\(\int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx\) [1235]

Optimal result

Integrand size = 20, antiderivative size = 86 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {21}{5 (2+3 x)^5}+\frac {309}{4 (2+3 x)^4}+\frac {1020}{(2+3 x)^3}+\frac {12675}{(2+3 x)^2}+\frac {189375}{2+3 x}-\frac {6875}{2 (3+5 x)^2}+\frac {125000}{3+5 x}-1321875 \log (2+3 x)+1321875 \log (3+5 x) \]

[Out]

21/5/(2+3*x)^5+309/4/(2+3*x)^4+1020/(2+3*x)^3+12675/(2+3*x)^2+189375/(2+3*x)-6875/2/(3+5*x)^2+125000/(3+5*x)-1
321875*ln(2+3*x)+1321875*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {189375}{3 x+2}+\frac {125000}{5 x+3}+\frac {12675}{(3 x+2)^2}-\frac {6875}{2 (5 x+3)^2}+\frac {1020}{(3 x+2)^3}+\frac {309}{4 (3 x+2)^4}+\frac {21}{5 (3 x+2)^5}-1321875 \log (3 x+2)+1321875 \log (5 x+3) \]

[In]

Int[(1 - 2*x)/((2 + 3*x)^6*(3 + 5*x)^3),x]

[Out]

21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3*x)^2 + 189375/(2 + 3*x) - 6875/(2*(
3 + 5*x)^2) + 125000/(3 + 5*x) - 1321875*Log[2 + 3*x] + 1321875*Log[3 + 5*x]

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {63}{(2+3 x)^6}-\frac {927}{(2+3 x)^5}-\frac {9180}{(2+3 x)^4}-\frac {76050}{(2+3 x)^3}-\frac {568125}{(2+3 x)^2}-\frac {3965625}{2+3 x}+\frac {34375}{(3+5 x)^3}-\frac {625000}{(3+5 x)^2}+\frac {6609375}{3+5 x}\right ) \, dx \\ & = \frac {21}{5 (2+3 x)^5}+\frac {309}{4 (2+3 x)^4}+\frac {1020}{(2+3 x)^3}+\frac {12675}{(2+3 x)^2}+\frac {189375}{2+3 x}-\frac {6875}{2 (3+5 x)^2}+\frac {125000}{3+5 x}-1321875 \log (2+3 x)+1321875 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {21}{5 (2+3 x)^5}+\frac {309}{4 (2+3 x)^4}+\frac {1020}{(2+3 x)^3}+\frac {12675}{(2+3 x)^2}+\frac {189375}{2+3 x}-\frac {6875}{2 (3+5 x)^2}+\frac {125000}{3+5 x}-1321875 \log (2+3 x)+1321875 \log (-3 (3+5 x)) \]

[In]

Integrate[(1 - 2*x)/((2 + 3*x)^6*(3 + 5*x)^3),x]

[Out]

21/(5*(2 + 3*x)^5) + 309/(4*(2 + 3*x)^4) + 1020/(2 + 3*x)^3 + 12675/(2 + 3*x)^2 + 189375/(2 + 3*x) - 6875/(2*(
3 + 5*x)^2) + 125000/(3 + 5*x) - 1321875*Log[2 + 3*x] + 1321875*Log[-3*(3 + 5*x)]

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73

method	result	size
norman	\(\frac {535359375 x^{6}[phone] x^{2}[phone] x^{4}+\frac {1484332427}{4} x +\frac {4175803125}{2} x^{5}+\frac {11746762875}{4} x^{3}+\frac {401107483}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-1321875 \ln \left (2+3 x \right )+1321875 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {535359375 x^{6}[phone] x^{2}[phone] x^{4}+\frac {1484332427}{4} x +\frac {4175803125}{2} x^{5}+\frac {11746762875}{4} x^{3}+\frac {401107483}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-1321875 \ln \left (2+3 x \right )+1321875 \ln \left (3+5 x \right )\)	\(64\)
default	\(\frac {21}{5 \left (2+3 x \right )^{5}}+\frac {309}{4 \left (2+3 x \right )^{4}}+\frac {1020}{\left (2+3 x \right )^{3}}+\frac {12675}{\left (2+3 x \right )^{2}}+\frac {189375}{2+3 x}-\frac {6875}{2 \left (3+5 x \right )^{2}}+\frac {125000}{3+5 x}-1321875 \ln \left (2+3 x \right )+1321875 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(-\frac {182735999520 x -55125360000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-11877840000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-23127525000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-1096416000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-142077240000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-203625009000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-219625830000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-104844780000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{2880 \left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}\)	\(185\)

[In]

int((1-2*x)/(2+3*x)^6/(3+5*x)^3,x,method=_RETURNVERBOSE)

[Out]

(535359375*x^6+1429766790*x^2+3391402500*x^4+1484332427/4*x+4175803125/2*x^5+11746762875/4*x^3+401107483/10)/(
2+3*x)^5/(3+5*x)^2-1321875*ln(2+3*x)+1321875*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.80 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 26437500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 26437500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 7421662135 \, x + 802214966}{20 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]

[In]

integrate((1-2*x)/(2+3*x)^6/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^3 + 28595335800*x^2 + 26437500*(6075
*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(5*x + 3) - 26437500*(6075
*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)*log(3*x + 2) + 7421662135*x +
802214966)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.97 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=- \frac {- 10707187500 x^{6} - 41758031250 x^{5} - 67828050000 x^{4} - 58733814375 x^{3} - 28595335800 x^{2} - 7421662135 x - 802214966}{121500 x^{7} + 550800 x^{6} + 1069740 x^{5} + 1153800 x^{4} + 746400 x^{3} + 289600 x^{2} + 62400 x + 5760} + 1321875 \log {\left (x + \frac {3}{5} \right )} - 1321875 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)/(2+3*x)**6/(3+5*x)**3,x)

[Out]

-(-10707187500*x**6 - 41758031250*x**5 - 67828050000*x**4 - 58733814375*x**3 - 28595335800*x**2 - 7421662135*x
- 802214966)/(121500*x**7 + 550800*x**6 + 1069740*x**5 + 1153800*x**4 + 746400*x**3 + 289600*x**2 + 62400*x +
5760) + 1321875*log(x + 3/5) - 1321875*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 7421662135 \, x + 802214966}{20 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 1321875 \, \log \left (5 \, x + 3\right ) - 1321875 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)/(2+3*x)^6/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^3 + 28595335800*x^2 + 7421662135*x +
802214966)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) + 1321875*lo
g(5*x + 3) - 1321875*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {10707187500 \, x^{6} + 41758031250 \, x^{5} + 67828050000 \, x^{4} + 58733814375 \, x^{3} + 28595335800 \, x^{2} + 7421662135 \, x + 802214966}{20 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 1321875 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 1321875 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

integrate((1-2*x)/(2+3*x)^6/(3+5*x)^3,x, algorithm="giac")

[Out]

1/20*(10707187500*x^6 + 41758031250*x^5 + 67828050000*x^4 + 58733814375*x^3 + 28595335800*x^2 + 7421662135*x +
802214966)/((5*x + 3)^2*(3*x + 2)^5) + 1321875*log(abs(5*x + 3)) - 1321875*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 1.24 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {1-2 x}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {88125\,x^6+\frac {687375\,x^5}{2}+\frac {5024300\,x^4}{9}+\frac {52207835\,x^3}{108}+\frac {95317786\,x^2}{405}+\frac {1484332427\,x}{24300}+\frac {401107483}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-2643750\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

int(-(2*x - 1)/((3*x + 2)^6*(5*x + 3)^3),x)

[Out]

((1484332427*x)/24300 + (95317786*x^2)/405 + (52207835*x^3)/108 + (5024300*x^4)/9 + (687375*x^5)/2 + 88125*x^6
+ 401107483/60750)/((208*x)/405 + (2896*x^2)/1215 + (2488*x^3)/405 + (1282*x^4)/135 + (1981*x^5)/225 + (68*x^
6)/15 + x^7 + 32/675) - 2643750*atanh(30*x + 19)