3.81.7 \(\int \frac {1}{9} (4 e^2 x^3+400 e x^7+7500 x^{11}) \, dx\)

Optimal. Leaf size=17 \[ \frac {1}{9} \left (e x^2+25 x^6\right )^2 \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.53, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12} \begin {gather*} \frac {625 x^{12}}{9}+\frac {50 e x^8}{9}+\frac {e^2 x^4}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^2*x^3 + 400*E*x^7 + 7500*x^11)/9,x]

[Out]

(E^2*x^4)/9 + (50*E*x^8)/9 + (625*x^12)/9

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{9} \int \left (4 e^2 x^3+400 e x^7+7500 x^{11}\right ) \, dx\\ &=\frac {e^2 x^4}{9}+\frac {50 e x^8}{9}+\frac {625 x^{12}}{9}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.76 \begin {gather*} \frac {4}{9} \left (\frac {e^2 x^4}{4}+\frac {25 e x^8}{2}+\frac {625 x^{12}}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^2*x^3 + 400*E*x^7 + 7500*x^11)/9,x]

[Out]

(4*((E^2*x^4)/4 + (25*E*x^8)/2 + (625*x^12)/4))/9

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fricas [A]  time = 0.59, size = 20, normalized size = 1.18 \begin {gather*} \frac {625}{9} \, x^{12} + \frac {50}{9} \, x^{8} e + \frac {1}{9} \, x^{4} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/9*x*exp(log(x)+1)^2+400/9*x^6*exp(log(x)+1)+2500/3*x^11,x, algorithm="fricas")

[Out]

625/9*x^12 + 50/9*x^8*e + 1/9*x^4*e^2

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giac [A]  time = 0.17, size = 20, normalized size = 1.18 \begin {gather*} \frac {625}{9} \, x^{12} + \frac {50}{9} \, x^{8} e + \frac {1}{9} \, x^{4} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/9*x*exp(log(x)+1)^2+400/9*x^6*exp(log(x)+1)+2500/3*x^11,x, algorithm="giac")

[Out]

625/9*x^12 + 50/9*x^8*e + 1/9*x^4*e^2

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maple [A]  time = 0.03, size = 21, normalized size = 1.24

method	result	size
default	\(\frac {625 x^{12}}{9}+\frac {x^{4} {\mathrm e}^{2}}{9}+\frac {50 x^{8} {\mathrm e}}{9}\)	\(21\)
risch	\(\frac {625 x^{12}}{9}+\frac {x^{4} {\mathrm e}^{2}}{9}+\frac {50 x^{8} {\mathrm e}}{9}\)	\(21\)
norman	\(\frac {625 x^{12}}{9}+\frac {x^{4} {\mathrm e}^{2}}{9}+\frac {50 x^{8} {\mathrm e}}{9}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4/9*x*exp(ln(x)+1)^2+400/9*x^6*exp(ln(x)+1)+2500/3*x^11,x,method=_RETURNVERBOSE)

[Out]

625/9*x^12+1/9*x^4*exp(2)+50/9*x^8*exp(1)

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maxima [A]  time = 0.36, size = 20, normalized size = 1.18 \begin {gather*} \frac {625}{9} \, x^{12} + \frac {50}{9} \, x^{8} e + \frac {1}{9} \, x^{4} e^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/9*x*exp(log(x)+1)^2+400/9*x^6*exp(log(x)+1)+2500/3*x^11,x, algorithm="maxima")

[Out]

625/9*x^12 + 50/9*x^8*e + 1/9*x^4*e^2

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mupad [B]  time = 5.62, size = 15, normalized size = 0.88 \begin {gather*} \frac {x^4\,{\left (25\,x^4+\mathrm {e}\right )}^2}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2500*x^11)/3 + (4*x*exp(2*log(x) + 2))/9 + (400*x^6*exp(log(x) + 1))/9,x)

[Out]

(x^4*(exp(1) + 25*x^4)^2)/9

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sympy [A]  time = 0.06, size = 24, normalized size = 1.41 \begin {gather*} \frac {625 x^{12}}{9} + \frac {50 e x^{8}}{9} + \frac {x^{4} e^{2}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4/9*x*exp(ln(x)+1)**2+400/9*x**6*exp(ln(x)+1)+2500/3*x**11,x)

[Out]

625*x**12/9 + 50*E*x**8/9 + x**4*exp(2)/9

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