\(\int \frac {300-6 x^3-300 x^4}{(25 x+5 x^3+x^4+25 x^5) \log ^3(\frac {25+5 x^2+x^3+25 x^4}{x^2})} \, dx\) [5464]

Optimal result

Integrand size = 54, antiderivative size = 18 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{\log ^2\left (5+\frac {25}{x^2}+x+25 x^2\right )} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {6818} \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{\log ^2\left (\frac {25 x^4+x^3+5 x^2+25}{x^2}\right )} \]

[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 6818

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{\log ^2\left (5+\frac {25}{x^2}+x+25 x^2\right )} \]

[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

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39

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

[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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{\log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2}} \]

[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

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.22 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{\log {\left (\frac {25 x^{4} + x^{3} + 5 x^{2} + 25}{x^{2}} \right )}^{2}} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (18) = 36\).

Time = 0.22 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.72 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{\log \left (25 \, x^{4} + x^{3} + 5 \, x^{2} + 25\right )^{2} - 4 \, \log \left (25 \, x^{4} + x^{3} + 5 \, x^{2} + 25\right ) \log \left (x\right ) + 4 \, \log \left (x\right )^{2}} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 92, normalized size of antiderivative = 5.11 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3 \, {\left (50 \, x^{4} + x^{3} - 50\right )}}{50 \, x^{4} \log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2} + x^{3} \log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2} - 50 \, \log \left (\frac {25 \, x^{4} + x^{3} + 5 \, x^{2} + 25}{x^{2}}\right )^{2}} \]

[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)

Mupad [B] (verification not implemented)

Time = 11.90 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.33 \[ \int \frac {300-6 x^3-300 x^4}{\left (25 x+5 x^3+x^4+25 x^5\right ) \log ^3\left (\frac {25+5 x^2+x^3+25 x^4}{x^2}\right )} \, dx=\frac {3}{{\ln \left (\frac {25\,x^4+x^3+5\,x^2+25}{x^2}\right )}^2} \]

[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