\(\int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx\) [1431]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {343}{15 (2+3 x)^5}+\frac {1617}{4 (2+3 x)^4}+\frac {5236}{(2+3 x)^3}+\frac {64317}{(2+3 x)^2}+\frac {953535}{2+3 x}-\frac {33275}{2 (3+5 x)^2}+\frac {617100}{3+5 x}-6618975 \log (2+3 x)+6618975 \log (3+5 x) \]

[Out]

343/15/(2+3*x)^5+1617/4/(2+3*x)^4+5236/(2+3*x)^3+64317/(2+3*x)^2+953535/(2+3*x)-33275/2/(3+5*x)^2+617100/(3+5*
x)-6618975*ln(2+3*x)+6618975*ln(3+5*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {953535}{3 x+2}+\frac {617100}{5 x+3}+\frac {64317}{(3 x+2)^2}-\frac {33275}{2 (5 x+3)^2}+\frac {5236}{(3 x+2)^3}+\frac {1617}{4 (3 x+2)^4}+\frac {343}{15 (3 x+2)^5}-6618975 \log (3 x+2)+6618975 \log (5 x+3) \]

[In]

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

[Out]

343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/
(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6618975*Log[2 + 3*x] + 6618975*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 {343}{(2+3 x)^6}-\frac {4851}{(2+3 x)^5}-\frac {47124}{(2+3 x)^4}-\frac {385902}{(2+3 x)^3}-\frac {2860605}{(2+3 x)^2}-\frac {19856925}{2+3 x}+\frac {166375}{(3+5 x)^3}-\frac {3085500}{(3+5 x)^2}+\frac {33094875}{3+5 x}\right ) \, dx \\ & = \frac {343}{15 (2+3 x)^5}+\frac {1617}{4 (2+3 x)^4}+\frac {5236}{(2+3 x)^3}+\frac {64317}{(2+3 x)^2}+\frac {953535}{2+3 x}-\frac {33275}{2 (3+5 x)^2}+\frac {617100}{3+5 x}-6618975 \log (2+3 x)+6618975 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {343}{15 (2+3 x)^5}+\frac {1617}{4 (2+3 x)^4}+\frac {5236}{(2+3 x)^3}+\frac {64317}{(2+3 x)^2}+\frac {953535}{2+3 x}-\frac {33275}{2 (3+5 x)^2}+\frac {617100}{3+5 x}-6618975 \log (5 (2+3 x))+6618975 \log (3+5 x) \]

[In]

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

[Out]

343/(15*(2 + 3*x)^5) + 1617/(4*(2 + 3*x)^4) + 5236/(2 + 3*x)^3 + 64317/(2 + 3*x)^2 + 953535/(2 + 3*x) - 33275/
(2*(3 + 5*x)^2) + 617100/(3 + 5*x) - 6618975*Log[5*(2 + 3*x)] + 6618975*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.49 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73

method	result	size
norman	\(\frac {2680684875 x^{6}[phone] x^{4}+\frac {7432441967}{4} x +\frac {20909342025}{2} x^{5}+\frac {21477652514}{3} x^{2}+\frac {58819124199}{4} x^{3}+\frac {2008450423}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-6618975 \ln \left (2+3 x \right )+6618975 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {2680684875 x^{6}[phone] x^{4}+\frac {7432441967}{4} x +\frac {20909342025}{2} x^{5}+\frac {21477652514}{3} x^{2}+\frac {58819124199}{4} x^{3}+\frac {2008450423}{10}}{\left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}-6618975 \ln \left (2+3 x \right )+6618975 \ln \left (3+5 x \right )\)	\(64\)
default	\(\frac {343}{15 \left (2+3 x \right )^{5}}+\frac {1617}{4 \left (2+3 x \right )^{4}}+\frac {5236}{\left (2+3 x \right )^{3}}+\frac {64317}{\left (2+3 x \right )^{2}}+\frac {953535}{2+3 x}-\frac {33275}{2 \left (3+5 x \right )^{2}}+\frac {617100}{3+5 x}-6618975 \ln \left (2+3 x \right )+6618975 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(-\frac {915007103520 x -276027143040000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-59475461760000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}+1099724163120000 \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-115805586600000 \ln \left (x +\frac {3}{5}\right ) x^{7}-5490042624000 \ln \left (x +\frac {3}{5}\right )+1019603853576000 \ln \left (\frac {2}{3}+x \right ) x^{5}-711418023360000 \ln \left (x +\frac {3}{5}\right ) x^{3}-1019603853576000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1099724163120000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-524985325920000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{2880 \left (2+3 x \right )^{5} \left (3+5 x \right )^{2}}\)	\(185\)

[In]

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

[Out]

(2680684875*x^6+16981642260*x^4+7432441967/4*x+20909342025/2*x^5+21477652514/3*x^2+58819124199/4*x^3+200845042
3/10)/(2+3*x)^5/(3+5*x)^2-6618975*ln(2+3*x)+6618975*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 397138500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (5 \, x + 3\right ) - 397138500 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )} \log \left (3 \, x + 2\right ) + 111486629505 \, x + 12050702538}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} \]

[In]

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

[Out]

1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 882286862985*x^3 + 429553050280*x^2 + 39713850
0*(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) - 3971385
00*(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) + 111486
629505*x + 12050702538)/(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 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=- \frac {- 160841092500 x^{6} - 627280260750 x^{5} - 1018898535600 x^{4} - 882286862985 x^{3} - 429553050280 x^{2} - 111486629505 x - 12050702538}{364500 x^{7} + 1652400 x^{6} + 3209220 x^{5} + 3461400 x^{4} + 2239200 x^{3} + 868800 x^{2} + 187200 x + 17280} + 6618975 \log {\left (x + \frac {3}{5} \right )} - 6618975 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

-(-160841092500*x**6 - 627280260750*x**5 - 1018898535600*x**4 - 882286862985*x**3 - 429553050280*x**2 - 111486
629505*x - 12050702538)/(364500*x**7 + 1652400*x**6 + 3209220*x**5 + 3461400*x**4 + 2239200*x**3 + 868800*x**2
+ 187200*x + 17280) + 6618975*log(x + 3/5) - 6618975*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 111486629505 \, x + 12050702538}{60 \, {\left (6075 \, x^{7} + 27540 \, x^{6} + 53487 \, x^{5} + 57690 \, x^{4} + 37320 \, x^{3} + 14480 \, x^{2} + 3120 \, x + 288\right )}} + 6618975 \, \log \left (5 \, x + 3\right ) - 6618975 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 882286862985*x^3 + 429553050280*x^2 + 11148662
9505*x + 12050702538)/(6075*x^7 + 27540*x^6 + 53487*x^5 + 57690*x^4 + 37320*x^3 + 14480*x^2 + 3120*x + 288) +
6618975*log(5*x + 3) - 6618975*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {160841092500 \, x^{6} + 627280260750 \, x^{5} + 1018898535600 \, x^{4} + 882286862985 \, x^{3} + 429553050280 \, x^{2} + 111486629505 \, x + 12050702538}{60 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{5}} + 6618975 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 6618975 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

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

[Out]

1/60*(160841092500*x^6 + 627280260750*x^5 + 1018898535600*x^4 + 882286862985*x^3 + 429553050280*x^2 + 11148662
9505*x + 12050702538)/((5*x + 3)^2*(3*x + 2)^5) + 6618975*log(abs(5*x + 3)) - 6618975*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^3} \, dx=\frac {441265\,x^6+\frac {3441867\,x^5}{2}+\frac {377369828\,x^4}{135}+\frac {19606374733\,x^3}{8100}+\frac {21477652514\,x^2}{18225}+\frac {7432441967\,x}{24300}+\frac {2008450423}{60750}}{x^7+\frac {68\,x^6}{15}+\frac {1981\,x^5}{225}+\frac {1282\,x^4}{135}+\frac {2488\,x^3}{405}+\frac {2896\,x^2}{1215}+\frac {208\,x}{405}+\frac {32}{675}}-13237950\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((7432441967*x)/24300 + (21477652514*x^2)/18225 + (19606374733*x^3)/8100 + (377369828*x^4)/135 + (3441867*x^5)
/2 + 441265*x^6 + 2008450423/60750)/((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) - 13237950*atanh(30*x + 19)