \, e^{3} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((52*x^7*exp(3)^4+(208*x^7+160*x^6)*exp(3)^3+(312*x^7+480*x^6+96*x^5)*exp(3)^2+(208*x^7+480*x^6+192*x
^5)*exp(3)+52*x^7+160*x^6+96*x^5)/(169*x^4*exp(3)^4+(676*x^4+832*x^3)*exp(3)^3+(1014*x^4+2496*x^3+1440*x^2)*ex
p(3)^2+(676*x^4+2496*x^3+2880*x^2+1024*x)*exp(3)+169*x^4+832*x^3+1440*x^2+1024*x+256),x, algorithm="maxima")

[Out]

-4096/371293*(2550*x*(e^3 + 1) + 1769)/(13*x^2*(e^18 + 6*e^15 + 15*e^12 + 20*e^9 + 15*e^6 + 6*e^3 + 1) + 32*x*
(e^15 + 5*e^12 + 10*e^9 + 10*e^6 + 5*e^3 + 1) + 16*e^12 + 64*e^9 + 96*e^6 + 64*e^3 + 16) + 1/28561*(2197*x^4*(
e^9 + 3*e^6 + 3*e^3 + 1) - 5408*x^3*(e^6 + 2*e^3 + 1) + 10608*x^2*(e^3 + 1) - 19456*x)/(e^9 + 3*e^6 + 3*e^3 +
1)

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mupad [B]  time = 7.48, size = 133, normalized size = 5.54 \begin {gather*} \frac {816\,x^2}{2197\,{\left ({\mathrm {e}}^3+1\right )}^2}-\frac {32\,x^3}{169\,\left ({\mathrm {e}}^3+1\right )}+x\,\left (\frac {33792}{28561\,{\left ({\mathrm {e}}^3+1\right )}^3}-\frac {4\,\left (1024\,{\mathrm {e}}^3+1024\right )}{2197\,{\left ({\mathrm {e}}^3+1\right )}^4}\right )+\frac {x^4}{13}-\frac {\frac {10444800\,x}{13}+\frac {7245824}{13\,\left ({\mathrm {e}}^3+1\right )}}{\left (1856465\,{\mathrm {e}}^3+3712930\,{\mathrm {e}}^6+3712930\,{\mathrm {e}}^9+1856465\,{\mathrm {e}}^{12}+371293\,{\mathrm {e}}^{15}+371293\right )\,x^2+\left (3655808\,{\mathrm {e}}^3+5483712\,{\mathrm {e}}^6+3655808\,{\mathrm {e}}^9+913952\,{\mathrm {e}}^{12}+913952\right )\,x+1370928\,{\mathrm {e}}^3+1370928\,{\mathrm {e}}^6+456976\,{\mathrm {e}}^9+456976} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(9)*(160*x^6 + 208*x^7) + 52*x^7*exp(12) + exp(3)*(192*x^5 + 480*x^6 + 208*x^7) + exp(6)*(96*x^5 + 480
*x^6 + 312*x^7) + 96*x^5 + 160*x^6 + 52*x^7)/(1024*x + exp(9)*(832*x^3 + 676*x^4) + 169*x^4*exp(12) + exp(3)*(
1024*x + 2880*x^2 + 2496*x^3 + 676*x^4) + exp(6)*(1440*x^2 + 2496*x^3 + 1014*x^4) + 1440*x^2 + 832*x^3 + 169*x
^4 + 256),x)

[Out]

(816*x^2)/(2197*(exp(3) + 1)^2) - (32*x^3)/(169*(exp(3) + 1)) + x*(33792/(28561*(exp(3) + 1)^3) - (4*(1024*exp
(3) + 1024))/(2197*(exp(3) + 1)^4)) + x^4/13 - ((10444800*x)/13 + 7245824/(13*(exp(3) + 1)))/(1370928*exp(3) +
1370928*exp(6) + 456976*exp(9) + x*(3655808*exp(3) + 5483712*exp(6) + 3655808*exp(9) + 913952*exp(12) + 91395
2) + x^2*(1856465*exp(3) + 3712930*exp(6) + 3712930*exp(9) + 1856465*exp(12) + 371293*exp(15) + 371293) + 4569
76)

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sympy [B]  time = 4.28, size = 151, normalized size = 6.29 \begin {gather*} \frac {x^{4}}{13} - \frac {32 x^{3}}{169 + 169 e^{3}} + \frac {816 x^{2}}{2197 + 4394 e^{3} + 2197 e^{6}} - \frac {19456 x}{28561 + 85683 e^{3} + 85683 e^{6} + 28561 e^{9}} + \frac {x \left (- 10444800 e^{3} - 10444800\right ) - 7245824}{x^{2} \left (4826809 + 28960854 e^{3} + 72402135 e^{6} + 96536180 e^{9} + 72402135 e^{12} + 28960854 e^{15} + 4826809 e^{18}\right ) + x \left (11881376 + 59406880 e^{3} + 118813760 e^{6} + 118813760 e^{9} + 59406880 e^{12} + 11881376 e^{15}\right ) + 5940688 + 23762752 e^{3} + 35644128 e^{6} + 23762752 e^{9} + 5940688 e^{12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((52*x**7*exp(3)**4+(208*x**7+160*x**6)*exp(3)**3+(312*x**7+480*x**6+96*x**5)*exp(3)**2+(208*x**7+480
*x**6+192*x**5)*exp(3)+52*x**7+160*x**6+96*x**5)/(169*x**4*exp(3)**4+(676*x**4+832*x**3)*exp(3)**3+(1014*x**4+
2496*x**3+1440*x**2)*exp(3)**2+(676*x**4+2496*x**3+2880*x**2+1024*x)*exp(3)+169*x**4+832*x**3+1440*x**2+1024*x
+256),x)

[Out]

x**4/13 - 32*x**3/(169 + 169*exp(3)) + 816*x**2/(2197 + 4394*exp(3) + 2197*exp(6)) - 19456*x/(28561 + 85683*ex
p(3) + 85683*exp(6) + 28561*exp(9)) + (x*(-10444800*exp(3) - 10444800) - 7245824)/(x**2*(4826809 + 28960854*ex
p(3) + 72402135*exp(6) + 96536180*exp(9) + 72402135*exp(12) + 28960854*exp(15) + 4826809*exp(18)) + x*(1188137
6 + 59406880*exp(3) + 118813760*exp(6) + 118813760*exp(9) + 59406880*exp(12) + 11881376*exp(15)) + 5940688 + 2
3762752*exp(3) + 35644128*exp(6) + 23762752*exp(9) + 5940688*exp(12))

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