3.94.51 \(\int (-4 e^{x^4} x^3+e (2+50 x)) \, dx\)

Optimal. Leaf size=20 \[ -e^{x^4}+e \left (5+2 x+25 x^2\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2209} \begin {gather*} \frac {1}{25} e (25 x+1)^2-e^{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-4*E^x^4*x^3 + E*(2 + 50*x),x]

[Out]

-E^x^4 + (E*(1 + 25*x)^2)/25

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} e (1+25 x)^2-4 \int e^{x^4} x^3 \, dx\\ &=-e^{x^4}+\frac {1}{25} e (1+25 x)^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 23, normalized size = 1.15 \begin {gather*} 2 \left (-\frac {e^{x^4}}{2}+e x+\frac {25 e x^2}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-4*E^x^4*x^3 + E*(2 + 50*x),x]

[Out]

2*(-1/2*E^x^4 + E*x + (25*E*x^2)/2)

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fricas [A]  time = 0.53, size = 19, normalized size = 0.95 \begin {gather*} {\left (25 \, x^{2} + 2 \, x\right )} e - e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x^3*exp(x^4)+(50*x+2)*exp(1),x, algorithm="fricas")

[Out]

(25*x^2 + 2*x)*e - e^(x^4)

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giac [A]  time = 0.14, size = 19, normalized size = 0.95 \begin {gather*} {\left (25 \, x^{2} + 2 \, x\right )} e - e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x^3*exp(x^4)+(50*x+2)*exp(1),x, algorithm="giac")

[Out]

(25*x^2 + 2*x)*e - e^(x^4)

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maple [A]  time = 0.04, size = 20, normalized size = 1.00

method	result	size
default	\({\mathrm e} \left (25 x^{2}+2 x \right )-{\mathrm e}^{x^{4}}\)	\(20\)
norman	\(2 x \,{\mathrm e}+25 x^{2} {\mathrm e}-{\mathrm e}^{x^{4}}\)	\(20\)
risch	\(2 x \,{\mathrm e}+25 x^{2} {\mathrm e}-{\mathrm e}^{x^{4}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-4*x^3*exp(x^4)+(50*x+2)*exp(1),x,method=_RETURNVERBOSE)

[Out]

exp(1)*(25*x^2+2*x)-exp(x^4)

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maxima [A]  time = 0.35, size = 19, normalized size = 0.95 \begin {gather*} {\left (25 \, x^{2} + 2 \, x\right )} e - e^{\left (x^{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x^3*exp(x^4)+(50*x+2)*exp(1),x, algorithm="maxima")

[Out]

(25*x^2 + 2*x)*e - e^(x^4)

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mupad [B]  time = 7.19, size = 19, normalized size = 0.95 \begin {gather*} 2\,x\,\mathrm {e}-{\mathrm {e}}^{x^4}+25\,x^2\,\mathrm {e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(1)*(50*x + 2) - 4*x^3*exp(x^4),x)

[Out]

2*x*exp(1) - exp(x^4) + 25*x^2*exp(1)

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sympy [A]  time = 0.09, size = 19, normalized size = 0.95 \begin {gather*} 25 e x^{2} + 2 e x - e^{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-4*x**3*exp(x**4)+(50*x+2)*exp(1),x)

[Out]

25*E*x**2 + 2*E*x - exp(x**4)

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