\(\int \frac {-25 x-15 x^2-2 x^3+(5+27 x+15 x^2+2 x^3) \log (1+5 x+x^2)}{(1+5 x+x^2) \log ^2(1+5 x+x^2)} \, dx\) [3905]

Optimal result

Integrand size = 61, antiderivative size = 16 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x (5+x)}{\log \left (x \left (5+\frac {1}{x}+x\right )\right )} \]

[Out]

(5+x)/ln((x+1/x+5)*x)*x

Rubi [F]

\[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx \]

[In]

Int[(-25*x - 15*x^2 - 2*x^3 + (5 + 27*x + 15*x^2 + 2*x^3)*Log[1 + 5*x + x^2])/((1 + 5*x + x^2)*Log[1 + 5*x + x
^2]^2),x]

[Out]

-Log[1 + 5*x + x^2]^(-1) - 5*Defer[Int][Log[1 + 5*x + x^2]^(-2), x] - 2*Defer[Int][x/Log[1 + 5*x + x^2]^2, x]
+ 5*Defer[Int][Log[1 + 5*x + x^2]^(-1), x] + 2*Defer[Int][x/Log[1 + 5*x + x^2], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {x \left (25+15 x+2 x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )}+\frac {5+2 x}{\log \left (1+5 x+x^2\right )}\right ) \, dx \\ & = -\int \frac {x \left (25+15 x+2 x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx+\int \frac {5+2 x}{\log \left (1+5 x+x^2\right )} \, dx \\ & = -\int \left (\frac {5}{\log ^2\left (1+5 x+x^2\right )}+\frac {2 x}{\log ^2\left (1+5 x+x^2\right )}+\frac {-5-2 x}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )}\right ) \, dx+\int \left (\frac {5}{\log \left (1+5 x+x^2\right )}+\frac {2 x}{\log \left (1+5 x+x^2\right )}\right ) \, dx \\ & = -\left (2 \int \frac {x}{\log ^2\left (1+5 x+x^2\right )} \, dx\right )+2 \int \frac {x}{\log \left (1+5 x+x^2\right )} \, dx-5 \int \frac {1}{\log ^2\left (1+5 x+x^2\right )} \, dx+5 \int \frac {1}{\log \left (1+5 x+x^2\right )} \, dx-\int \frac {-5-2 x}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx \\ & = -\frac {1}{\log \left (1+5 x+x^2\right )}-2 \int \frac {x}{\log ^2\left (1+5 x+x^2\right )} \, dx+2 \int \frac {x}{\log \left (1+5 x+x^2\right )} \, dx-5 \int \frac {1}{\log ^2\left (1+5 x+x^2\right )} \, dx+5 \int \frac {1}{\log \left (1+5 x+x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x (5+x)}{\log \left (1+5 x+x^2\right )} \]

[In]

Integrate[(-25*x - 15*x^2 - 2*x^3 + (5 + 27*x + 15*x^2 + 2*x^3)*Log[1 + 5*x + x^2])/((1 + 5*x + x^2)*Log[1 + 5
*x + x^2]^2),x]

[Out]

(x*(5 + x))/Log[1 + 5*x + x^2]

Maple [A] (verified)

Time = 3.41 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06

method	result	size
risch	\(\frac {\left (5+x \right ) x}{\ln \left (x^{2}+5 x +1\right )}\)	\(17\)
norman	\(\frac {x^{2}+5 x}{\ln \left (x^{2}+5 x +1\right )}\)	\(20\)
parallelrisch	\(\frac {x^{2}+5 x}{\ln \left (x^{2}+5 x +1\right )}\)	\(20\)
default	\(\frac {x^{2}+5 x +1}{\ln \left (x^{2}+5 x +1\right )}-\frac {1}{\ln \left (x^{2}+5 x +1\right )}\)	\(35\)
parts	\(\frac {x^{2}+5 x +1}{\ln \left (x^{2}+5 x +1\right )}-\frac {1}{\ln \left (x^{2}+5 x +1\right )}\)	\(35\)

[In]

int(((2*x^3+15*x^2+27*x+5)*ln(x^2+5*x+1)-2*x^3-15*x^2-25*x)/(x^2+5*x+1)/ln(x^2+5*x+1)^2,x,method=_RETURNVERBOS
E)

[Out]

(5+x)*x/ln(x^2+5*x+1)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x^{2} + 5 \, x}{\log \left (x^{2} + 5 \, x + 1\right )} \]

[In]

integrate(((2*x^3+15*x^2+27*x+5)*log(x^2+5*x+1)-2*x^3-15*x^2-25*x)/(x^2+5*x+1)/log(x^2+5*x+1)^2,x, algorithm="
fricas")

[Out]

(x^2 + 5*x)/log(x^2 + 5*x + 1)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x^{2} + 5 x}{\log {\left (x^{2} + 5 x + 1 \right )}} \]

[In]

integrate(((2*x**3+15*x**2+27*x+5)*ln(x**2+5*x+1)-2*x**3-15*x**2-25*x)/(x**2+5*x+1)/ln(x**2+5*x+1)**2,x)

[Out]

(x**2 + 5*x)/log(x**2 + 5*x + 1)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x^{2} + 5 \, x}{\log \left (x^{2} + 5 \, x + 1\right )} \]

[In]

integrate(((2*x^3+15*x^2+27*x+5)*log(x^2+5*x+1)-2*x^3-15*x^2-25*x)/(x^2+5*x+1)/log(x^2+5*x+1)^2,x, algorithm="
maxima")

[Out]

(x^2 + 5*x)/log(x^2 + 5*x + 1)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x^{2} + 5 \, x}{\log \left (x^{2} + 5 \, x + 1\right )} \]

[In]

integrate(((2*x^3+15*x^2+27*x+5)*log(x^2+5*x+1)-2*x^3-15*x^2-25*x)/(x^2+5*x+1)/log(x^2+5*x+1)^2,x, algorithm="
giac")

[Out]

(x^2 + 5*x)/log(x^2 + 5*x + 1)

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-25 x-15 x^2-2 x^3+\left (5+27 x+15 x^2+2 x^3\right ) \log \left (1+5 x+x^2\right )}{\left (1+5 x+x^2\right ) \log ^2\left (1+5 x+x^2\right )} \, dx=\frac {x\,\left (x+5\right )}{\ln \left (x^2+5\,x+1\right )} \]

[In]

int(-(25*x - log(5*x + x^2 + 1)*(27*x + 15*x^2 + 2*x^3 + 5) + 15*x^2 + 2*x^3)/(log(5*x + x^2 + 1)^2*(5*x + x^2
+ 1)),x)

[Out]

(x*(x + 5))/log(5*x + x^2 + 1)