3.9.69 \(\int (-1+486 x^5+e^{7 x} (2430 x^4+4374 x^5+1134 x^6)+e^{14 x} (2916 x^3+12636 x^4+7290 x^5+1134 x^6)) \, dx\)

Optimal. Leaf size=26 \[ -1-x+81 x^4 \left (-e^{7 x} (-3-x)+x\right )^2 \]

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Rubi [B]  time = 0.52, antiderivative size = 59, normalized size of antiderivative = 2.27, number of steps used = 46, number of rules used = 4, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1594, 2196, 2176, 2194} \begin {gather*} 162 e^{7 x} x^6+81 e^{14 x} x^6+81 x^6+486 e^{7 x} x^5+486 e^{14 x} x^5+729 e^{14 x} x^4-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + 486*x^5 + E^(7*x)*(2430*x^4 + 4374*x^5 + 1134*x^6) + E^(14*x)*(2916*x^3 + 12636*x^4 + 7290*x^5 + 1134
*x^6),x]

[Out]

-x + 729*E^(14*x)*x^4 + 486*E^(7*x)*x^5 + 486*E^(14*x)*x^5 + 81*x^6 + 162*E^(7*x)*x^6 + 81*E^(14*x)*x^6

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 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} &=-x+81 x^6+\int e^{7 x} \left (2430 x^4+4374 x^5+1134 x^6\right ) \, dx+\int e^{14 x} \left (2916 x^3+12636 x^4+7290 x^5+1134 x^6\right ) \, dx\\ &=-x+81 x^6+\int e^{7 x} x^4 \left (2430+4374 x+1134 x^2\right ) \, dx+\int \left (2916 e^{14 x} x^3+12636 e^{14 x} x^4+7290 e^{14 x} x^5+1134 e^{14 x} x^6\right ) \, dx\\ &=-x+81 x^6+1134 \int e^{14 x} x^6 \, dx+2916 \int e^{14 x} x^3 \, dx+7290 \int e^{14 x} x^5 \, dx+12636 \int e^{14 x} x^4 \, dx+\int \left (2430 e^{7 x} x^4+4374 e^{7 x} x^5+1134 e^{7 x} x^6\right ) \, dx\\ &=-x+\frac {1458}{7} e^{14 x} x^3+\frac {6318}{7} e^{14 x} x^4+\frac {3645}{7} e^{14 x} x^5+81 x^6+81 e^{14 x} x^6-486 \int e^{14 x} x^5 \, dx-\frac {4374}{7} \int e^{14 x} x^2 \, dx+1134 \int e^{7 x} x^6 \, dx+2430 \int e^{7 x} x^4 \, dx-\frac {18225}{7} \int e^{14 x} x^4 \, dx-\frac {25272}{7} \int e^{14 x} x^3 \, dx+4374 \int e^{7 x} x^5 \, dx\\ &=-x-\frac {2187}{49} e^{14 x} x^2-\frac {2430}{49} e^{14 x} x^3+\frac {2430}{7} e^{7 x} x^4+\frac {70227}{98} e^{14 x} x^4+\frac {4374}{7} e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6+\frac {4374}{49} \int e^{14 x} x \, dx+\frac {1215}{7} \int e^{14 x} x^4 \, dx+\frac {36450}{49} \int e^{14 x} x^3 \, dx+\frac {37908}{49} \int e^{14 x} x^2 \, dx-972 \int e^{7 x} x^5 \, dx-\frac {9720}{7} \int e^{7 x} x^3 \, dx-\frac {21870}{7} \int e^{7 x} x^4 \, dx\\ &=-x+\frac {2187}{343} e^{14 x} x+\frac {3645}{343} e^{14 x} x^2-\frac {9720}{49} e^{7 x} x^3+\frac {1215}{343} e^{14 x} x^3-\frac {4860}{49} e^{7 x} x^4+729 e^{14 x} x^4+486 e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6-\frac {2187}{343} \int e^{14 x} \, dx-\frac {2430}{49} \int e^{14 x} x^3 \, dx-\frac {37908}{343} \int e^{14 x} x \, dx-\frac {54675}{343} \int e^{14 x} x^2 \, dx+\frac {29160}{49} \int e^{7 x} x^2 \, dx+\frac {4860}{7} \int e^{7 x} x^4 \, dx+\frac {87480}{49} \int e^{7 x} x^3 \, dx\\ &=-\frac {2187 e^{14 x}}{4802}-x-\frac {3645 e^{14 x} x}{2401}+\frac {29160}{343} e^{7 x} x^2-\frac {3645 e^{14 x} x^2}{4802}+\frac {19440}{343} e^{7 x} x^3+729 e^{14 x} x^4+486 e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6+\frac {18954 \int e^{14 x} \, dx}{2401}+\frac {3645}{343} \int e^{14 x} x^2 \, dx+\frac {54675 \int e^{14 x} x \, dx}{2401}-\frac {58320}{343} \int e^{7 x} x \, dx-\frac {19440}{49} \int e^{7 x} x^3 \, dx-\frac {262440}{343} \int e^{7 x} x^2 \, dx\\ &=\frac {3645 e^{14 x}}{33614}-x-\frac {58320 e^{7 x} x}{2401}+\frac {3645 e^{14 x} x}{33614}-\frac {58320 e^{7 x} x^2}{2401}+729 e^{14 x} x^4+486 e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6-\frac {3645 \int e^{14 x} x \, dx}{2401}-\frac {54675 \int e^{14 x} \, dx}{33614}+\frac {58320 \int e^{7 x} \, dx}{2401}+\frac {58320}{343} \int e^{7 x} x^2 \, dx+\frac {524880 \int e^{7 x} x \, dx}{2401}\\ &=\frac {58320 e^{7 x}}{16807}-\frac {3645 e^{14 x}}{470596}-x+\frac {116640 e^{7 x} x}{16807}+729 e^{14 x} x^4+486 e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6+\frac {3645 \int e^{14 x} \, dx}{33614}-\frac {524880 \int e^{7 x} \, dx}{16807}-\frac {116640 \int e^{7 x} x \, dx}{2401}\\ &=-\frac {116640 e^{7 x}}{117649}-x+729 e^{14 x} x^4+486 e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6+\frac {116640 \int e^{7 x} \, dx}{16807}\\ &=-x+729 e^{14 x} x^4+486 e^{7 x} x^5+486 e^{14 x} x^5+81 x^6+162 e^{7 x} x^6+81 e^{14 x} x^6\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 52, normalized size = 2.00 \begin {gather*} -x+81 x^6+162 e^{14 x} \left (\frac {9 x^4}{2}+3 x^5+\frac {x^6}{2}\right )+162 e^{7 x} \left (3 x^5+x^6\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + 486*x^5 + E^(7*x)*(2430*x^4 + 4374*x^5 + 1134*x^6) + E^(14*x)*(2916*x^3 + 12636*x^4 + 7290*x^5
+ 1134*x^6),x]

[Out]

-x + 81*x^6 + 162*E^(14*x)*((9*x^4)/2 + 3*x^5 + x^6/2) + 162*E^(7*x)*(3*x^5 + x^6)

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fricas [A]  time = 0.54, size = 44, normalized size = 1.69 \begin {gather*} 81 \, x^{6} + 81 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4}\right )} e^{\left (14 \, x\right )} + 162 \, {\left (x^{6} + 3 \, x^{5}\right )} e^{\left (7 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1134*x^6+7290*x^5+12636*x^4+2916*x^3)*exp(7*x)^2+(1134*x^6+4374*x^5+2430*x^4)*exp(7*x)+486*x^5-1,x,
algorithm="fricas")

[Out]

81*x^6 + 81*(x^6 + 6*x^5 + 9*x^4)*e^(14*x) + 162*(x^6 + 3*x^5)*e^(7*x) - x

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giac [A]  time = 0.35, size = 44, normalized size = 1.69 \begin {gather*} 81 \, x^{6} + 81 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4}\right )} e^{\left (14 \, x\right )} + 162 \, {\left (x^{6} + 3 \, x^{5}\right )} e^{\left (7 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1134*x^6+7290*x^5+12636*x^4+2916*x^3)*exp(7*x)^2+(1134*x^6+4374*x^5+2430*x^4)*exp(7*x)+486*x^5-1,x,
algorithm="giac")

[Out]

81*x^6 + 81*(x^6 + 6*x^5 + 9*x^4)*e^(14*x) + 162*(x^6 + 3*x^5)*e^(7*x) - x

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maple [A]  time = 0.05, size = 47, normalized size = 1.81

method	result	size
risch	\(\left (81 x^{6}+486 x^{5}+729 x^{4}\right ) {\mathrm e}^{14 x}+\left (162 x^{6}+486 x^{5}\right ) {\mathrm e}^{7 x}+81 x^{6}-x\)	\(47\)
derivativedivides	\(-x +81 x^{6}+162 \,{\mathrm e}^{7 x} x^{6}+486 \,{\mathrm e}^{7 x} x^{5}+81 \,{\mathrm e}^{14 x} x^{6}+486 \,{\mathrm e}^{14 x} x^{5}+729 \,{\mathrm e}^{14 x} x^{4}\)	\(61\)
default	\(-x +81 x^{6}+162 \,{\mathrm e}^{7 x} x^{6}+486 \,{\mathrm e}^{7 x} x^{5}+81 \,{\mathrm e}^{14 x} x^{6}+486 \,{\mathrm e}^{14 x} x^{5}+729 \,{\mathrm e}^{14 x} x^{4}\)	\(61\)
norman	\(-x +81 x^{6}+162 \,{\mathrm e}^{7 x} x^{6}+486 \,{\mathrm e}^{7 x} x^{5}+81 \,{\mathrm e}^{14 x} x^{6}+486 \,{\mathrm e}^{14 x} x^{5}+729 \,{\mathrm e}^{14 x} x^{4}\)	\(61\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1134*x^6+7290*x^5+12636*x^4+2916*x^3)*exp(7*x)^2+(1134*x^6+4374*x^5+2430*x^4)*exp(7*x)+486*x^5-1,x,method
=_RETURNVERBOSE)

[Out]

(81*x^6+486*x^5+729*x^4)*exp(14*x)+(162*x^6+486*x^5)*exp(7*x)+81*x^6-x

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maxima [A]  time = 0.37, size = 44, normalized size = 1.69 \begin {gather*} 81 \, x^{6} + 81 \, {\left (x^{6} + 6 \, x^{5} + 9 \, x^{4}\right )} e^{\left (14 \, x\right )} + 162 \, {\left (x^{6} + 3 \, x^{5}\right )} e^{\left (7 \, x\right )} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1134*x^6+7290*x^5+12636*x^4+2916*x^3)*exp(7*x)^2+(1134*x^6+4374*x^5+2430*x^4)*exp(7*x)+486*x^5-1,x,
algorithm="maxima")

[Out]

81*x^6 + 81*(x^6 + 6*x^5 + 9*x^4)*e^(14*x) + 162*(x^6 + 3*x^5)*e^(7*x) - x

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mupad [B]  time = 0.63, size = 54, normalized size = 2.08 \begin {gather*} 486\,x^5\,{\mathrm {e}}^{7\,x}-x+162\,x^6\,{\mathrm {e}}^{7\,x}+729\,x^4\,{\mathrm {e}}^{14\,x}+486\,x^5\,{\mathrm {e}}^{14\,x}+81\,x^6\,{\mathrm {e}}^{14\,x}+81\,x^6 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(7*x)*(2430*x^4 + 4374*x^5 + 1134*x^6) + exp(14*x)*(2916*x^3 + 12636*x^4 + 7290*x^5 + 1134*x^6) + 486*x
^5 - 1,x)

[Out]

486*x^5*exp(7*x) - x + 162*x^6*exp(7*x) + 729*x^4*exp(14*x) + 486*x^5*exp(14*x) + 81*x^6*exp(14*x) + 81*x^6

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sympy [B]  time = 0.13, size = 41, normalized size = 1.58 \begin {gather*} 81 x^{6} - x + \left (162 x^{6} + 486 x^{5}\right ) e^{7 x} + \left (81 x^{6} + 486 x^{5} + 729 x^{4}\right ) e^{14 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1134*x**6+7290*x**5+12636*x**4+2916*x**3)*exp(7*x)**2+(1134*x**6+4374*x**5+2430*x**4)*exp(7*x)+486*
x**5-1,x)

[Out]

81*x**6 - x + (162*x**6 + 486*x**5)*exp(7*x) + (81*x**6 + 486*x**5 + 729*x**4)*exp(14*x)

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