3.88.85 \(\int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+(30 x-20 x^2-5 x^3) \log (4)-50 \log ^2(4)}{2 x^5} \, dx\)

Optimal. Leaf size=27 \[ \left (\frac {-1+x}{x}+\frac {x}{2}+x^2+\frac {5 \log (4)}{2 x^2}\right )^2 \]

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Rubi [B]  time = 0.04, antiderivative size = 58, normalized size of antiderivative = 2.15, number of steps used = 3, number of rules used = 2, integrand size = 61, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {12, 14} \begin {gather*} x^4+\frac {25 \log ^2(4)}{4 x^4}+x^3-\frac {5 \log (4)}{x^3}+\frac {9 x^2}{4}+\frac {1+5 \log (4)}{x^2}-x-\frac {4-5 \log (4)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^2 + 4*x^3 - 2*x^5 + 9*x^6 + 6*x^7 + 8*x^8 + (30*x - 20*x^2 - 5*x^3)*Log[4] - 50*Log[4]^2)/(2*x^5),x]

[Out]

-x + (9*x^2)/4 + x^3 + x^4 - (4 - 5*Log[4])/(2*x) - (5*Log[4])/x^3 + (25*Log[4]^2)/(4*x^4) + (1 + 5*Log[4])/x^
2

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {-4 x^2+4 x^3-2 x^5+9 x^6+6 x^7+8 x^8+\left (30 x-20 x^2-5 x^3\right ) \log (4)-50 \log ^2(4)}{x^5} \, dx\\ &=\frac {1}{2} \int \left (-2+9 x+6 x^2+8 x^3+\frac {4-5 \log (4)}{x^2}+\frac {30 \log (4)}{x^4}-\frac {50 \log ^2(4)}{x^5}-\frac {4 (1+5 \log (4))}{x^3}\right ) \, dx\\ &=-x+\frac {9 x^2}{4}+x^3+x^4-\frac {4-5 \log (4)}{2 x}-\frac {5 \log (4)}{x^3}+\frac {25 \log ^2(4)}{4 x^4}+\frac {1+5 \log (4)}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 64, normalized size = 2.37 \begin {gather*} \frac {1}{2} \left (-2 x+\frac {9 x^2}{2}+2 x^3+2 x^4-\frac {10 \log (4)}{x^3}+\frac {25 \log ^2(4)}{2 x^4}+\frac {-4+5 \log (4)}{x}+\frac {2 (1+5 \log (4))}{x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^2 + 4*x^3 - 2*x^5 + 9*x^6 + 6*x^7 + 8*x^8 + (30*x - 20*x^2 - 5*x^3)*Log[4] - 50*Log[4]^2)/(2*x
^5),x]

[Out]

(-2*x + (9*x^2)/2 + 2*x^3 + 2*x^4 - (10*Log[4])/x^3 + (25*Log[4]^2)/(2*x^4) + (-4 + 5*Log[4])/x + (2*(1 + 5*Lo
g[4]))/x^2)/2

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fricas [B]  time = 0.64, size = 58, normalized size = 2.15 \begin {gather*} \frac {4 \, x^{8} + 4 \, x^{7} + 9 \, x^{6} - 4 \, x^{5} - 8 \, x^{3} + 4 \, x^{2} + 20 \, {\left (x^{3} + 2 \, x^{2} - 2 \, x\right )} \log \relax (2) + 100 \, \log \relax (2)^{2}}{4 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-200*log(2)^2+2*(-5*x^3-20*x^2+30*x)*log(2)+8*x^8+6*x^7+9*x^6-2*x^5+4*x^3-4*x^2)/x^5,x, algorit
hm="fricas")

[Out]

1/4*(4*x^8 + 4*x^7 + 9*x^6 - 4*x^5 - 8*x^3 + 4*x^2 + 20*(x^3 + 2*x^2 - 2*x)*log(2) + 100*log(2)^2)/x^4

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giac [B]  time = 0.15, size = 53, normalized size = 1.96 \begin {gather*} x^{4} + x^{3} + \frac {9}{4} \, x^{2} - x + \frac {5 \, x^{3} \log \relax (2) - 2 \, x^{3} + 10 \, x^{2} \log \relax (2) + x^{2} - 10 \, x \log \relax (2) + 25 \, \log \relax (2)^{2}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-200*log(2)^2+2*(-5*x^3-20*x^2+30*x)*log(2)+8*x^8+6*x^7+9*x^6-2*x^5+4*x^3-4*x^2)/x^5,x, algorit
hm="giac")

[Out]

x^4 + x^3 + 9/4*x^2 - x + (5*x^3*log(2) - 2*x^3 + 10*x^2*log(2) + x^2 - 10*x*log(2) + 25*log(2)^2)/x^4

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maple [B]  time = 0.07, size = 53, normalized size = 1.96

method	result	size
norman	\(\frac {x^{7}+x^{8}+\left (5 \ln \relax (2)-2\right ) x^{3}+\left (10 \ln \relax (2)+1\right ) x^{2}-x^{5}+\frac {9 x^{6}}{4}+25 \ln \relax (2)^{2}-10 x \ln \relax (2)}{x^{4}}\)	\(53\)
risch	\(x^{4}+x^{3}+\frac {9 x^{2}}{4}-x +\frac {\left (10 \ln \relax (2)-4\right ) x^{3}+\left (2+20 \ln \relax (2)\right ) x^{2}-20 x \ln \relax (2)+50 \ln \relax (2)^{2}}{2 x^{4}}\)	\(53\)
default	\(x^{4}+x^{3}+\frac {9 x^{2}}{4}-x +\frac {25 \ln \relax (2)^{2}}{x^{4}}-\frac {-40 \ln \relax (2)-4}{4 x^{2}}-\frac {-10 \ln \relax (2)+4}{2 x}-\frac {10 \ln \relax (2)}{x^{3}}\)	\(54\)
gosper	\(\frac {4 x^{8}+4 x^{7}+9 x^{6}-4 x^{5}+20 x^{3} \ln \relax (2)+40 x^{2} \ln \relax (2)-8 x^{3}+100 \ln \relax (2)^{2}-40 x \ln \relax (2)+4 x^{2}}{4 x^{4}}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(-200*ln(2)^2+2*(-5*x^3-20*x^2+30*x)*ln(2)+8*x^8+6*x^7+9*x^6-2*x^5+4*x^3-4*x^2)/x^5,x,method=_RETURNVE
RBOSE)

[Out]

(x^7+x^8+(5*ln(2)-2)*x^3+(10*ln(2)+1)*x^2-x^5+9/4*x^6+25*ln(2)^2-10*x*ln(2))/x^4

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maxima [A]  time = 0.35, size = 51, normalized size = 1.89 \begin {gather*} x^{4} + x^{3} + \frac {9}{4} \, x^{2} - x + \frac {x^{3} {\left (5 \, \log \relax (2) - 2\right )} + x^{2} {\left (10 \, \log \relax (2) + 1\right )} - 10 \, x \log \relax (2) + 25 \, \log \relax (2)^{2}}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-200*log(2)^2+2*(-5*x^3-20*x^2+30*x)*log(2)+8*x^8+6*x^7+9*x^6-2*x^5+4*x^3-4*x^2)/x^5,x, algorit
hm="maxima")

[Out]

x^4 + x^3 + 9/4*x^2 - x + (x^3*(5*log(2) - 2) + x^2*(10*log(2) + 1) - 10*x*log(2) + 25*log(2)^2)/x^4

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mupad [B]  time = 5.56, size = 49, normalized size = 1.81 \begin {gather*} \frac {\left (\ln \left (32\right )-2\right )\,x^3+\left (10\,\ln \relax (2)+1\right )\,x^2-10\,\ln \relax (2)\,x+25\,{\ln \relax (2)}^2}{x^4}-x+\frac {9\,x^2}{4}+x^3+x^4 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(2)*(20*x^2 - 30*x + 5*x^3) + 100*log(2)^2 + 2*x^2 - 2*x^3 + x^5 - (9*x^6)/2 - 3*x^7 - 4*x^8)/x^5,x)

[Out]

(x^3*(log(32) - 2) - 10*x*log(2) + x^2*(10*log(2) + 1) + 25*log(2)^2)/x^4 - x + (9*x^2)/4 + x^3 + x^4

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sympy [B]  time = 0.54, size = 53, normalized size = 1.96 \begin {gather*} x^{4} + x^{3} + \frac {9 x^{2}}{4} - x + \frac {x^{3} \left (-4 + 10 \log {\relax (2 )}\right ) + x^{2} \left (2 + 20 \log {\relax (2 )}\right ) - 20 x \log {\relax (2 )} + 50 \log {\relax (2 )}^{2}}{2 x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(-200*ln(2)**2+2*(-5*x**3-20*x**2+30*x)*ln(2)+8*x**8+6*x**7+9*x**6-2*x**5+4*x**3-4*x**2)/x**5,x)

[Out]

x**4 + x**3 + 9*x**2/4 - x + (x**3*(-4 + 10*log(2)) + x**2*(2 + 20*log(2)) - 20*x*log(2) + 50*log(2)**2)/(2*x*
*4)

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