3.80.45 \(\int \frac {e^x (192 x^2-23 x^4+9 x^5)}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx\)

Optimal. Leaf size=24 \[ \frac {e^x x}{4 \left (5+\frac {4 (4-x)^2}{x^2}\right )} \]

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Rubi [C]  time = 1.29, antiderivative size = 544, normalized size of antiderivative = 22.67, number of steps used = 30, number of rules used = 9, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1594, 6688, 12, 6742, 2194, 2176, 2177, 2178, 2270} \begin {gather*} \frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{3645}-\frac {176 \left (2-i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{3645}+\frac {7 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )}{81 \sqrt {5}}+\frac {8}{405} e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2-i \sqrt {5}\right )\right )\right )+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{3645}-\frac {176 \left (2+i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{3645}-\frac {7 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )}{81 \sqrt {5}}+\frac {8}{405} e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (9 x-8 \left (2+i \sqrt {5}\right )\right )\right )+\frac {e^x x}{36}+\frac {8 e^x}{81}-\frac {176 \left (2-i \sqrt {5}\right ) e^x}{405 \left (-9 x+8 \left (2-i \sqrt {5}\right )\right )}+\frac {8 e^x}{45 \left (-9 x+8 \left (2-i \sqrt {5}\right )\right )}-\frac {176 \left (2+i \sqrt {5}\right ) e^x}{405 \left (-9 x+8 \left (2+i \sqrt {5}\right )\right )}+\frac {8 e^x}{45 \left (-9 x+8 \left (2+i \sqrt {5}\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^x*(192*x^2 - 23*x^4 + 9*x^5))/(16384 - 16384*x + 8704*x^2 - 2304*x^3 + 324*x^4),x]

[Out]

(8*E^x)/81 + (8*E^x)/(45*(8*(2 - I*Sqrt[5]) - 9*x)) - (176*(2 - I*Sqrt[5])*E^x)/(405*(8*(2 - I*Sqrt[5]) - 9*x)
) + (8*E^x)/(45*(8*(2 + I*Sqrt[5]) - 9*x)) - (176*(2 + I*Sqrt[5])*E^x)/(405*(8*(2 + I*Sqrt[5]) - 9*x)) + (E^x*
x)/36 + (8*E^((8*(2 - I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/405 + (((7*I)/81)*E^((8*(2 -
I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/Sqrt[5] - (176*(2 - I*Sqrt[5])*E^((8*(2 - I*Sqrt[
5]))/9)*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/3645 + ((280 - (239*I)*Sqrt[5])*E^((8*(2 - I*Sqrt[5]))/9)
*ExpIntegralEi[(-8*(2 - I*Sqrt[5]) + 9*x)/9])/3645 + (8*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I*Sqr
t[5]) + 9*x)/9])/405 - (((7*I)/81)*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I*Sqrt[5]) + 9*x)/9])/Sqrt
[5] - (176*(2 + I*Sqrt[5])*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I*Sqrt[5]) + 9*x)/9])/3645 + ((280
+ (239*I)*Sqrt[5])*E^((8*(2 + I*Sqrt[5]))/9)*ExpIntegralEi[(-8*(2 + I*Sqrt[5]) + 9*x)/9])/3645

Rule 12

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

Rule 1594

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2270

Int[((F_)^((g_.)*((d_.) + (e_.)*(x_))^(n_.))*(u_)^(m_.))/((a_.) + (b_.)*(x_) + (c_)*(x_)^2), x_Symbol] :> Int[
ExpandIntegrand[F^(g*(d + e*x)^n), u^m/(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e, g, n}, x] && Poly
nomialQ[u, x] && IntegerQ[m]

Rule 6688

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x x^2 \left (192-23 x^2+9 x^3\right )}{16384-16384 x+8704 x^2-2304 x^3+324 x^4} \, dx\\ &=\int \frac {e^x x^2 \left (192-23 x^2+9 x^3\right )}{4 \left (64-32 x+9 x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {e^x x^2 \left (192-23 x^2+9 x^3\right )}{\left (64-32 x+9 x^2\right )^2} \, dx\\ &=\frac {1}{4} \int \left (\frac {41 e^x}{81}+\frac {e^x x}{9}+\frac {2048 e^x (-4+11 x)}{81 \left (64-32 x+9 x^2\right )^2}+\frac {64 e^x (-39+7 x)}{81 \left (64-32 x+9 x^2\right )}\right ) \, dx\\ &=\frac {1}{36} \int e^x x \, dx+\frac {41 \int e^x \, dx}{324}+\frac {16}{81} \int \frac {e^x (-39+7 x)}{64-32 x+9 x^2} \, dx+\frac {512}{81} \int \frac {e^x (-4+11 x)}{\left (64-32 x+9 x^2\right )^2} \, dx\\ &=\frac {41 e^x}{324}+\frac {e^x x}{36}-\frac {\int e^x \, dx}{36}+\frac {16}{81} \int \left (\frac {\left (7+\frac {239 i}{8 \sqrt {5}}\right ) e^x}{-32-16 i \sqrt {5}+18 x}+\frac {\left (7-\frac {239 i}{8 \sqrt {5}}\right ) e^x}{-32+16 i \sqrt {5}+18 x}\right ) \, dx+\frac {512}{81} \int \left (-\frac {4 e^x}{\left (64-32 x+9 x^2\right )^2}+\frac {11 e^x x}{\left (64-32 x+9 x^2\right )^2}\right ) \, dx\\ &=\frac {8 e^x}{81}+\frac {e^x x}{36}-\frac {2048}{81} \int \frac {e^x}{\left (64-32 x+9 x^2\right )^2} \, dx+\frac {5632}{81} \int \frac {e^x x}{\left (64-32 x+9 x^2\right )^2} \, dx+\frac {1}{405} \left (2 \left (280-239 i \sqrt {5}\right )\right ) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx+\frac {1}{405} \left (2 \left (280+239 i \sqrt {5}\right )\right ) \int \frac {e^x}{-32-16 i \sqrt {5}+18 x} \, dx\\ &=\frac {8 e^x}{81}+\frac {e^x x}{36}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}-\frac {2048}{81} \int \left (-\frac {81 e^x}{320 \left (32+16 i \sqrt {5}-18 x\right )^2}+\frac {81 i e^x}{5120 \sqrt {5} \left (32+16 i \sqrt {5}-18 x\right )}-\frac {81 e^x}{320 \left (-32+16 i \sqrt {5}+18 x\right )^2}+\frac {81 i e^x}{5120 \sqrt {5} \left (-32+16 i \sqrt {5}+18 x\right )}\right ) \, dx+\frac {5632}{81} \int \left (-\frac {9 \left (32+16 i \sqrt {5}\right ) e^x}{640 \left (32+16 i \sqrt {5}-18 x\right )^2}+\frac {9 i e^x}{320 \sqrt {5} \left (32+16 i \sqrt {5}-18 x\right )}-\frac {9 \left (32-16 i \sqrt {5}\right ) e^x}{640 \left (-32+16 i \sqrt {5}+18 x\right )^2}+\frac {9 i e^x}{320 \sqrt {5} \left (-32+16 i \sqrt {5}+18 x\right )}\right ) \, dx\\ &=\frac {8 e^x}{81}+\frac {e^x x}{36}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {32}{5} \int \frac {e^x}{\left (32+16 i \sqrt {5}-18 x\right )^2} \, dx+\frac {32}{5} \int \frac {e^x}{\left (-32+16 i \sqrt {5}+18 x\right )^2} \, dx-\frac {(2 i) \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx}{5 \sqrt {5}}-\frac {(2 i) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx}{5 \sqrt {5}}+\frac {(88 i) \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx}{45 \sqrt {5}}+\frac {(88 i) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx}{45 \sqrt {5}}-\frac {1}{45} \left (704 \left (2-i \sqrt {5}\right )\right ) \int \frac {e^x}{\left (-32+16 i \sqrt {5}+18 x\right )^2} \, dx-\frac {1}{45} \left (704 \left (2+i \sqrt {5}\right )\right ) \int \frac {e^x}{\left (32+16 i \sqrt {5}-18 x\right )^2} \, dx\\ &=\frac {8 e^x}{81}+\frac {8 e^x}{45 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2-i \sqrt {5}\right ) e^x}{405 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}+\frac {8 e^x}{45 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2+i \sqrt {5}\right ) e^x}{405 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}+\frac {e^x x}{36}+\frac {7 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}-\frac {7 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}-\frac {16}{45} \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx+\frac {16}{45} \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx-\frac {1}{405} \left (352 \left (2-i \sqrt {5}\right )\right ) \int \frac {e^x}{-32+16 i \sqrt {5}+18 x} \, dx+\frac {1}{405} \left (352 \left (2+i \sqrt {5}\right )\right ) \int \frac {e^x}{32+16 i \sqrt {5}-18 x} \, dx\\ &=\frac {8 e^x}{81}+\frac {8 e^x}{45 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2-i \sqrt {5}\right ) e^x}{405 \left (8 \left (2-i \sqrt {5}\right )-9 x\right )}+\frac {8 e^x}{45 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}-\frac {176 \left (2+i \sqrt {5}\right ) e^x}{405 \left (8 \left (2+i \sqrt {5}\right )-9 x\right )}+\frac {e^x x}{36}+\frac {8}{405} e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )+\frac {7 i e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}-\frac {176 \left (2-i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280-239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2-i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2-i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {8}{405} e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )-\frac {7 i e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{81 \sqrt {5}}-\frac {176 \left (2+i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}+\frac {\left (280+239 i \sqrt {5}\right ) e^{\frac {8}{9} \left (2+i \sqrt {5}\right )} \text {Ei}\left (\frac {1}{9} \left (-8 \left (2+i \sqrt {5}\right )+9 x\right )\right )}{3645}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.30, size = 22, normalized size = 0.92 \begin {gather*} \frac {e^x x^3}{4 \left (64-32 x+9 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(192*x^2 - 23*x^4 + 9*x^5))/(16384 - 16384*x + 8704*x^2 - 2304*x^3 + 324*x^4),x]

[Out]

(E^x*x^3)/(4*(64 - 32*x + 9*x^2))

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fricas [A]  time = 0.72, size = 19, normalized size = 0.79 \begin {gather*} \frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x+16384),x, algorithm="fricas")

[Out]

1/4*x^3*e^x/(9*x^2 - 32*x + 64)

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giac [A]  time = 0.20, size = 19, normalized size = 0.79 \begin {gather*} \frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x+16384),x, algorithm="giac")

[Out]

1/4*x^3*e^x/(9*x^2 - 32*x + 64)

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maple [A]  time = 0.07, size = 20, normalized size = 0.83

method	result	size
gosper	\(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\)	\(20\)
norman	\(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\)	\(20\)
risch	\(\frac {{\mathrm e}^{x} x^{3}}{36 x^{2}-128 x +256}\)	\(20\)
default	\(-\frac {8 \,{\mathrm e}^{x} \left (x +16\right )}{15 \left (9 x^{2}-32 x +64\right )}-\frac {23 \,{\mathrm e}^{x}}{324}+\frac {184 \,{\mathrm e}^{x} \left (79 x -176\right )}{3645 \left (9 x^{2}-32 x +64\right )}+\frac {\left (9 x +55\right ) {\mathrm e}^{x}}{324}-\frac {128 \,{\mathrm e}^{x} \left (59 x -316\right )}{3645 \left (9 x^{2}-32 x +64\right )}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x+16384),x,method=_RETURNVERBOSE)

[Out]

1/4*x^3*exp(x)/(9*x^2-32*x+64)

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maxima [A]  time = 0.39, size = 19, normalized size = 0.79 \begin {gather*} \frac {x^{3} e^{x}}{4 \, {\left (9 \, x^{2} - 32 \, x + 64\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^5-23*x^4+192*x^2)*exp(x)/(324*x^4-2304*x^3+8704*x^2-16384*x+16384),x, algorithm="maxima")

[Out]

1/4*x^3*e^x/(9*x^2 - 32*x + 64)

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mupad [B]  time = 5.14, size = 18, normalized size = 0.75 \begin {gather*} \frac {x^3\,{\mathrm {e}}^x}{4\,\left (9\,x^2-32\,x+64\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(192*x^2 - 23*x^4 + 9*x^5))/(8704*x^2 - 16384*x - 2304*x^3 + 324*x^4 + 16384),x)

[Out]

(x^3*exp(x))/(4*(9*x^2 - 32*x + 64))

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sympy [A]  time = 0.11, size = 15, normalized size = 0.62 \begin {gather*} \frac {x^{3} e^{x}}{36 x^{2} - 128 x + 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**5-23*x**4+192*x**2)*exp(x)/(324*x**4-2304*x**3+8704*x**2-16384*x+16384),x)

[Out]

x**3*exp(x)/(36*x**2 - 128*x + 256)

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