3.60.96 \(\int \frac {(-3-\log (\frac {3}{x})) \sqrt [4]{4 x+x \log (\frac {3}{x})}}{50176 x+12544 x \log (\frac {3}{x})+\sqrt {4 x+x \log (\frac {3}{x})} (16 x+4 x \log (\frac {3}{x}))+\sqrt [4]{4 x+x \log (\frac {3}{x})} (1792 x+448 x \log (\frac {3}{x}))} \, dx\)

Optimal. Leaf size=20 \[ 2+\frac {1}{56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}} \]

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Rubi [A]  time = 0.45, antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 3, number of rules used = 3, integrand size = 102, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {6688, 12, 6686} \begin {gather*} \frac {1}{\sqrt [4]{x \left (\log \left (\frac {3}{x}\right )+4\right )}+56} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-3 - Log[3/x])*(4*x + x*Log[3/x])^(1/4))/(50176*x + 12544*x*Log[3/x] + Sqrt[4*x + x*Log[3/x]]*(16*x + 4*
x*Log[3/x]) + (4*x + x*Log[3/x])^(1/4)*(1792*x + 448*x*Log[3/x])),x]

[Out]

(56 + (x*(4 + Log[3/x]))^(1/4))^(-1)

Rule 12

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3-\log \left (\frac {3}{x}\right )}{4 \left (4 x+x \log \left (\frac {3}{x}\right )\right )^{3/4} \left (56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}\right )^2} \, dx\\ &=\frac {1}{4} \int \frac {-3-\log \left (\frac {3}{x}\right )}{\left (4 x+x \log \left (\frac {3}{x}\right )\right )^{3/4} \left (56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}\right )^2} \, dx\\ &=\frac {1}{56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.57, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{56+\sqrt [4]{x \left (4+\log \left (\frac {3}{x}\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-3 - Log[3/x])*(4*x + x*Log[3/x])^(1/4))/(50176*x + 12544*x*Log[3/x] + Sqrt[4*x + x*Log[3/x]]*(16*
x + 4*x*Log[3/x]) + (4*x + x*Log[3/x])^(1/4)*(1792*x + 448*x*Log[3/x])),x]

[Out]

(56 + (x*(4 + Log[3/x]))^(1/4))^(-1)

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fricas [B]  time = 0.60, size = 64, normalized size = 3.20 \begin {gather*} \frac {{\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {3}{4}} - 56 \, \sqrt {x \log \left (\frac {3}{x}\right ) + 4 \, x} + 3136 \, {\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {1}{4}} - 175616}{x \log \left (\frac {3}{x}\right ) + 4 \, x - 9834496} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log(3/x)+4*x)^(1/2)+(448*x*log(3/x)+179
2*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*log(3/x)+50176*x),x, algorithm="fricas")

[Out]

((x*log(3/x) + 4*x)^(3/4) - 56*sqrt(x*log(3/x) + 4*x) + 3136*(x*log(3/x) + 4*x)^(1/4) - 175616)/(x*log(3/x) +
4*x - 9834496)

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giac [A]  time = 0.16, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{{\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {1}{4}} + 56} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log(3/x)+4*x)^(1/2)+(448*x*log(3/x)+179
2*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*log(3/x)+50176*x),x, algorithm="giac")

[Out]

1/((x*log(3/x) + 4*x)^(1/4) + 56)

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maple [F]  time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (-\ln \left (\frac {3}{x}\right )-3\right ) \left (x \ln \left (\frac {3}{x}\right )+4 x \right )^{\frac {1}{4}}}{\left (4 x \ln \left (\frac {3}{x}\right )+16 x \right ) \sqrt {x \ln \left (\frac {3}{x}\right )+4 x}+\left (448 x \ln \left (\frac {3}{x}\right )+1792 x \right ) \left (x \ln \left (\frac {3}{x}\right )+4 x \right )^{\frac {1}{4}}+12544 x \ln \left (\frac {3}{x}\right )+50176 x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(3/x)-3)*(x*ln(3/x)+4*x)^(1/4)/((4*x*ln(3/x)+16*x)*(x*ln(3/x)+4*x)^(1/2)+(448*x*ln(3/x)+1792*x)*(x*ln(
3/x)+4*x)^(1/4)+12544*x*ln(3/x)+50176*x),x)

[Out]

int((-ln(3/x)-3)*(x*ln(3/x)+4*x)^(1/4)/((4*x*ln(3/x)+16*x)*(x*ln(3/x)+4*x)^(1/2)+(448*x*ln(3/x)+1792*x)*(x*ln(
3/x)+4*x)^(1/4)+12544*x*ln(3/x)+50176*x),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {1}{4} \, \int \frac {{\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {1}{4}} {\left (\log \left (\frac {3}{x}\right ) + 3\right )}}{3136 \, x \log \left (\frac {3}{x}\right ) + {\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {3}{2}} + 112 \, {\left (x \log \left (\frac {3}{x}\right ) + 4 \, x\right )}^{\frac {5}{4}} + 12544 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(3/x)-3)*(x*log(3/x)+4*x)^(1/4)/((4*x*log(3/x)+16*x)*(x*log(3/x)+4*x)^(1/2)+(448*x*log(3/x)+179
2*x)*(x*log(3/x)+4*x)^(1/4)+12544*x*log(3/x)+50176*x),x, algorithm="maxima")

[Out]

-1/4*integrate((x*log(3/x) + 4*x)^(1/4)*(log(3/x) + 3)/(3136*x*log(3/x) + (x*log(3/x) + 4*x)^(3/2) + 112*(x*lo
g(3/x) + 4*x)^(5/4) + 12544*x), x)

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mupad [B]  time = 5.36, size = 18, normalized size = 0.90 \begin {gather*} \frac {1}{{\left (4\,x+x\,\ln \left (\frac {3}{x}\right )\right )}^{1/4}+56} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((log(3/x) + 3)*(4*x + x*log(3/x))^(1/4))/(50176*x + (4*x + x*log(3/x))^(1/2)*(16*x + 4*x*log(3/x)) + (4*
x + x*log(3/x))^(1/4)*(1792*x + 448*x*log(3/x)) + 12544*x*log(3/x)),x)

[Out]

1/((4*x + x*log(3/x))^(1/4) + 56)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(3/x)-3)*(x*ln(3/x)+4*x)**(1/4)/((4*x*ln(3/x)+16*x)*(x*ln(3/x)+4*x)**(1/2)+(448*x*ln(3/x)+1792*x
)*(x*ln(3/x)+4*x)**(1/4)+12544*x*ln(3/x)+50176*x),x)

[Out]

Timed out

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