3.55.22 \(\int (300 x^3+e^x (100 x^3+25 x^4)) \, dx\)

Optimal. Leaf size=10 \[ 25 \left (3+e^x\right ) x^4 \]

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Rubi [A]  time = 0.11, antiderivative size = 14, normalized size of antiderivative = 1.40, number of steps used = 13, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {1593, 2196, 2176, 2194} \begin {gather*} 25 e^x x^4+75 x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[300*x^3 + E^x*(100*x^3 + 25*x^4),x]

[Out]

75*x^4 + 25*E^x*x^4

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - 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} &=75 x^4+\int e^x \left (100 x^3+25 x^4\right ) \, dx\\ &=75 x^4+\int e^x x^3 (100+25 x) \, dx\\ &=75 x^4+\int \left (100 e^x x^3+25 e^x x^4\right ) \, dx\\ &=75 x^4+25 \int e^x x^4 \, dx+100 \int e^x x^3 \, dx\\ &=100 e^x x^3+75 x^4+25 e^x x^4-100 \int e^x x^3 \, dx-300 \int e^x x^2 \, dx\\ &=-300 e^x x^2+75 x^4+25 e^x x^4+300 \int e^x x^2 \, dx+600 \int e^x x \, dx\\ &=600 e^x x+75 x^4+25 e^x x^4-600 \int e^x \, dx-600 \int e^x x \, dx\\ &=-600 e^x+75 x^4+25 e^x x^4+600 \int e^x \, dx\\ &=75 x^4+25 e^x x^4\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 10, normalized size = 1.00 \begin {gather*} 25 \left (3+e^x\right ) x^4 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[300*x^3 + E^x*(100*x^3 + 25*x^4),x]

[Out]

25*(3 + E^x)*x^4

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fricas [A]  time = 0.75, size = 13, normalized size = 1.30 \begin {gather*} 25 \, x^{4} e^{x} + 75 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^4+100*x^3)*exp(x)+300*x^3,x, algorithm="fricas")

[Out]

25*x^4*e^x + 75*x^4

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giac [A]  time = 0.20, size = 13, normalized size = 1.30 \begin {gather*} 25 \, x^{4} e^{x} + 75 \, x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^4+100*x^3)*exp(x)+300*x^3,x, algorithm="giac")

[Out]

25*x^4*e^x + 75*x^4

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maple [A]  time = 0.03, size = 14, normalized size = 1.40

method	result	size
default	\(25 \,{\mathrm e}^{x} x^{4}+75 x^{4}\)	\(14\)
norman	\(25 \,{\mathrm e}^{x} x^{4}+75 x^{4}\)	\(14\)
risch	\(25 \,{\mathrm e}^{x} x^{4}+75 x^{4}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*x^4+100*x^3)*exp(x)+300*x^3,x,method=_RETURNVERBOSE)

[Out]

25*exp(x)*x^4+75*x^4

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maxima [B]  time = 0.35, size = 45, normalized size = 4.50 \begin {gather*} 75 \, x^{4} + 25 \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + 100 \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^4+100*x^3)*exp(x)+300*x^3,x, algorithm="maxima")

[Out]

75*x^4 + 25*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x + 100*(x^3 - 3*x^2 + 6*x - 6)*e^x

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mupad [B]  time = 3.45, size = 9, normalized size = 0.90 \begin {gather*} 25\,x^4\,\left ({\mathrm {e}}^x+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x)*(100*x^3 + 25*x^4) + 300*x^3,x)

[Out]

25*x^4*(exp(x) + 3)

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sympy [A]  time = 0.08, size = 12, normalized size = 1.20 \begin {gather*} 25 x^{4} e^{x} + 75 x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x**4+100*x**3)*exp(x)+300*x**3,x)

[Out]

25*x**4*exp(x) + 75*x**4

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