3.19.63 \(\int \frac {1}{40} (200+5 x^4-2 x^3 \log (4)) \, dx\)

Optimal. Leaf size=25 \[ 5 x-\frac {1}{80} x^3 \left (-x^2+x (-x+\log (4))\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 0.80, 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 = {12} \begin {gather*} \frac {x^5}{40}-\frac {1}{80} x^4 \log (4)+5 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(200 + 5*x^4 - 2*x^3*Log[4])/40,x]

[Out]

5*x + x^5/40 - (x^4*Log[4])/80

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}{40} \int \left (200+5 x^4-2 x^3 \log (4)\right ) \, dx\\ &=5 x+\frac {x^5}{40}-\frac {1}{80} x^4 \log (4)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} 5 x+\frac {x^5}{40}-\frac {1}{40} x^4 \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(200 + 5*x^4 - 2*x^3*Log[4])/40,x]

[Out]

5*x + x^5/40 - (x^4*Log[2])/40

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fricas [A]  time = 0.79, size = 16, normalized size = 0.64 \begin {gather*} \frac {1}{40} \, x^{5} - \frac {1}{40} \, x^{4} \log \relax (2) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/10*x^3*log(2)+1/8*x^4+5,x, algorithm="fricas")

[Out]

1/40*x^5 - 1/40*x^4*log(2) + 5*x

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giac [A]  time = 0.16, size = 16, normalized size = 0.64 \begin {gather*} \frac {1}{40} \, x^{5} - \frac {1}{40} \, x^{4} \log \relax (2) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/10*x^3*log(2)+1/8*x^4+5,x, algorithm="giac")

[Out]

1/40*x^5 - 1/40*x^4*log(2) + 5*x

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maple [A]  time = 0.02, size = 17, normalized size = 0.68

method	result	size
gosper	\(-\frac {x \left (x^{3} \ln \relax (2)-x^{4}-200\right )}{40}\)	\(17\)
default	\(-\frac {x^{4} \ln \relax (2)}{40}+\frac {x^{5}}{40}+5 x\)	\(17\)
norman	\(-\frac {x^{4} \ln \relax (2)}{40}+\frac {x^{5}}{40}+5 x\)	\(17\)
risch	\(-\frac {x^{4} \ln \relax (2)}{40}+\frac {x^{5}}{40}+5 x\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/10*x^3*ln(2)+1/8*x^4+5,x,method=_RETURNVERBOSE)

[Out]

-1/40*x*(x^3*ln(2)-x^4-200)

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maxima [A]  time = 0.42, size = 16, normalized size = 0.64 \begin {gather*} \frac {1}{40} \, x^{5} - \frac {1}{40} \, x^{4} \log \relax (2) + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/10*x^3*log(2)+1/8*x^4+5,x, algorithm="maxima")

[Out]

1/40*x^5 - 1/40*x^4*log(2) + 5*x

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mupad [B]  time = 0.03, size = 15, normalized size = 0.60 \begin {gather*} \frac {x\,\left (x^4-\ln \relax (2)\,x^3+200\right )}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/8 - (x^3*log(2))/10 + 5,x)

[Out]

(x*(x^4 - x^3*log(2) + 200))/40

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sympy [A]  time = 0.06, size = 15, normalized size = 0.60 \begin {gather*} \frac {x^{5}}{40} - \frac {x^{4} \log {\relax (2 )}}{40} + 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-1/10*x**3*ln(2)+1/8*x**4+5,x)

[Out]

x**5/40 - x**4*log(2)/40 + 5*x

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