\(\int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx\) [1401]

Optimal result

Integrand size = 22, antiderivative size = 81 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {343}{162 (2+3 x)^6}+\frac {1421}{135 (2+3 x)^5}+\frac {7189}{108 (2+3 x)^4}+\frac {1331}{3 (2+3 x)^3}+\frac {6655}{2 (2+3 x)^2}+\frac {33275}{2+3 x}-166375 \log (2+3 x)+166375 \log (3+5 x) \]

[Out]

343/162/(2+3*x)^6+1421/135/(2+3*x)^5+7189/108/(2+3*x)^4+1331/3/(2+3*x)^3+6655/2/(2+3*x)^2+33275/(2+3*x)-166375
*ln(2+3*x)+166375*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 81, 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} \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {33275}{3 x+2}+\frac {6655}{2 (3 x+2)^2}+\frac {1331}{3 (3 x+2)^3}+\frac {7189}{108 (3 x+2)^4}+\frac {1421}{135 (3 x+2)^5}+\frac {343}{162 (3 x+2)^6}-166375 \log (3 x+2)+166375 \log (5 x+3) \]

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)),x]

[Out]

343/(162*(2 + 3*x)^6) + 1421/(135*(2 + 3*x)^5) + 7189/(108*(2 + 3*x)^4) + 1331/(3*(2 + 3*x)^3) + 6655/(2*(2 +
3*x)^2) + 33275/(2 + 3*x) - 166375*Log[2 + 3*x] + 166375*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}{9 (2+3 x)^7}-\frac {1421}{9 (2+3 x)^6}-\frac {7189}{9 (2+3 x)^5}-\frac {3993}{(2+3 x)^4}-\frac {19965}{(2+3 x)^3}-\frac {99825}{(2+3 x)^2}-\frac {499125}{2+3 x}+\frac {831875}{3+5 x}\right ) \, dx \\ & = \frac {343}{162 (2+3 x)^6}+\frac {1421}{135 (2+3 x)^5}+\frac {7189}{108 (2+3 x)^4}+\frac {1331}{3 (2+3 x)^3}+\frac {6655}{2 (2+3 x)^2}+\frac {33275}{2+3 x}-166375 \log (2+3 x)+166375 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {3430+17052 (2+3 x)+107835 (2+3 x)^2+718740 (2+3 x)^3+5390550 (2+3 x)^4+53905500 (2+3 x)^5}{1620 (2+3 x)^6}-166375 \log (5 (2+3 x))+166375 \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)),x]

[Out]

(3430 + 17052*(2 + 3*x) + 107835*(2 + 3*x)^2 + 718740*(2 + 3*x)^3 + 5390550*(2 + 3*x)^4 + 53905500*(2 + 3*x)^5
)/(1620*(2 + 3*x)^6) - 166375*Log[5*(2 + 3*x)] + 166375*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63

method	result	size
norman	\(\frac {8085825 x^{5}[phone] x^{3}+\frac {54444555}{2} x^{4}+\frac {296415565}{12} x^{2}+\frac {374500906}{45} x +\frac {908721797}{810}}{\left (2+3 x \right )^{6}}-166375 \ln \left (2+3 x \right )+166375 \ln \left (3+5 x \right )\)	\(51\)
risch	\(\frac {8085825 x^{5}[phone] x^{3}+\frac {54444555}{2} x^{4}+\frac {296415565}{12} x^{2}+\frac {374500906}{45} x +\frac {908721797}{810}}{\left (2+3 x \right )^{6}}-166375 \ln \left (2+3 x \right )+166375 \ln \left (3+5 x \right )\)	\(52\)
default	\(\frac {343}{162 \left (2+3 x \right )^{6}}+\frac {1421}{135 \left (2+3 x \right )^{5}}+\frac {7189}{108 \left (2+3 x \right )^{4}}+\frac {1331}{3 \left (2+3 x \right )^{3}}+\frac {6655}{2 \left (2+3 x \right )^{2}}+\frac {33275}{2+3 x}-166375 \ln \left (2+3 x \right )+166375 \ln \left (3+5 x \right )\)	\(72\)
parallelrisch	\(-\frac {3407359680 x -689990400000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-183997440000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )-20444160000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-1379980800000 \ln \left (x +\frac {3}{5}\right ) x^{3}-931487040000 \ln \left (x +\frac {3}{5}\right ) x^{5}-1552478400000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-232871760000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{1920 \left (2+3 x \right )^{6}}\)	\(155\)

[In]

int((1-2*x)^3/(2+3*x)^7/(3+5*x),x,method=_RETURNVERBOSE)

[Out]

(8085825*x^5+36667719*x^3+54444555/2*x^4+296415565/12*x^2+374500906/45*x+908721797/810)/(2+3*x)^6-166375*ln(2+
3*x)+166375*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {13099036500 \, x^{5} + 44100089550 \, x^{4} + 59401704780 \, x^{3} + 40016101275 \, x^{2} + 269527500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 269527500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 13482032616 \, x + 1817443594}{1620 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x),x, algorithm="fricas")

[Out]

1/1620*(13099036500*x^5 + 44100089550*x^4 + 59401704780*x^3 + 40016101275*x^2 + 269527500*(729*x^6 + 2916*x^5
+ 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*log(5*x + 3) - 269527500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*
x^3 + 2160*x^2 + 576*x + 64)*log(3*x + 2) + 13482032616*x + 1817443594)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*
x^3 + 2160*x^2 + 576*x + 64)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=- \frac {- 13099036500 x^{5} - 44100089550 x^{4} - 59401704780 x^{3} - 40016101275 x^{2} - 13482032616 x - 1817443594}{1180980 x^{6} + 4723920 x^{5} + 7873200 x^{4} + 6998400 x^{3} + 3499200 x^{2} + 933120 x + 103680} + 166375 \log {\left (x + \frac {3}{5} \right )} - 166375 \log {\left (x + \frac {2}{3} \right )} \]

[In]

integrate((1-2*x)**3/(2+3*x)**7/(3+5*x),x)

[Out]

-(-13099036500*x**5 - 44100089550*x**4 - 59401704780*x**3 - 40016101275*x**2 - 13482032616*x - 1817443594)/(11
80980*x**6 + 4723920*x**5 + 7873200*x**4 + 6998400*x**3 + 3499200*x**2 + 933120*x + 103680) + 166375*log(x + 3
/5) - 166375*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {13099036500 \, x^{5} + 44100089550 \, x^{4} + 59401704780 \, x^{3} + 40016101275 \, x^{2} + 13482032616 \, x + 1817443594}{1620 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 166375 \, \log \left (5 \, x + 3\right ) - 166375 \, \log \left (3 \, x + 2\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x),x, algorithm="maxima")

[Out]

1/1620*(13099036500*x^5 + 44100089550*x^4 + 59401704780*x^3 + 40016101275*x^2 + 13482032616*x + 1817443594)/(7
29*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 166375*log(5*x + 3) - 166375*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {13099036500 \, x^{5} + 44100089550 \, x^{4} + 59401704780 \, x^{3} + 40016101275 \, x^{2} + 13482032616 \, x + 1817443594}{1620 \, {\left (3 \, x + 2\right )}^{6}} + 166375 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 166375 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \]

[In]

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x),x, algorithm="giac")

[Out]

1/1620*(13099036500*x^5 + 44100089550*x^4 + 59401704780*x^3 + 40016101275*x^2 + 13482032616*x + 1817443594)/(3
*x + 2)^6 + 166375*log(abs(5*x + 3)) - 166375*log(abs(3*x + 2))

Mupad [B] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)} \, dx=\frac {\frac {33275\,x^5}{3}+\frac {672155\,x^4}{18}+\frac {4074191\,x^3}{81}+\frac {296415565\,x^2}{8748}+\frac {374500906\,x}{32805}+\frac {908721797}{590490}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}}-332750\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

int(-(2*x - 1)^3/((3*x + 2)^7*(5*x + 3)),x)

[Out]

((374500906*x)/32805 + (296415565*x^2)/8748 + (4074191*x^3)/81 + (672155*x^4)/18 + (33275*x^5)/3 + 908721797/5
90490)/((64*x)/81 + (80*x^2)/27 + (160*x^3)/27 + (20*x^4)/3 + 4*x^5 + x^6 + 64/729) - 332750*atanh(30*x + 19)