3.712 \(\int (2-7 e^{x^4})^5 x^3 \, dx\)

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

[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

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Rubi [A]  time = 0.0857752, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6715, 2282, 43} \[ 8 x^4-140 e^{x^4}+490 e^{2 x^4}-\frac{3430 e^{3 x^4}}{3}+\frac{12005 e^{4 x^4}}{8}-\frac{16807 e^{5 x^4}}{20} \]

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 6715

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]

Rule 2282

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 43

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])

Rubi steps

\begin{align*} \int \left (2-7 e^{x^4}\right )^5 x^3 \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \left (2-7 e^x\right )^5 \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{(2-7 x)^5}{x} \, dx,x,e^{x^4}\right )\\ &=\frac{1}{4} \operatorname{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*}

Mathematica [A]  time = 0.0289462, size = 55, normalized size = 1. \[ 8 x^4-140 e^{x^4}+490 e^{2 x^4}-\frac{3430 e^{3 x^4}}{3}+\frac{12005 e^{4 x^4}}{8}-\frac{16807 e^{5 x^4}}{20} \]

Antiderivative was successfully verified.

[In]

Integrate[(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

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Maple [A]  time = 0.021, size = 47, normalized size = 0.9 \begin{align*} -{\frac{16807\, \left ({{\rm e}^{{x}^{4}}} \right ) ^{5}}{20}}+{\frac{12005\, \left ({{\rm e}^{{x}^{4}}} \right ) ^{4}}{8}}-{\frac{3430\, \left ({{\rm e}^{{x}^{4}}} \right ) ^{3}}{3}}+490\, \left ({{\rm e}^{{x}^{4}}} \right ) ^{2}-140\,{{\rm e}^{{x}^{4}}}+8\,\ln \left ({{\rm e}^{{x}^{4}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.974356, size = 59, normalized size = 1.07 \begin{align*} 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{align*}

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 = 0.925768, size = 131, normalized size = 2.38 \begin{align*} 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{align*}

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 = 0.141958, size = 49, normalized size = 0.89 \begin{align*} 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{align*}

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 = 1.26594, size = 59, normalized size = 1.07 \begin{align*} 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{align*}

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)