3.38.70 \(\int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6)}{8 x^4} \, dx\)

Optimal. Leaf size=29 \[ \frac {\left (5-\frac {7 x}{2}\right ) \left (e^{x/4}+x\right ) \left (\frac {2}{x}+x\right )^2}{x} \]

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Rubi [B]  time = 0.29, antiderivative size = 95, normalized size of antiderivative = 3.28, number of steps used = 23, number of rules used = 7, integrand size = 69, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.101, Rules used = {12, 14, 2199, 2194, 2177, 2178, 2176} \begin {gather*} -\frac {7 x^3}{2}+\frac {20 e^{x/4}}{x^3}-\frac {7}{2} e^{x/4} x^2+5 x^2-\frac {14 e^{x/4}}{x^2}+\frac {20}{x^2}+5 e^{x/4} x-14 x-14 e^{x/4}+\frac {20 e^{x/4}}{x}-\frac {14}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-320*x + 112*x^2 - 112*x^4 + 80*x^5 - 84*x^6 + E^(x/4)*(-480 + 264*x - 188*x^2 + 40*x^3 + 12*x^4 - 46*x^5
- 7*x^6))/(8*x^4),x]

[Out]

-14*E^(x/4) + (20*E^(x/4))/x^3 + 20/x^2 - (14*E^(x/4))/x^2 - 14/x + (20*E^(x/4))/x - 14*x + 5*E^(x/4)*x + 5*x^
2 - (7*E^(x/4)*x^2)/2 - (7*x^3)/2

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{8} \int \frac {-320 x+112 x^2-112 x^4+80 x^5-84 x^6+e^{x/4} \left (-480+264 x-188 x^2+40 x^3+12 x^4-46 x^5-7 x^6\right )}{x^4} \, dx\\ &=\frac {1}{8} \int \left (-\frac {e^{x/4} \left (2+x^2\right ) \left (240-132 x-26 x^2+46 x^3+7 x^4\right )}{x^4}-\frac {4 \left (80-28 x+28 x^3-20 x^4+21 x^5\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {e^{x/4} \left (2+x^2\right ) \left (240-132 x-26 x^2+46 x^3+7 x^4\right )}{x^4} \, dx\right )-\frac {1}{2} \int \frac {80-28 x+28 x^3-20 x^4+21 x^5}{x^3} \, dx\\ &=-\left (\frac {1}{8} \int \left (-12 e^{x/4}+\frac {480 e^{x/4}}{x^4}-\frac {264 e^{x/4}}{x^3}+\frac {188 e^{x/4}}{x^2}-\frac {40 e^{x/4}}{x}+46 e^{x/4} x+7 e^{x/4} x^2\right ) \, dx\right )-\frac {1}{2} \int \left (28+\frac {80}{x^3}-\frac {28}{x^2}-20 x+21 x^2\right ) \, dx\\ &=\frac {20}{x^2}-\frac {14}{x}-14 x+5 x^2-\frac {7 x^3}{2}-\frac {7}{8} \int e^{x/4} x^2 \, dx+\frac {3}{2} \int e^{x/4} \, dx+5 \int \frac {e^{x/4}}{x} \, dx-\frac {23}{4} \int e^{x/4} x \, dx-\frac {47}{2} \int \frac {e^{x/4}}{x^2} \, dx+33 \int \frac {e^{x/4}}{x^3} \, dx-60 \int \frac {e^{x/4}}{x^4} \, dx\\ &=6 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {33 e^{x/4}}{2 x^2}-\frac {14}{x}+\frac {47 e^{x/4}}{2 x}-14 x-23 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}+5 \text {Ei}\left (\frac {x}{4}\right )+\frac {33}{8} \int \frac {e^{x/4}}{x^2} \, dx-5 \int \frac {e^{x/4}}{x^3} \, dx-\frac {47}{8} \int \frac {e^{x/4}}{x} \, dx+7 \int e^{x/4} x \, dx+23 \int e^{x/4} \, dx\\ &=98 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {14 e^{x/4}}{x^2}-\frac {14}{x}+\frac {155 e^{x/4}}{8 x}-14 x+5 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}-\frac {7 \text {Ei}\left (\frac {x}{4}\right )}{8}-\frac {5}{8} \int \frac {e^{x/4}}{x^2} \, dx+\frac {33}{32} \int \frac {e^{x/4}}{x} \, dx-28 \int e^{x/4} \, dx\\ &=-14 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {14 e^{x/4}}{x^2}-\frac {14}{x}+\frac {20 e^{x/4}}{x}-14 x+5 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}+\frac {5 \text {Ei}\left (\frac {x}{4}\right )}{32}-\frac {5}{32} \int \frac {e^{x/4}}{x} \, dx\\ &=-14 e^{x/4}+\frac {20 e^{x/4}}{x^3}+\frac {20}{x^2}-\frac {14 e^{x/4}}{x^2}-\frac {14}{x}+\frac {20 e^{x/4}}{x}-14 x+5 e^{x/4} x+5 x^2-\frac {7}{2} e^{x/4} x^2-\frac {7 x^3}{2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.19, size = 50, normalized size = 1.72 \begin {gather*} -\frac {e^{x/4} (-10+7 x) \left (2+x^2\right )^2+x \left (-40+28 x+28 x^3-10 x^4+7 x^5\right )}{2 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-320*x + 112*x^2 - 112*x^4 + 80*x^5 - 84*x^6 + E^(x/4)*(-480 + 264*x - 188*x^2 + 40*x^3 + 12*x^4 -
46*x^5 - 7*x^6))/(8*x^4),x]

[Out]

-1/2*(E^(x/4)*(-10 + 7*x)*(2 + x^2)^2 + x*(-40 + 28*x + 28*x^3 - 10*x^4 + 7*x^5))/x^3

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fricas [B]  time = 0.66, size = 59, normalized size = 2.03 \begin {gather*} -\frac {7 \, x^{6} - 10 \, x^{5} + 28 \, x^{4} + 28 \, x^{2} + {\left (7 \, x^{5} - 10 \, x^{4} + 28 \, x^{3} - 40 \, x^{2} + 28 \, x - 40\right )} e^{\left (\frac {1}{4} \, x\right )} - 40 \, x}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)-84*x^6+80*x^5-112*x^4+112*x^2-320*x)
/x^4,x, algorithm="fricas")

[Out]

-1/2*(7*x^6 - 10*x^5 + 28*x^4 + 28*x^2 + (7*x^5 - 10*x^4 + 28*x^3 - 40*x^2 + 28*x - 40)*e^(1/4*x) - 40*x)/x^3

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giac [B]  time = 0.22, size = 78, normalized size = 2.69 \begin {gather*} -\frac {7 \, x^{6} + 7 \, x^{5} e^{\left (\frac {1}{4} \, x\right )} - 10 \, x^{5} - 10 \, x^{4} e^{\left (\frac {1}{4} \, x\right )} + 28 \, x^{4} + 28 \, x^{3} e^{\left (\frac {1}{4} \, x\right )} - 40 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} + 28 \, x^{2} + 28 \, x e^{\left (\frac {1}{4} \, x\right )} - 40 \, x - 40 \, e^{\left (\frac {1}{4} \, x\right )}}{2 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)-84*x^6+80*x^5-112*x^4+112*x^2-320*x)
/x^4,x, algorithm="giac")

[Out]

-1/2*(7*x^6 + 7*x^5*e^(1/4*x) - 10*x^5 - 10*x^4*e^(1/4*x) + 28*x^4 + 28*x^3*e^(1/4*x) - 40*x^2*e^(1/4*x) + 28*
x^2 + 28*x*e^(1/4*x) - 40*x - 40*e^(1/4*x))/x^3

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maple [B]  time = 0.11, size = 59, normalized size = 2.03

method	result	size
risch	\(-\frac {7 x^{3}}{2}+5 x^{2}-14 x +\frac {-112 x +160}{8 x^{2}}-\frac {\left (7 x^{5}-10 x^{4}+28 x^{3}-40 x^{2}+28 x -40\right ) {\mathrm e}^{\frac {x}{4}}}{2 x^{3}}\)	\(59\)
derivativedivides	\(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 x \,{\mathrm e}^{\frac {x}{4}}-14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 x^{2} {\mathrm e}^{\frac {x}{4}}}{2}\)	\(74\)
default	\(5 x^{2}-14 x +\frac {20}{x^{2}}-\frac {14}{x}-\frac {7 x^{3}}{2}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}-\frac {14 \,{\mathrm e}^{\frac {x}{4}}}{x^{2}}+\frac {20 \,{\mathrm e}^{\frac {x}{4}}}{x}+5 x \,{\mathrm e}^{\frac {x}{4}}-14 \,{\mathrm e}^{\frac {x}{4}}-\frac {7 x^{2} {\mathrm e}^{\frac {x}{4}}}{2}\)	\(74\)
norman	\(\frac {20 x -14 x^{2}-14 x^{4}+5 x^{5}-\frac {7 x^{6}}{2}-14 x \,{\mathrm e}^{\frac {x}{4}}+20 x^{2} {\mathrm e}^{\frac {x}{4}}-14 \,{\mathrm e}^{\frac {x}{4}} x^{3}+5 \,{\mathrm e}^{\frac {x}{4}} x^{4}-\frac {7 \,{\mathrm e}^{\frac {x}{4}} x^{5}}{2}+20 \,{\mathrm e}^{\frac {x}{4}}}{x^{3}}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)-84*x^6+80*x^5-112*x^4+112*x^2-320*x)/x^4,x
,method=_RETURNVERBOSE)

[Out]

-7/2*x^3+5*x^2-14*x+1/8*(-112*x+160)/x^2-1/2*(7*x^5-10*x^4+28*x^3-40*x^2+28*x-40)/x^3*exp(1/4*x)

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maxima [C]  time = 0.63, size = 80, normalized size = 2.76 \begin {gather*} -\frac {7}{2} \, x^{3} + 5 \, x^{2} - \frac {7}{2} \, {\left (x^{2} - 8 \, x + 32\right )} e^{\left (\frac {1}{4} \, x\right )} - 23 \, {\left (x - 4\right )} e^{\left (\frac {1}{4} \, x\right )} - 14 \, x - \frac {14}{x} + \frac {20}{x^{2}} + 5 \, {\rm Ei}\left (\frac {1}{4} \, x\right ) + 6 \, e^{\left (\frac {1}{4} \, x\right )} - \frac {47}{8} \, \Gamma \left (-1, -\frac {1}{4} \, x\right ) - \frac {33}{16} \, \Gamma \left (-2, -\frac {1}{4} \, x\right ) - \frac {15}{16} \, \Gamma \left (-3, -\frac {1}{4} \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-7*x^6-46*x^5+12*x^4+40*x^3-188*x^2+264*x-480)*exp(1/4*x)-84*x^6+80*x^5-112*x^4+112*x^2-320*x)
/x^4,x, algorithm="maxima")

[Out]

-7/2*x^3 + 5*x^2 - 7/2*(x^2 - 8*x + 32)*e^(1/4*x) - 23*(x - 4)*e^(1/4*x) - 14*x - 14/x + 20/x^2 + 5*Ei(1/4*x)
+ 6*e^(1/4*x) - 47/8*gamma(-1, -1/4*x) - 33/16*gamma(-2, -1/4*x) - 15/16*gamma(-3, -1/4*x)

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mupad [B]  time = 0.09, size = 69, normalized size = 2.38 \begin {gather*} \frac {20\,{\mathrm {e}}^{x/4}+x^2\,\left (20\,{\mathrm {e}}^{x/4}-14\right )-x\,\left (14\,{\mathrm {e}}^{x/4}-20\right )}{x^3}-x^2\,\left (\frac {7\,{\mathrm {e}}^{x/4}}{2}-5\right )-14\,{\mathrm {e}}^{x/4}+x\,\left (5\,{\mathrm {e}}^{x/4}-14\right )-\frac {7\,x^3}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(40*x - 14*x^2 + 14*x^4 - 10*x^5 + (21*x^6)/2 + (exp(x/4)*(188*x^2 - 264*x - 40*x^3 - 12*x^4 + 46*x^5 + 7
*x^6 + 480))/8)/x^4,x)

[Out]

(20*exp(x/4) + x^2*(20*exp(x/4) - 14) - x*(14*exp(x/4) - 20))/x^3 - x^2*((7*exp(x/4))/2 - 5) - 14*exp(x/4) + x
*(5*exp(x/4) - 14) - (7*x^3)/2

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sympy [B]  time = 0.15, size = 60, normalized size = 2.07 \begin {gather*} - \frac {7 x^{3}}{2} + 5 x^{2} - 14 x - \frac {28 x - 40}{2 x^{2}} + \frac {\left (- 7 x^{5} + 10 x^{4} - 28 x^{3} + 40 x^{2} - 28 x + 40\right ) e^{\frac {x}{4}}}{2 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/8*((-7*x**6-46*x**5+12*x**4+40*x**3-188*x**2+264*x-480)*exp(1/4*x)-84*x**6+80*x**5-112*x**4+112*x*
*2-320*x)/x**4,x)

[Out]

-7*x**3/2 + 5*x**2 - 14*x - (28*x - 40)/(2*x**2) + (-7*x**5 + 10*x**4 - 28*x**3 + 40*x**2 - 28*x + 40)*exp(x/4
)/(2*x**3)

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