\(\int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx\) [1417]

Optimal result

Integrand size = 22, antiderivative size = 97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {49}{9 (2+3 x)^7}-\frac {1568}{27 (2+3 x)^6}-\frac {2541}{5 (2+3 x)^5}-\frac {8349}{2 (2+3 x)^4}-\frac {34485}{(2+3 x)^3}-\frac {308550}{(2+3 x)^2}-\frac {3584625}{2+3 x}-\frac {831875}{3+5 x}[phone] \log (2+3 x)-20418750 \log (3+5 x) \]

[Out]

-49/9/(2+3*x)^7-1568/27/(2+3*x)^6-2541/5/(2+3*x)^5-8349/2/(2+3*x)^4-34485/(2+3*x)^3-308550/(2+3*x)^2-3584625/(
2+3*x)-831875/(3+5*x)[phone]*ln(2+3*x)-20418750*ln(3+5*x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, 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} \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {3584625}{3 x+2}-\frac {831875}{5 x+3}-\frac {308550}{(3 x+2)^2}-\frac {34485}{(3 x+2)^3}-\frac {8349}{2 (3 x+2)^4}-\frac {2541}{5 (3 x+2)^5}-\frac {1568}{27 (3 x+2)^6}-\frac {49}{9 (3 x+2)^7}[phone] \log (3 x+2)-20418750 \log (5 x+3) \]

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[2 + 3*x] - 20418750*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}{3 (2+3 x)^8}+\frac {3136}{3 (2+3 x)^7}+\frac {7623}{(2+3 x)^6}+\frac {50094}{(2+3 x)^5}+\frac {310365}{(2+3 x)^4}+\frac {1851300}{(2+3 x)^3}+\frac {10753875}{(2+3 x)^2}+\frac {61256250}{2+3 x}+\frac {4159375}{(3+5 x)^2}-\frac {102093750}{3+5 x}\right ) \, dx \\ & = -\frac {49}{9 (2+3 x)^7}-\frac {1568}{27 (2+3 x)^6}-\frac {2541}{5 (2+3 x)^5}-\frac {8349}{2 (2+3 x)^4}-\frac {34485}{(2+3 x)^3}-\frac {308550}{(2+3 x)^2}-\frac {3584625}{2+3 x}-\frac {831875}{3+5 x}[phone] \log (2+3 x)-20418750 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {49}{9 (2+3 x)^7}-\frac {1568}{27 (2+3 x)^6}-\frac {2541}{5 (2+3 x)^5}-\frac {8349}{2 (2+3 x)^4}-\frac {34485}{(2+3 x)^3}-\frac {308550}{(2+3 x)^2}-\frac {3584625}{2+3 x}-\frac {831875}{3+5 x}[phone] \log (5 (2+3 x))-20418750 \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[5*(2 + 3*x)] - 20418750*Log[3 + 5*x
]

Maple [A] (verified)

Time = 2.46 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70

method	result	size
norman	\(\frac {-136933446375 x^{5}-68968411875 x^{6}-14885268750 x^{7}-\frac {1784716642958}{45} x^{2}-\frac {1180468875937}{135} x -\frac {302041582425}{2} x^{4}-\frac {199840400397}{2} x^{3}-\frac {37174982903}{45}}{\left (2+3 x \right )^{7} \left (3+5 x \right )}[phone] \ln \left (2+3 x \right )-20418750 \ln \left (3+5 x \right )\)	\(68\)
risch	\(\frac {-136933446375 x^{5}-68968411875 x^{6}-14885268750 x^{7}-\frac {1784716642958}{45} x^{2}-\frac {1180468875937}{135} x -\frac {302041582425}{2} x^{4}-\frac {199840400397}{2} x^{3}-\frac {37174982903}{45}}{\left (2+3 x \right )^{7} \left (3+5 x \right )}[phone] \ln \left (2+3 x \right )-20418750 \ln \left (3+5 x \right )\)	\(69\)
default	\(-\frac {49}{9 \left (2+3 x \right )^{7}}-\frac {1568}{27 \left (2+3 x \right )^{6}}-\frac {2541}{5 \left (2+3 x \right )^{5}}-\frac {8349}{2 \left (2+3 x \right )^{4}}-\frac {34485}{\left (2+3 x \right )^{3}}-\frac {308550}{\left (2+3 x \right )^{2}}-\frac {3584625}{2+3 x}-\frac {831875}{3+5 x}[phone] \ln \left (2+3 x \right )-20418750 \ln \left (3+5 x \right )\)	\(90\)
parallelrisch	\(\frac {2509056000320 x -974768256000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+2963822400000000 \ln \left (\frac {2}{3}+x \right ) x^{3}-183161088000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] x^{8}+5631262560000000 \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )+2257797564000000 \ln \left (\frac {2}{3}+x \right ) x^{7}-2257797564000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-15054336000000 \ln \left (x +\frac {3}{5}\right )+6846429744000000 \ln \left (\frac {2}{3}+x \right ) x^{5}-2963822400000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-6846429744000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-5631262560000000 \ln \left (x +\frac {3}{5}\right ) x^{4}+5201508312000000 \ln \left (\frac {2}{3}+x \right ) x^{6}-5201508312000000 \ln \left (x +\frac {3}{5}\right ) x^{6}[phone] \ln \left (\frac {2}{3}+x \right ) x^{8}-428695740000000 \ln \left (x +\frac {3}{5}\right ) x^{8}}{1920 \left (2+3 x \right )^{7} \left (3+5 x \right )}\)	\(208\)

[In]

int((1-2*x)^3/(2+3*x)^8/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

(-136933446375*x^5-68968411875*x^6-14885268750*x^7-1784716642958/45*x^2-1180468875937/135*x-302041582425/2*x^4
-199840400397/2*x^3-37174982903/45)/(2+3*x)^7/(3+5*x)[phone]*ln(2+3*x)-20418750*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {4019022562500 \, x^{7} + 18621471206250 \, x^{6} + 36972030521250 \, x^{5} + 40775613627375 \, x^{4} + 26978454053595 \, x^{3} + 10708299857748 \, x^{2} + 5513062500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (5 \, x + 3\right ) - 5513062500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (3 \, x + 2\right ) + 2360937751874 \, x + 223049897418}{270 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} \]

[In]

integrate((1-2*x)^3/(2+3*x)^8/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/270*(4019022562500*x^7 + 18621471206250*x^6 + 36972030521250*x^5 + 40775613627375*x^4 + 26978454053595*x^3
+ 10708299857748*x^2 + 5513062500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 +
24864*x^2 + 4672*x + 384)*log(5*x + 3) - 5513062500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*
x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)*log(3*x + 2) + 2360937751874*x + 223049897418)/(10935*x^8 + 57591*
x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=- \frac {4019022562500 x^{7} + 18621471206250 x^{6} + 36972030521250 x^{5} + 40775613627375 x^{4} + 26978454053595 x^{3} + 10708299857748 x^{2} + 2360937751874 x + 223049897418}{2952450 x^{8} + 15549570 x^{7} + 35823060 x^{6} + 47151720 x^{5} + 38782800 x^{4} + 20412000 x^{3} + 6713280 x^{2} + 1261440 x + 103680} - 20418750 \log {\left (x + \frac {3}{5} \right )} + 20418750 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**3/(2+3*x)**8/(3+5*x)**2,x)

[Out]

-(4019022562500*x**7 + 18621471206250*x**6 + 36972030521250*x**5 + 40775613627375*x**4 + 26978454053595*x**3 +
10708299857748*x**2 + 2360937751874*x + 223049897418)/(2952450*x**8 + 15549570*x**7 + 35823060*x**6 + 4715172
0*x**5 + 38782800*x**4 + 20412000*x**3 + 6713280*x**2 + 1261440*x + 103680) - 20418750*log(x + 3/5) + 20418750
*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.99 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {4019022562500 \, x^{7} + 18621471206250 \, x^{6} + 36972030521250 \, x^{5} + 40775613627375 \, x^{4} + 26978454053595 \, x^{3} + 10708299857748 \, x^{2} + 2360937751874 \, x + 223049897418}{270 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} - 20418750 \, \log \left (5 \, x + 3\right ) + 20418750 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^8/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/270*(4019022562500*x^7 + 18621471206250*x^6 + 36972030521250*x^5 + 40775613627375*x^4 + 26978454053595*x^3
+ 10708299857748*x^2 + 2360937751874*x + 223049897418)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 1436
40*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384) - 20418750*log(5*x + 3) + 20418750*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=-\frac {831875}{5 \, x + 3} + \frac {625 \, {\left (\frac {537521373}{5 \, x + 3} + \frac {489712095}{{\left (5 \, x + 3\right )}^{2}} + \frac {241051911}{{\left (5 \, x + 3\right )}^{3}} + \frac {67932770}{{\left (5 \, x + 3\right )}^{4}} + \frac {10476370}{{\left (5 \, x + 3\right )}^{5}} + \frac {701580}{{\left (5 \, x + 3\right )}^{6}} + 248285331\right )}}{2 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{7}} + 20418750 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^8/(3+5*x)^2,x, algorithm="giac")

[Out]

-831875/(5*x + 3) + 625/2*(537521373/(5*x + 3) + 489712095/(5*x + 3)^2 + 241051911/(5*x + 3)^3 + 67932770/(5*x
+ 3)^4 + 10476370/(5*x + 3)^5 + 701580/(5*x + 3)^6 + 248285331)/(1/(5*x + 3) + 3)^7 + 20418750*log(abs(-1/(5*
x + 3) - 3))

Mupad [B] (verification not implemented)

Time = 1.23 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx=40837500\,\mathrm {atanh}\left (30\,x+19\right )-\frac {1361250\,x^7+6307125\,x^6+\frac {37567475\,x^5}{3}+\frac {248593895\,x^4}{18}+\frac {274129493\,x^3}{30}+\frac {1784716642958\,x^2}{492075}+\frac {1180468875937\,x}{1476225}+\frac {37174982903}{492075}}{x^8+\frac {79\,x^7}{15}+\frac {182\,x^6}{15}+\frac {2156\,x^5}{135}+\frac {1064\,x^4}{81}+\frac {560\,x^3}{81}+\frac {8288\,x^2}{3645}+\frac {4672\,x}{10935}+\frac {128}{3645}} \]

[In]

int(-(2*x - 1)^3/((3*x + 2)^8*(5*x + 3)^2),x)

[Out]

40837500*atanh(30*x + 19) - ((1180468875937*x)/1476225 + (1784716642958*x^2)/492075 + (274129493*x^3)/30 + (24
8593895*x^4)/18 + (37567475*x^5)/3 + 6307125*x^6 + 1361250*x^7 + 37174982903/492075)/((4672*x)/10935 + (8288*x
^2)/3645 + (560*x^3)/81 + (1064*x^4)/81 + (2156*x^5)/135 + (182*x^6)/15 + (79*x^7)/15 + x^8 + 128/3645)