3.36.21 \(\int \frac {16 x^3+20 x^4+6 x^5-56 x^6-64 x^7-18 x^8+40 x^9+44 x^{10}+12 x^{11}}{4+\log (4)} \, dx\)

Optimal. Leaf size=24 \[ \frac {x^2 (2+x)^2 \left (x-x^4\right )^2}{4+\log (4)} \]

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Rubi [B]  time = 0.02, antiderivative size = 98, normalized size of antiderivative = 4.08, number of steps used = 2, number of rules used = 1, integrand size = 53, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {12} \begin {gather*} \frac {x^{12}}{4+\log (4)}+\frac {4 x^{11}}{4+\log (4)}+\frac {4 x^{10}}{4+\log (4)}-\frac {2 x^9}{4+\log (4)}-\frac {8 x^8}{4+\log (4)}-\frac {8 x^7}{4+\log (4)}+\frac {x^6}{4+\log (4)}+\frac {4 x^5}{4+\log (4)}+\frac {4 x^4}{4+\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16*x^3 + 20*x^4 + 6*x^5 - 56*x^6 - 64*x^7 - 18*x^8 + 40*x^9 + 44*x^10 + 12*x^11)/(4 + Log[4]),x]

[Out]

(4*x^4)/(4 + Log[4]) + (4*x^5)/(4 + Log[4]) + x^6/(4 + Log[4]) - (8*x^7)/(4 + Log[4]) - (8*x^8)/(4 + Log[4]) -
(2*x^9)/(4 + Log[4]) + (4*x^10)/(4 + Log[4]) + (4*x^11)/(4 + Log[4]) + x^12/(4 + Log[4])

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 {\int \left (16 x^3+20 x^4+6 x^5-56 x^6-64 x^7-18 x^8+40 x^9+44 x^{10}+12 x^{11}\right ) \, dx}{4+\log (4)}\\ &=\frac {4 x^4}{4+\log (4)}+\frac {4 x^5}{4+\log (4)}+\frac {x^6}{4+\log (4)}-\frac {8 x^7}{4+\log (4)}-\frac {8 x^8}{4+\log (4)}-\frac {2 x^9}{4+\log (4)}+\frac {4 x^{10}}{4+\log (4)}+\frac {4 x^{11}}{4+\log (4)}+\frac {x^{12}}{4+\log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 58, normalized size = 2.42 \begin {gather*} \frac {2 \left (2 x^4+2 x^5+\frac {x^6}{2}-4 x^7-4 x^8-x^9+2 x^{10}+2 x^{11}+\frac {x^{12}}{2}\right )}{4+\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x^3 + 20*x^4 + 6*x^5 - 56*x^6 - 64*x^7 - 18*x^8 + 40*x^9 + 44*x^10 + 12*x^11)/(4 + Log[4]),x]

[Out]

(2*(2*x^4 + 2*x^5 + x^6/2 - 4*x^7 - 4*x^8 - x^9 + 2*x^10 + 2*x^11 + x^12/2))/(4 + Log[4])

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fricas [A]  time = 0.56, size = 50, normalized size = 2.08 \begin {gather*} \frac {x^{12} + 4 \, x^{11} + 4 \, x^{10} - 2 \, x^{9} - 8 \, x^{8} - 8 \, x^{7} + x^{6} + 4 \, x^{5} + 4 \, x^{4}}{2 \, {\left (\log \relax (2) + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^11+44*x^10+40*x^9-18*x^8-64*x^7-56*x^6+6*x^5+20*x^4+16*x^3)/(4+2*log(2)),x, algorithm="fricas"
)

[Out]

1/2*(x^12 + 4*x^11 + 4*x^10 - 2*x^9 - 8*x^8 - 8*x^7 + x^6 + 4*x^5 + 4*x^4)/(log(2) + 2)

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giac [A]  time = 0.23, size = 50, normalized size = 2.08 \begin {gather*} \frac {x^{12} + 4 \, x^{11} + 4 \, x^{10} - 2 \, x^{9} - 8 \, x^{8} - 8 \, x^{7} + x^{6} + 4 \, x^{5} + 4 \, x^{4}}{2 \, {\left (\log \relax (2) + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^11+44*x^10+40*x^9-18*x^8-64*x^7-56*x^6+6*x^5+20*x^4+16*x^3)/(4+2*log(2)),x, algorithm="giac")

[Out]

1/2*(x^12 + 4*x^11 + 4*x^10 - 2*x^9 - 8*x^8 - 8*x^7 + x^6 + 4*x^5 + 4*x^4)/(log(2) + 2)

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maple [A]  time = 0.05, size = 40, normalized size = 1.67

method	result	size
gosper	\(\frac {x^{4} \left (x^{5}+4 x^{4}+4 x^{3}-x^{2}-4 x -4\right ) \left (x^{3}-1\right )}{4+2 \ln \relax (2)}\)	\(40\)
default	\(\frac {x^{12}+4 x^{11}+4 x^{10}-2 x^{9}-8 x^{8}-8 x^{7}+x^{6}+4 x^{5}+4 x^{4}}{4+2 \ln \relax (2)}\)	\(52\)
norman	\(\frac {2 x^{4}}{\ln \relax (2)+2}+\frac {2 x^{5}}{\ln \relax (2)+2}+\frac {x^{6}}{4+2 \ln \relax (2)}-\frac {4 x^{7}}{\ln \relax (2)+2}-\frac {4 x^{8}}{\ln \relax (2)+2}-\frac {x^{9}}{\ln \relax (2)+2}+\frac {2 x^{10}}{\ln \relax (2)+2}+\frac {2 x^{11}}{\ln \relax (2)+2}+\frac {x^{12}}{4+2 \ln \relax (2)}\)	\(101\)
risch	\(\frac {x^{12}}{4+2 \ln \relax (2)}+\frac {4 x^{11}}{4+2 \ln \relax (2)}+\frac {4 x^{10}}{4+2 \ln \relax (2)}-\frac {2 x^{9}}{4+2 \ln \relax (2)}-\frac {8 x^{8}}{4+2 \ln \relax (2)}-\frac {8 x^{7}}{4+2 \ln \relax (2)}+\frac {x^{6}}{4+2 \ln \relax (2)}+\frac {4 x^{5}}{4+2 \ln \relax (2)}+\frac {4 x^{4}}{4+2 \ln \relax (2)}\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^11+44*x^10+40*x^9-18*x^8-64*x^7-56*x^6+6*x^5+20*x^4+16*x^3)/(4+2*ln(2)),x,method=_RETURNVERBOSE)

[Out]

1/2*x^4*(x^5+4*x^4+4*x^3-x^2-4*x-4)*(x^3-1)/(ln(2)+2)

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maxima [A]  time = 0.42, size = 50, normalized size = 2.08 \begin {gather*} \frac {x^{12} + 4 \, x^{11} + 4 \, x^{10} - 2 \, x^{9} - 8 \, x^{8} - 8 \, x^{7} + x^{6} + 4 \, x^{5} + 4 \, x^{4}}{2 \, {\left (\log \relax (2) + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^11+44*x^10+40*x^9-18*x^8-64*x^7-56*x^6+6*x^5+20*x^4+16*x^3)/(4+2*log(2)),x, algorithm="maxima"
)

[Out]

1/2*(x^12 + 4*x^11 + 4*x^10 - 2*x^9 - 8*x^8 - 8*x^7 + x^6 + 4*x^5 + 4*x^4)/(log(2) + 2)

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mupad [B]  time = 2.06, size = 102, normalized size = 4.25 \begin {gather*} \frac {x^{12}}{2\,\left (\ln \relax (2)+2\right )}+\frac {2\,x^{11}}{\ln \relax (2)+2}+\frac {2\,x^{10}}{\ln \relax (2)+2}-\frac {x^9}{\ln \relax (2)+2}-\frac {4\,x^8}{\ln \relax (2)+2}-\frac {4\,x^7}{\ln \relax (2)+2}+\frac {x^6}{2\,\left (\ln \relax (2)+2\right )}+\frac {2\,x^5}{\ln \relax (2)+2}+\frac {2\,x^4}{\ln \relax (2)+2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^3 + 20*x^4 + 6*x^5 - 56*x^6 - 64*x^7 - 18*x^8 + 40*x^9 + 44*x^10 + 12*x^11)/(2*log(2) + 4),x)

[Out]

(2*x^4)/(log(2) + 2) + (2*x^5)/(log(2) + 2) + x^6/(2*(log(2) + 2)) - (4*x^7)/(log(2) + 2) - (4*x^8)/(log(2) +
2) - x^9/(log(2) + 2) + (2*x^10)/(log(2) + 2) + (2*x^11)/(log(2) + 2) + x^12/(2*(log(2) + 2))

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sympy [B]  time = 0.07, size = 88, normalized size = 3.67 \begin {gather*} \frac {x^{12}}{2 \log {\relax (2 )} + 4} + \frac {2 x^{11}}{\log {\relax (2 )} + 2} + \frac {2 x^{10}}{\log {\relax (2 )} + 2} - \frac {x^{9}}{\log {\relax (2 )} + 2} - \frac {4 x^{8}}{\log {\relax (2 )} + 2} - \frac {4 x^{7}}{\log {\relax (2 )} + 2} + \frac {x^{6}}{2 \log {\relax (2 )} + 4} + \frac {2 x^{5}}{\log {\relax (2 )} + 2} + \frac {2 x^{4}}{\log {\relax (2 )} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**11+44*x**10+40*x**9-18*x**8-64*x**7-56*x**6+6*x**5+20*x**4+16*x**3)/(4+2*ln(2)),x)

[Out]

x**12/(2*log(2) + 4) + 2*x**11/(log(2) + 2) + 2*x**10/(log(2) + 2) - x**9/(log(2) + 2) - 4*x**8/(log(2) + 2) -
4*x**7/(log(2) + 2) + x**6/(2*log(2) + 4) + 2*x**5/(log(2) + 2) + 2*x**4/(log(2) + 2)

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