\(\int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx\) [1413]

Optimal result

Integrand size = 22, antiderivative size = 57 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {343}{27 (2+3 x)^3}-\frac {1568}{9 (2+3 x)^2}-\frac {2541}{2+3 x}-\frac {1331}{3+5 x}+16698 \log (2+3 x)-16698 \log (3+5 x) \]

[Out]

-343/27/(2+3*x)^3-1568/9/(2+3*x)^2-2541/(2+3*x)-1331/(3+5*x)+16698*ln(2+3*x)-16698*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, 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} \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {2541}{3 x+2}-\frac {1331}{5 x+3}-\frac {1568}{9 (3 x+2)^2}-\frac {343}{27 (3 x+2)^3}+16698 \log (3 x+2)-16698 \log (5 x+3) \]

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

-343/(27*(2 + 3*x)^3) - 1568/(9*(2 + 3*x)^2) - 2541/(2 + 3*x) - 1331/(3 + 5*x) + 16698*Log[2 + 3*x] - 16698*Lo
g[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 {343}{3 (2+3 x)^4}+\frac {3136}{3 (2+3 x)^3}+\frac {7623}{(2+3 x)^2}+\frac {50094}{2+3 x}+\frac {6655}{(3+5 x)^2}-\frac {83490}{3+5 x}\right ) \, dx \\ & = -\frac {343}{27 (2+3 x)^3}-\frac {1568}{9 (2+3 x)^2}-\frac {2541}{2+3 x}-\frac {1331}{3+5 x}+16698 \log (2+3 x)-16698 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {343}{27 (2+3 x)^3}-\frac {1568}{9 (2+3 x)^2}-\frac {2541}{2+3 x}-\frac {1331}{3+5 x}+16698 \log (5 (2+3 x))-16698 \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

-343/(27*(2 + 3*x)^3) - 1568/(9*(2 + 3*x)^2) - 2541/(2 + 3*x) - 1331/(3 + 5*x) + 16698*Log[5*(2 + 3*x)] - 1669
8*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84

method	result	size
norman	\(\frac {-150282 x^{3}-\frac {5226815}{27} x -\frac {886663}{3} x^{2}-\frac {380011}{9}}{\left (2+3 x \right )^{3} \left (3+5 x \right )}+16698 \ln \left (2+3 x \right )-16698 \ln \left (3+5 x \right )\)	\(48\)
risch	\(\frac {-150282 x^{3}-\frac {5226815}{27} x -\frac {886663}{3} x^{2}-\frac {380011}{9}}{\left (2+3 x \right )^{3} \left (3+5 x \right )}+16698 \ln \left (2+3 x \right )-16698 \ln \left (3+5 x \right )\)	\(49\)
default	\(-\frac {343}{27 \left (2+3 x \right )^{3}}-\frac {1568}{9 \left (2+3 x \right )^{2}}-\frac {2541}{2+3 x}-\frac {1331}{3+5 x}+16698 \ln \left (2+3 x \right )-16698 \ln \left (3+5 x \right )\)	\(54\)
parallelrisch	\(\frac {54101520 \ln \left (\frac {2}{3}+x \right ) x^{4}-54101520 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-140663952 \ln \left (x +\frac {3}{5}\right ) x^{3}+5700165 x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-137057184 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -59311296 \ln \left (x +\frac {3}{5}\right ) x +7347114 x^{2}+9618048 \ln \left (\frac {2}{3}+x \right )-9618048 \ln \left (x +\frac {3}{5}\right )+1603012 x}{24 \left (2+3 x \right )^{3} \left (3+5 x \right )}\)	\(116\)

[In]

int((1-2*x)^3/(2+3*x)^4/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

(-150282*x^3-5226815/27*x-886663/3*x^2-380011/9)/(2+3*x)^3/(3+5*x)+16698*ln(2+3*x)-16698*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.67 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {4057614 \, x^{3} + 7979967 \, x^{2} + 450846 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (5 \, x + 3\right ) - 450846 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (3 \, x + 2\right ) + 5226815 \, x + 1140033}{27 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

[In]

integrate((1-2*x)^3/(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/27*(4057614*x^3 + 7979967*x^2 + 450846*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(5*x + 3) - 450846*(13
5*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(3*x + 2) + 5226815*x + 1140033)/(135*x^4 + 351*x^3 + 342*x^2 + 148
*x + 24)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=- \frac {4057614 x^{3} + 7979967 x^{2} + 5226815 x + 1140033}{3645 x^{4} + 9477 x^{3} + 9234 x^{2} + 3996 x + 648} - 16698 \log {\left (x + \frac {3}{5} \right )} + 16698 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**3/(2+3*x)**4/(3+5*x)**2,x)

[Out]

-(4057614*x**3 + 7979967*x**2 + 5226815*x + 1140033)/(3645*x**4 + 9477*x**3 + 9234*x**2 + 3996*x + 648) - 1669
8*log(x + 3/5) + 16698*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {4057614 \, x^{3} + 7979967 \, x^{2} + 5226815 \, x + 1140033}{27 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} - 16698 \, \log \left (5 \, x + 3\right ) + 16698 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/27*(4057614*x^3 + 7979967*x^2 + 5226815*x + 1140033)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24) - 16698*log
(5*x + 3) + 16698*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {1331}{5 \, x + 3} + \frac {35 \, {\left (\frac {11119}{5 \, x + 3} + \frac {2244}{{\left (5 \, x + 3\right )}^{2}} + 14386\right )}}{{\left (\frac {1}{5 \, x + 3} + 3\right )}^{3}} + 16698 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

-1331/(5*x + 3) + 35*(11119/(5*x + 3) + 2244/(5*x + 3)^2 + 14386)/(1/(5*x + 3) + 3)^3 + 16698*log(abs(-1/(5*x
+ 3) - 3))

Mupad [B] (verification not implemented)

Time = 1.36 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)^2} \, dx=33396\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {5566\,x^3}{5}+\frac {886663\,x^2}{405}+\frac {1045363\,x}{729}+\frac {380011}{1215}}{x^4+\frac {13\,x^3}{5}+\frac {38\,x^2}{15}+\frac {148\,x}{135}+\frac {8}{45}} \]

[In]

int(-(2*x - 1)^3/((3*x + 2)^4*(5*x + 3)^2),x)

[Out]

33396*atanh(30*x + 19) - ((1045363*x)/729 + (886663*x^2)/405 + (5566*x^3)/5 + 380011/1215)/((148*x)/135 + (38*
x^2)/15 + (13*x^3)/5 + x^4 + 8/45)