\(\int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx\) [1315]

Optimal result

Integrand size = 22, antiderivative size = 46 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {49}{6 (2+3 x)^2}-\frac {154}{2+3 x}-\frac {121}{3+5 x}+1133 \log (2+3 x)-1133 \log (3+5 x) \]

[Out]

-49/6/(2+3*x)^2-154/(2+3*x)-121/(3+5*x)+1133*ln(2+3*x)-1133*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {154}{3 x+2}-\frac {121}{5 x+3}-\frac {49}{6 (3 x+2)^2}+1133 \log (3 x+2)-1133 \log (5 x+3) \]

[In]

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

[Out]

-49/(6*(2 + 3*x)^2) - 154/(2 + 3*x) - 121/(3 + 5*x) + 1133*Log[2 + 3*x] - 1133*Log[3 + 5*x]

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {49}{(2+3 x)^3}+\frac {462}{(2+3 x)^2}+\frac {3399}{2+3 x}+\frac {605}{(3+5 x)^2}-\frac {5665}{3+5 x}\right ) \, dx \\ & = -\frac {49}{6 (2+3 x)^2}-\frac {154}{2+3 x}-\frac {121}{3+5 x}+1133 \log (2+3 x)-1133 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {49}{6 (2+3 x)^2}-\frac {154}{2+3 x}-\frac {121}{3+5 x}+1133 \log (5 (2+3 x))-1133 \log (3+5 x) \]

[In]

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

[Out]

-49/(6*(2 + 3*x)^2) - 154/(2 + 3*x) - 121/(3 + 5*x) + 1133*Log[5*(2 + 3*x)] - 1133*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {-3399 x^{2}-\frac {26513}{6} x -\frac {2865}{2}}{\left (2+3 x \right )^{2} \left (3+5 x \right )}+1133 \ln \left (2+3 x \right )-1133 \ln \left (3+5 x \right )\)	\(44\)
default	\(-\frac {49}{6 \left (2+3 x \right )^{2}}-\frac {154}{2+3 x}-\frac {121}{3+5 x}+1133 \ln \left (2+3 x \right )-1133 \ln \left (3+5 x \right )\)	\(45\)
norman	\(\frac {\frac {42975}{8} x^{3}+\frac {55893}{8} x^{2}+\frac {13597}{6} x}{\left (2+3 x \right )^{2} \left (3+5 x \right )}+1133 \ln \left (2+3 x \right )-1133 \ln \left (3+5 x \right )\)	\(47\)
parallelrisch	\(\frac {1223640 \ln \left (\frac {2}{3}+x \right ) x^{3}-1223640 \ln \left (x +\frac {3}{5}\right ) x^{3}+2365704 \ln \left (\frac {2}{3}+x \right ) x^{2}-2365704 \ln \left (x +\frac {3}{5}\right ) x^{2}+128925 x^{3}+1522752 \ln \left (\frac {2}{3}+x \right ) x -1522752 \ln \left (x +\frac {3}{5}\right ) x +167679 x^{2}+326304 \ln \left (\frac {2}{3}+x \right )-326304 \ln \left (x +\frac {3}{5}\right )+54388 x}{24 \left (2+3 x \right )^{2} \left (3+5 x \right )}\)	\(93\)

[In]

int((1-2*x)^2/(2+3*x)^3/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

45*(-1133/15*x^2-26513/270*x-191/6)/(2+3*x)^2/(3+5*x)+1133*ln(2+3*x)-1133*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.63 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {20394 \, x^{2} + 6798 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (5 \, x + 3\right ) - 6798 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )} \log \left (3 \, x + 2\right ) + 26513 \, x + 8595}{6 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} \]

[In]

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

[Out]

-1/6*(20394*x^2 + 6798*(45*x^3 + 87*x^2 + 56*x + 12)*log(5*x + 3) - 6798*(45*x^3 + 87*x^2 + 56*x + 12)*log(3*x
+ 2) + 26513*x + 8595)/(45*x^3 + 87*x^2 + 56*x + 12)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=\frac {- 20394 x^{2} - 26513 x - 8595}{270 x^{3} + 522 x^{2} + 336 x + 72} - 1133 \log {\left (x + \frac {3}{5} \right )} + 1133 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**2/(2+3*x)**3/(3+5*x)**2,x)

[Out]

(-20394*x**2 - 26513*x - 8595)/(270*x**3 + 522*x**2 + 336*x + 72) - 1133*log(x + 3/5) + 1133*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {20394 \, x^{2} + 26513 \, x + 8595}{6 \, {\left (45 \, x^{3} + 87 \, x^{2} + 56 \, x + 12\right )}} - 1133 \, \log \left (5 \, x + 3\right ) + 1133 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/6*(20394*x^2 + 26513*x + 8595)/(45*x^3 + 87*x^2 + 56*x + 12) - 1133*log(5*x + 3) + 1133*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=-\frac {121}{5 \, x + 3} + \frac {35 \, {\left (\frac {202}{5 \, x + 3} + 501\right )}}{2 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{2}} + 1133 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

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

[Out]

-121/(5*x + 3) + 35/2*(202/(5*x + 3) + 501)/(1/(5*x + 3) + 3)^2 + 1133*log(abs(-1/(5*x + 3) - 3))

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^2}{(2+3 x)^3 (3+5 x)^2} \, dx=2266\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {1133\,x^2}{15}+\frac {26513\,x}{270}+\frac {191}{6}}{x^3+\frac {29\,x^2}{15}+\frac {56\,x}{45}+\frac {4}{15}} \]

[In]

int((2*x - 1)^2/((3*x + 2)^3*(5*x + 3)^2),x)

[Out]

2266*atanh(30*x + 19) - ((26513*x)/270 + (1133*x^2)/15 + 191/6)/((56*x)/45 + (29*x^2)/15 + x^3 + 4/15)