∫ −1032∕5+6 5√_3 ex∕5x3+1960x4+ex∕5(−840x4−84x5)+e2x∕5(90x4+18x5) _______________________________________________________________________________________________

3.101.3 $\int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} (-840 x^4-84 x^5)+e^{2 x/5} (90 x^4+18 x^5)}{5 x^3} \, dx$

Optimal. Leaf size=28 \[ \left (-5+3 \left (-3+e^{x/5}\right )+\frac {\sqrt [5]{3}}{x^2}\right )^2 x^2 \]

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Rubi [A] time = 0.18, antiderivative size = 53, normalized size of antiderivative = 1.89, number of steps used = 20, number of rules used = 5, integrand size = 75, $\frac {\text {number of rules}}{\text {integrand size}}$ = 0.067, Rules used = {12, 14, 2196, 2176, 2194} \begin {gather*} -84 e^{x/5} x^2+9 e^{2 x/5} x^2+196 x^2+\frac {3^{2/5}}{x^2}+6 \sqrt [5]{3} e^{x/5} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(-10*3^(2/5) + 6*3^(1/5)*E^(x/5)*x^3 + 1960*x^4 + E^(x/5)*(-840*x^4 - 84*x^5) + E^((2*x)/5)*(90*x^4 + 18*x

^5))/(5*x^3),x]

[Out]

6*3^(1/5)*E^(x/5) + 3^(2/5)/x^2 + 196*x^2 - 84*E^(x/5)*x^2 + 9*E^((2*x)/5)*x^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 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 2196

Int[(F_)^((c_.)*(v_))*(u_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), u, x], x] /; FreeQ[{F, c

}, x] && PolynomialQ[u, x] && LinearQ[v, x] && !$UseGamma === True

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{5} \int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} \left (-840 x^4-84 x^5\right )+e^{2 x/5} \left (90 x^4+18 x^5\right )}{x^3} \, dx\\ &=\frac {1}{5} \int \left (18 e^{2 x/5} x (5+x)+6 e^{x/5} \left (\sqrt [5]{3}-140 x-14 x^2\right )-\frac {10 \left (3^{2/5}-196 x^4\right )}{x^3}\right ) \, dx\\ &=\frac {6}{5} \int e^{x/5} \left (\sqrt [5]{3}-140 x-14 x^2\right ) \, dx-2 \int \frac {3^{2/5}-196 x^4}{x^3} \, dx+\frac {18}{5} \int e^{2 x/5} x (5+x) \, dx\\ &=\frac {6}{5} \int \left (\sqrt [5]{3} e^{x/5}-140 e^{x/5} x-14 e^{x/5} x^2\right ) \, dx-2 \int \left (\frac {3^{2/5}}{x^3}-196 x\right ) \, dx+\frac {18}{5} \int \left (5 e^{2 x/5} x+e^{2 x/5} x^2\right ) \, dx\\ &=\frac {3^{2/5}}{x^2}+196 x^2+\frac {18}{5} \int e^{2 x/5} x^2 \, dx-\frac {84}{5} \int e^{x/5} x^2 \, dx+18 \int e^{2 x/5} x \, dx-168 \int e^{x/5} x \, dx+\frac {1}{5} \left (6 \sqrt [5]{3}\right ) \int e^{x/5} \, dx\\ &=6 \sqrt [5]{3} e^{x/5}+\frac {3^{2/5}}{x^2}-840 e^{x/5} x+45 e^{2 x/5} x+196 x^2-84 e^{x/5} x^2+9 e^{2 x/5} x^2-18 \int e^{2 x/5} x \, dx-45 \int e^{2 x/5} \, dx+168 \int e^{x/5} x \, dx+840 \int e^{x/5} \, dx\\ &=4200 e^{x/5}+6 \sqrt [5]{3} e^{x/5}-\frac {225}{2} e^{2 x/5}+\frac {3^{2/5}}{x^2}+196 x^2-84 e^{x/5} x^2+9 e^{2 x/5} x^2+45 \int e^{2 x/5} \, dx-840 \int e^{x/5} \, dx\\ &=6 \sqrt [5]{3} e^{x/5}+\frac {3^{2/5}}{x^2}+196 x^2-84 e^{x/5} x^2+9 e^{2 x/5} x^2\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.06, size = 56, normalized size = 2.00 \begin {gather*} \frac {2}{5} \left (\frac {5\ 3^{2/5}}{2 x^2}+490 x^2+\frac {45}{2} e^{2 x/5} x^2+15 e^{x/5} \left (\sqrt [5]{3}-14 x^2\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-10*3^(2/5) + 6*3^(1/5)*E^(x/5)*x^3 + 1960*x^4 + E^(x/5)*(-840*x^4 - 84*x^5) + E^((2*x)/5)*(90*x^4

+ 18*x^5))/(5*x^3),x]

[Out]

(2*((5*3^(2/5))/(2*x^2) + 490*x^2 + (45*E^((2*x)/5)*x^2)/2 + 15*E^(x/5)*(3^(1/5) - 14*x^2)))/5

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fricas [A] time = 0.90, size = 42, normalized size = 1.50 \begin {gather*} \frac {9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} + 196 \, x^{4} - 6 \, {\left (14 \, x^{4} - 3^{\frac {1}{5}} x^{2}\right )} e^{\left (\frac {1}{5} \, x\right )} + 3^{\frac {2}{5}}}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-10*3^(2/5)+6*x^3*exp(1/5*x)*3^(1/5)+(18*x^5+90*x^4)*exp(1/5*x)^2+(-84*x^5-840*x^4)*exp(1/5*x)+

1960*x^4)/x^3,x, algorithm="fricas")

[Out]

(9*x^4*e^(2/5*x) + 196*x^4 - 6*(14*x^4 - 3^(1/5)*x^2)*e^(1/5*x) + 3^(2/5))/x^2

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giac [B] time = 0.16, size = 43, normalized size = 1.54 \begin {gather*} \frac {9 \, x^{4} e^{\left (\frac {2}{5} \, x\right )} - 84 \, x^{4} e^{\left (\frac {1}{5} \, x\right )} + 196 \, x^{4} + 6 \cdot 3^{\frac {1}{5}} x^{2} e^{\left (\frac {1}{5} \, x\right )} + 3^{\frac {2}{5}}}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-10*3^(2/5)+6*x^3*exp(1/5*x)*3^(1/5)+(18*x^5+90*x^4)*exp(1/5*x)^2+(-84*x^5-840*x^4)*exp(1/5*x)+

1960*x^4)/x^3,x, algorithm="giac")

[Out]

(9*x^4*e^(2/5*x) - 84*x^4*e^(1/5*x) + 196*x^4 + 6*3^(1/5)*x^2*e^(1/5*x) + 3^(2/5))/x^2

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maple [A] time = 0.06, size = 40, normalized size = 1.43


method	result	size

risch	$196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+\frac {\left (30 \,3^{\frac {1}{5}}-420 x^{2}\right ) {\mathrm e}^{\frac {x}{5}}}{5}$	$40$
derivativedivides	$196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 x^{2} {\mathrm e}^{\frac {x}{5}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}$	$43$
default	$196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 x^{2} {\mathrm e}^{\frac {x}{5}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}$	$43$
norman	$\frac {3^{\frac {2}{5}}+196 x^{4}-84 x^{4} {\mathrm e}^{\frac {x}{5}}+9 x^{4} {\mathrm e}^{\frac {2 x}{5}}+6 x^{2} {\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}}{x^{2}}$	$46$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/5*(-10*3^(2/5)+6*x^3*exp(1/5*x)*3^(1/5)+(18*x^5+90*x^4)*exp(1/5*x)^2+(-84*x^5-840*x^4)*exp(1/5*x)+1960*x

^4)/x^3,x,method=_RETURNVERBOSE)

[Out]

196*x^2+3^(2/5)/x^2+9*x^2*exp(2/5*x)+1/5*(30*3^(1/5)-420*x^2)*exp(1/5*x)

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maxima [B] time = 0.60, size = 72, normalized size = 2.57 \begin {gather*} 196 \, x^{2} + \frac {9}{2} \, {\left (2 \, x^{2} - 10 \, x + 25\right )} e^{\left (\frac {2}{5} \, x\right )} + \frac {45}{2} \, {\left (2 \, x - 5\right )} e^{\left (\frac {2}{5} \, x\right )} - 84 \, {\left (x^{2} - 10 \, x + 50\right )} e^{\left (\frac {1}{5} \, x\right )} - 840 \, {\left (x - 5\right )} e^{\left (\frac {1}{5} \, x\right )} + 6 \cdot 3^{\frac {1}{5}} e^{\left (\frac {1}{5} \, x\right )} + \frac {3^{\frac {2}{5}}}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-10*3^(2/5)+6*x^3*exp(1/5*x)*3^(1/5)+(18*x^5+90*x^4)*exp(1/5*x)^2+(-84*x^5-840*x^4)*exp(1/5*x)+

1960*x^4)/x^3,x, algorithm="maxima")

[Out]

196*x^2 + 9/2*(2*x^2 - 10*x + 25)*e^(2/5*x) + 45/2*(2*x - 5)*e^(2/5*x) - 84*(x^2 - 10*x + 50)*e^(1/5*x) - 840*

(x - 5)*e^(1/5*x) + 6*3^(1/5)*e^(1/5*x) + 3^(2/5)/x^2

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mupad [B] time = 0.19, size = 35, normalized size = 1.25 \begin {gather*} \frac {3^{2/5}}{x^2}+x^2\,\left (9\,{\mathrm {e}}^{\frac {2\,x}{5}}-84\,{\mathrm {e}}^{x/5}+196\right )+6\,3^{1/5}\,{\mathrm {e}}^{x/5} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((exp((2*x)/5)*(90*x^4 + 18*x^5))/5 - 2*3^(2/5) - (exp(x/5)*(840*x^4 + 84*x^5))/5 + 392*x^4 + (6*3^(1/5)*x

^3*exp(x/5))/5)/x^3,x)

[Out]

3^(2/5)/x^2 + x^2*(9*exp((2*x)/5) - 84*exp(x/5) + 196) + 6*3^(1/5)*exp(x/5)

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sympy [B] time = 0.32, size = 41, normalized size = 1.46 \begin {gather*} 9 x^{2} e^{\frac {2 x}{5}} + 196 x^{2} + \left (- 84 x^{2} + 6 \sqrt [5]{3}\right ) e^{\frac {x}{5}} + \frac {3^{\frac {2}{5}}}{x^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/5*(-10*3**(2/5)+6*x**3*exp(1/5*x)*3**(1/5)+(18*x**5+90*x**4)*exp(1/5*x)**2+(-84*x**5-840*x**4)*exp

(1/5*x)+1960*x**4)/x**3,x)

[Out]

9*x**2*exp(2*x/5) + 196*x**2 + (-84*x**2 + 6*3**(1/5))*exp(x/5) + 3**(2/5)/x**2

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method	result	size

risch	\(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+\frac {\left (30 \,3^{\frac {1}{5}}-420 x^{2}\right ) {\mathrm e}^{\frac {x}{5}}}{5}\)	\(40\)
derivativedivides	\(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 x^{2} {\mathrm e}^{\frac {x}{5}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}\)	\(43\)
default	\(196 x^{2}+\frac {3^{\frac {2}{5}}}{x^{2}}-84 x^{2} {\mathrm e}^{\frac {x}{5}}+9 x^{2} {\mathrm e}^{\frac {2 x}{5}}+6 \,{\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}\)	\(43\)
norman	\(\frac {3^{\frac {2}{5}}+196 x^{4}-84 x^{4} {\mathrm e}^{\frac {x}{5}}+9 x^{4} {\mathrm e}^{\frac {2 x}{5}}+6 x^{2} {\mathrm e}^{\frac {x}{5}} 3^{\frac {1}{5}}}{x^{2}}\)	\(46\)

3.101.3 \(\int \frac {-10 3^{2/5}+6 \sqrt [5]{3} e^{x/5} x^3+1960 x^4+e^{x/5} (-840 x^4-84 x^5)+e^{2 x/5} (90 x^4+18 x^5)}{5 x^3} \, dx\)