3.55.64 \(\int \frac {300-6 x^3-300 x^4}{(25 x+5 x^3+x^4+25 x^5) \log ^3(\frac {25+5 x^2+x^3+25 x^4}{x^2})} \, dx\)

Optimal. Leaf size=18 \[ \frac {3}{\log ^2\left (5+\frac {25}{x^2}+x+25 x^2\right )} \]

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Rubi [A]  time = 0.09, antiderivative size = 24, normalized size of antiderivative = 1.33, number of steps used = 1, number of rules used = 1, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6686} \begin {gather*} \frac {3}{\log ^2\left (\frac {25 x^4+x^3+5 x^2+25}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(300 - 6*x^3 - 300*x^4)/((25*x + 5*x^3 + x^4 + 25*x^5)*Log[(25 + 5*x^2 + x^3 + 25*x^4)/x^2]^3),x]

[Out]

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

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {3}{\log ^2\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {3}{\log ^2\left (5+\frac {25}{x^2}+x+25 x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(300 - 6*x^3 - 300*x^4)/((25*x + 5*x^3 + x^4 + 25*x^5)*Log[(25 + 5*x^2 + x^3 + 25*x^4)/x^2]^3),x]

[Out]

3/Log[5 + 25/x^2 + x + 25*x^2]^2

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fricas [A]  time = 1.24, size = 24, normalized size = 1.33 \begin {gather*} \frac {3}{\log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-300*x^4-6*x^3+300)/(25*x^5+x^4+5*x^3+25*x)/log((25*x^4+x^3+5*x^2+25)/x^2)^3,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.21, size = 92, normalized size = 5.11 \begin {gather*} \frac {3 \, {\left (50 \, x^{4} + x^{3} - 50\right )}}{50 \, x^{4} \log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2} + x^{3} \log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2} - 50 \, \log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-300*x^4-6*x^3+300)/(25*x^5+x^4+5*x^3+25*x)/log((25*x^4+x^3+5*x^2+25)/x^2)^3,x, algorithm="giac")

[Out]

3*(50*x^4 + x^3 - 50)/(50*x^4*log((25*x^4 + x^3 + 5*x^2 + 25)/x^2)^2 + x^3*log((25*x^4 + x^3 + 5*x^2 + 25)/x^2
)^2 - 50*log((25*x^4 + x^3 + 5*x^2 + 25)/x^2)^2)

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maple [A]  time = 0.06, size = 25, normalized size = 1.39

method	result	size
norman	\(\frac {3}{\ln \left (\frac {25 x^{4}+x^{3}+5 x^{2}+25}{x^{2}}\right )^{2}}\)	\(25\)
risch	\(\frac {3}{\ln \left (\frac {25 x^{4}+x^{3}+5 x^{2}+25}{x^{2}}\right )^{2}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-300*x^4-6*x^3+300)/(25*x^5+x^4+5*x^3+25*x)/ln((25*x^4+x^3+5*x^2+25)/x^2)^3,x,method=_RETURNVERBOSE)

[Out]

3/ln((25*x^4+x^3+5*x^2+25)/x^2)^2

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maxima [B]  time = 0.41, size = 49, normalized size = 2.72 \begin {gather*} \frac {3}{\log \left (25 \, x^{4} + x^{3} + 5 \, x^{2} + 25\right )^{2} - 4 \, \log \left (25 \, x^{4} + x^{3} + 5 \, x^{2} + 25\right ) \log \relax (x) + 4 \, \log \relax (x)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-300*x^4-6*x^3+300)/(25*x^5+x^4+5*x^3+25*x)/log((25*x^4+x^3+5*x^2+25)/x^2)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 3.63, size = 24, normalized size = 1.33 \begin {gather*} \frac {3}{{\ln \left (\frac {25\,x^4+x^3+5\,x^2+25}{x^2}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x^3 + 300*x^4 - 300)/(log((5*x^2 + x^3 + 25*x^4 + 25)/x^2)^3*(25*x + 5*x^3 + x^4 + 25*x^5)),x)

[Out]

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

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sympy [A]  time = 0.17, size = 22, normalized size = 1.22 \begin {gather*} \frac {3}{\log {\left (\frac {25 x^{4} + x^{3} + 5 x^{2} + 25}{x^{2}} \right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-300*x**4-6*x**3+300)/(25*x**5+x**4+5*x**3+25*x)/ln((25*x**4+x**3+5*x**2+25)/x**2)**3,x)

[Out]

3/log((25*x**4 + x**3 + 5*x**2 + 25)/x**2)**2

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