\(\int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx\) [79]

Optimal result

Integrand size = 20, antiderivative size = 75 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {21}{4 (2+3 x)^4}+\frac {103}{(2+3 x)^3}+\frac {1530}{(2+3 x)^2}+\frac {25350}{2+3 x}-\frac {1375}{2 (3+5 x)^2}+\frac {20875}{3+5 x}-189375 \log (2+3 x)+189375 \log (3+5 x) \] Output:

21/4/(2+3*x)^4+103/(2+3*x)^3+1530/(2+3*x)^2+25350/(2+3*x)-1375/2/(3+5*x)^2 
+20875/(3+5*x)-189375*ln(2+3*x)+189375*ln(3+5*x)
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.03 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {21}{4 (2+3 x)^4}+\frac {103}{(2+3 x)^3}+\frac {1530}{(2+3 x)^2}+\frac {25350}{2+3 x}-\frac {1375}{2 (3+5 x)^2}+\frac {20875}{3+5 x}-189375 \log (2+3 x)+189375 \log (-3 (3+5 x)) \] Input:

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

Output:

21/(4*(2 + 3*x)^4) + 103/(2 + 3*x)^3 + 1530/(2 + 3*x)^2 + 25350/(2 + 3*x) 
- 1375/(2*(3 + 5*x)^2) + 20875/(3 + 5*x) - 189375*Log[2 + 3*x] + 189375*Lo 
g[-3*(3 + 5*x)]
 

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {86, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 86

\(\displaystyle \int \left (\frac {946875}{5 x+3}-\frac {104375}{(5 x+3)^2}+\frac {6875}{(5 x+3)^3}-\frac {568125}{3 x+2}-\frac {76050}{(3 x+2)^2}-\frac {9180}{(3 x+2)^3}-\frac {927}{(3 x+2)^4}-\frac {63}{(3 x+2)^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {25350}{3 x+2}+\frac {20875}{5 x+3}+\frac {1530}{(3 x+2)^2}-\frac {1375}{2 (5 x+3)^2}+\frac {103}{(3 x+2)^3}+\frac {21}{4 (3 x+2)^4}-189375 \log (3 x+2)+189375 \log (5 x+3)\)

Input:

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

Output:

21/(4*(2 + 3*x)^4) + 103/(2 + 3*x)^3 + 1530/(2 + 3*x)^2 + 25350/(2 + 3*x) 
- 1375/(2*(3 + 5*x)^2) + 20875/(3 + 5*x) - 189375*Log[2 + 3*x] + 189375*Lo 
g[3 + 5*x]
 

Defintions of rubi rules used

rule 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((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]))))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.77

method	result	size
norman	\(\frac {25565625 x^{5}[phone] x^{3}+\frac {44542717}{2} x +\frac {165324375}{2} x^{4}+\frac {276035525}{4} x^{2}+\frac {11492725}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-189375 \ln \left (2+3 x \right )+189375 \ln \left (3+5 x \right )\)	\(58\)
risch	\(\frac {25565625 x^{5}[phone] x^{3}+\frac {44542717}{2} x +\frac {165324375}{2} x^{4}+\frac {276035525}{4} x^{2}+\frac {11492725}{4}}{\left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}-189375 \ln \left (2+3 x \right )+189375 \ln \left (3+5 x \right )\)	\(59\)
default	\(\frac {21}{4 \left (2+3 x \right )^{4}}+\frac {103}{\left (2+3 x \right )^{3}}+\frac {1530}{\left (2+3 x \right )^{2}}+\frac {25350}{2+3 x}-\frac {1375}{2 \left (3+5 x \right )^{2}}+\frac {20875}{3+5 x}-189375 \ln \left (2+3 x \right )+189375 \ln \left (3+5 x \right )\)	\(72\)
parallelrisch	\(\frac {-2617919904 x [phone] \ln \left (x +\frac {3}{5}\right ) x^{2}-62854319400 x^{3}-97298349525 x^{4}-20288879800 x^{2}-75262236750 x^{5}-23272768125 x^{6}-854096400000 \ln \left (\frac {2}{3}+x \right ) x^{5}[phone] \ln \left (x +\frac {3}{5}\right ) x^{3}-1180681920000 \ln \left (\frac {2}{3}+x \right ) x^{3}-15707520000 \ln \left (\frac {2}{3}+x \right )-146603520000 \ln \left (\frac {2}{3}+x \right ) x -569833920000 \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (x +\frac {3}{5}\right ) x -1375389720000 \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (x +\frac {3}{5}\right ) x^{6}-220887000000 \ln \left (\frac {2}{3}+x \right ) x^{6}[phone] \ln \left (x +\frac {3}{5}\right ) x^{5}[phone] \ln \left (x +\frac {3}{5}\right )}{576 \left (2+3 x \right )^{4} \left (3+5 x \right )^{2}}\)	\(162\)

Input:

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

Output:

(25565625*x^5+106845375*x^3+44542717/2*x[phone]/2*x^4+276035525/4*x^2+1 
1492725/4)/(2+3*x)^4/(3+5*x)^2-189375*ln(2+3*x)+189375*ln(3+5*x)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.80 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {102262500 \, x^{5} + 330648750 \, x^{4} + 427381500 \, x^{3} + 276035525 \, x^{2} + 757500 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (5 \, x + 3\right ) - 757500 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )} \log \left (3 \, x + 2\right ) + 89085434 \, x + 11492725}{4 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} \] Input:

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

Output:

1/4*(102262500*x^5 + 330648750*x^4 + 427381500*x^3 + 276035525*x^2 + 75750 
0*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)* 
log(5*x + 3) - 757500*(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224* 
x^2 + 1344*x + 144)*log(3*x + 2) + 89085434*x + 11492725)/(2025*x^6 + 7830 
*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 + 1344*x + 144)
 

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.97 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=- \frac {- 102262500 x^{5} - 330648750 x^{4} - 427381500 x^{3} - 276035525 x^{2} - 89085434 x - 11492725}{8100 x^{6} + 31320 x^{5} + 50436 x^{4} + 43296 x^{3} + 20896 x^{2} + 5376 x + 576} + 189375 \log {\left (x + \frac {3}{5} \right )} - 189375 \log {\left (x + \frac {2}{3} \right )} \] Input:

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

Output:

-(-102262500*x**5 - 330648750*x**4 - 427381500*x**3 - 276035525*x**2 - 890 
85434*x - 11492725)/(8100*x**6 + 31320*x**5 + 50436*x**4 + 43296*x**3 + 20 
896*x**2 + 5376*x + 576) + 189375*log(x + 3/5) - 189375*log(x + 2/3)
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {102262500 \, x^{5} + 330648750 \, x^{4} + 427381500 \, x^{3} + 276035525 \, x^{2} + 89085434 \, x + 11492725}{4 \, {\left (2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144\right )}} + 189375 \, \log \left (5 \, x + 3\right ) - 189375 \, \log \left (3 \, x + 2\right ) \] Input:

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

Output:

1/4*(102262500*x^5 + 330648750*x^4 + 427381500*x^3 + 276035525*x^2 + 89085 
434*x + 11492725)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824*x^3 + 5224*x^2 
+ 1344*x + 144) + 189375*log(5*x + 3) - 189375*log(3*x + 2)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.01 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {25350}{3 \, x + 2} - \frac {9375 \, {\left (\frac {80}{3 \, x + 2} - 367\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + \frac {1530}{{\left (3 \, x + 2\right )}^{2}} + \frac {103}{{\left (3 \, x + 2\right )}^{3}} + \frac {21}{4 \, {\left (3 \, x + 2\right )}^{4}} + 189375 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \] Input:

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

Output:

25350/(3*x + 2) - 9375/2*(80/(3*x + 2) - 367)/(1/(3*x + 2) - 5)^2 + 1530/( 
3*x + 2)^2 + 103/(3*x + 2)^3 + 21/4/(3*x + 2)^4 + 189375*log(abs(-1/(3*x + 
 2) + 5))
 

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {12625\,x^5+\frac {244925\,x^4}{6}+\frac {1424605\,x^3}{27}+\frac {11041421\,x^2}{324}+\frac {44542717\,x}{4050}+\frac {459709}{324}}{x^6+\frac {58\,x^5}{15}+\frac {467\,x^4}{75}+\frac {3608\,x^3}{675}+\frac {5224\,x^2}{2025}+\frac {448\,x}{675}+\frac {16}{225}}-378750\,\mathrm {atanh}\left (30\,x+19\right ) \] Input:

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

Output:

((44542717*x)/4050 + (11041421*x^2)/324 + (1424605*x^3)/27 + (244925*x^4)/ 
6 + 12625*x^5 + 459709/324)/((448*x)/675 + (5224*x^2)/2025 + (3608*x^3)/67 
5 + (467*x^4)/75 + (58*x^5)/15 + x^6 + 16/225) - 378750*atanh(30*x + 19)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.69 \[ \int \frac {1-2 x}{(2+3 x)^5 (3+5 x)^3} \, dx=\frac {44484187500 \,\mathrm {log}\left (5 x +3\right ) x^{6}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{5}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{4}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{3}[phone] \,\mathrm {log}\left (5 x +3\right ) x^{2}[phone] \,\mathrm {log}\left (5 x +3\right ) x [phone] \,\mathrm {log}\left (5 x +3\right )-44484187500 \,\mathrm {log}\left (3 x +2\right ) x^{6}-172005525000 \,\mathrm {log}\left (3 x +2\right ) x^{5}-276988207500 \,\mathrm {log}\left (3 x +2\right ) x^{4}-237776220000 \,\mathrm {log}\left (3 x +2\right ) x^{3}-114758220000 \,\mathrm {log}\left (3 x +2\right ) x^{2}-29524320000 \,\mathrm {log}\left (3 x +2\right ) x -3163320000 \,\mathrm {log}\left (3 x +2\right )-766968750 x^{6}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]}{234900 x^{6}+908280 x^{5}+1462644 x^{4}+1255584 x^{3}+605984 x^{2}+155904 x +16704} \] Input:

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

Output:

(44484187500*log(5*x + 3)*x**6 + 172005525000*log(5*x + 3)*x**5 + 27698820 
7500*log(5*x + 3)*x**4 + 237776220000*log(5*x + 3)*x**3 + 114758220000*log 
(5*x + 3)*x**2 + 29524320000*log(5*x + 3)*x + 3163320000*log(5*x + 3) - 44 
484187500*log(3*x + 2)*x**6 - 172005525000*log(3*x + 2)*x**5 - 27698820750 
0*log(3*x + 2)*x**4 - 237776220000*log(3*x + 2)*x**3 - 114758220000*log(3* 
x + 2)*x**2 - 29524320000*log(3*x + 2)*x - 3163320000*log(3*x + 2) - 76696 
8750*x**6 + 4813155000*x**4 + 8294473500*x**3 + 6026440225*x**2 + 20744375 
86*x + 278749025)/(116*(2025*x**6 + 7830*x**5 + 12609*x**4 + 10824*x**3 + 
5224*x**2 + 1344*x + 144))