3.8.12 \(\int (2-7 e^{x^4})^5 x^3 \, dx\) [712]

Optimal. Leaf size=55 \[ -140 e^{x^4}+490 e^{2 x^4}-\frac {3430 e^{3 x^4}}{3}+\frac {12005 e^{4 x^4}}{8}-\frac {16807 e^{5 x^4}}{20}+8 x^4 \]

[Out]

-140*exp(x^4)+490*exp(2*x^4)-3430/3*exp(3*x^4)+12005/8*exp(4*x^4)-16807/20*exp(5*x^4)+8*x^4

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Rubi [A]
time = 0.06, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6847, 2320, 45} \begin {gather*} 8 x^4-140 e^{x^4}+490 e^{2 x^4}-\frac {3430 e^{3 x^4}}{3}+\frac {12005 e^{4 x^4}}{8}-\frac {16807 e^{5 x^4}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 7*E^x^4)^5*x^3,x]

[Out]

-140*E^x^4 + 490*E^(2*x^4) - (3430*E^(3*x^4))/3 + (12005*E^(4*x^4))/8 - (16807*E^(5*x^4))/20 + 8*x^4

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 6847

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rubi steps

\begin {align*} \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx &=\frac {1}{4} \text {Subst}\left (\int \left (2-7 e^x\right )^5 \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \frac {(2-7 x)^5}{x} \, dx,x,e^{x^4}\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (-560+\frac {32}{x}+3920 x-13720 x^2+24010 x^3-16807 x^4\right ) \, dx,x,e^{x^4}\right )\\ &=-140 e^{x^4}+490 e^{2 x^4}-\frac {3430 e^{3 x^4}}{3}+\frac {12005 e^{4 x^4}}{8}-\frac {16807 e^{5 x^4}}{20}+8 x^4\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 54, normalized size = 0.98 \begin {gather*} -\frac {7}{120} e^{x^4} \left (2400-8400 e^{x^4}+19600 e^{2 x^4}-25725 e^{3 x^4}+14406 e^{4 x^4}\right )+8 \log \left (e^{x^4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 7*E^x^4)^5*x^3,x]

[Out]

(-7*E^x^4*(2400 - 8400*E^x^4 + 19600*E^(2*x^4) - 25725*E^(3*x^4) + 14406*E^(4*x^4)))/120 + 8*Log[E^x^4]

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Maple [A]
time = 0.03, size = 47, normalized size = 0.85

method	result	size
norman	\(-140 \,{\mathrm e}^{x^{4}}+490 \,{\mathrm e}^{2 x^{4}}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+8 x^{4}\)	\(45\)
risch	\(-140 \,{\mathrm e}^{x^{4}}+490 \,{\mathrm e}^{2 x^{4}}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+8 x^{4}\)	\(45\)
derivativedivides	\(-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+490 \,{\mathrm e}^{2 x^{4}}-140 \,{\mathrm e}^{x^{4}}+8 \ln \left ({\mathrm e}^{x^{4}}\right )\)	\(47\)
default	\(-\frac {16807 \,{\mathrm e}^{5 x^{4}}}{20}+\frac {12005 \,{\mathrm e}^{4 x^{4}}}{8}-\frac {3430 \,{\mathrm e}^{3 x^{4}}}{3}+490 \,{\mathrm e}^{2 x^{4}}-140 \,{\mathrm e}^{x^{4}}+8 \ln \left ({\mathrm e}^{x^{4}}\right )\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2-7*exp(x^4))^5*x^3,x,method=_RETURNVERBOSE)

[Out]

-16807/20*exp(x^4)^5+12005/8*exp(x^4)^4-3430/3*exp(x^4)^3+490*exp(x^4)^2-140*exp(x^4)+8*ln(exp(x^4))

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Maxima [A]
time = 0.28, size = 44, normalized size = 0.80 \begin {gather*} 8 \, x^{4} - \frac {16807}{20} \, e^{\left (5 \, x^{4}\right )} + \frac {12005}{8} \, e^{\left (4 \, x^{4}\right )} - \frac {3430}{3} \, e^{\left (3 \, x^{4}\right )} + 490 \, e^{\left (2 \, x^{4}\right )} - 140 \, e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-7*exp(x^4))^5*x^3,x, algorithm="maxima")

[Out]

8*x^4 - 16807/20*e^(5*x^4) + 12005/8*e^(4*x^4) - 3430/3*e^(3*x^4) + 490*e^(2*x^4) - 140*e^(x^4)

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Fricas [A]
time = 0.39, size = 44, normalized size = 0.80 \begin {gather*} 8 \, x^{4} - \frac {16807}{20} \, e^{\left (5 \, x^{4}\right )} + \frac {12005}{8} \, e^{\left (4 \, x^{4}\right )} - \frac {3430}{3} \, e^{\left (3 \, x^{4}\right )} + 490 \, e^{\left (2 \, x^{4}\right )} - 140 \, e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-7*exp(x^4))^5*x^3,x, algorithm="fricas")

[Out]

8*x^4 - 16807/20*e^(5*x^4) + 12005/8*e^(4*x^4) - 3430/3*e^(3*x^4) + 490*e^(2*x^4) - 140*e^(x^4)

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Sympy [A]
time = 0.05, size = 49, normalized size = 0.89 \begin {gather*} 8 x^{4} - \frac {16807 e^{5 x^{4}}}{20} + \frac {12005 e^{4 x^{4}}}{8} - \frac {3430 e^{3 x^{4}}}{3} + 490 e^{2 x^{4}} - 140 e^{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-7*exp(x**4))**5*x**3,x)

[Out]

8*x**4 - 16807*exp(5*x**4)/20 + 12005*exp(4*x**4)/8 - 3430*exp(3*x**4)/3 + 490*exp(2*x**4) - 140*exp(x**4)

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Giac [A]
time = 4.68, size = 44, normalized size = 0.80 \begin {gather*} 8 \, x^{4} - \frac {16807}{20} \, e^{\left (5 \, x^{4}\right )} + \frac {12005}{8} \, e^{\left (4 \, x^{4}\right )} - \frac {3430}{3} \, e^{\left (3 \, x^{4}\right )} + 490 \, e^{\left (2 \, x^{4}\right )} - 140 \, e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2-7*exp(x^4))^5*x^3,x, algorithm="giac")

[Out]

8*x^4 - 16807/20*e^(5*x^4) + 12005/8*e^(4*x^4) - 3430/3*e^(3*x^4) + 490*e^(2*x^4) - 140*e^(x^4)

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Mupad [B]
time = 3.57, size = 44, normalized size = 0.80 \begin {gather*} 490\,{\mathrm {e}}^{2\,x^4}-140\,{\mathrm {e}}^{x^4}-\frac {3430\,{\mathrm {e}}^{3\,x^4}}{3}+\frac {12005\,{\mathrm {e}}^{4\,x^4}}{8}-\frac {16807\,{\mathrm {e}}^{5\,x^4}}{20}+8\,x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^3*(7*exp(x^4) - 2)^5,x)

[Out]

490*exp(2*x^4) - 140*exp(x^4) - (3430*exp(3*x^4))/3 + (12005*exp(4*x^4))/8 - (16807*exp(5*x^4))/20 + 8*x^4

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