\(\int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx\) [1319]

Optimal result

Integrand size = 22, antiderivative size = 90 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {49}{18 (2+3 x)^6}-\frac {154}{5 (2+3 x)^5}-\frac {1133}{4 (2+3 x)^4}-\frac {7480}{3 (2+3 x)^3}-\frac {46475}{2 (2+3 x)^2}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \log (2+3 x)-1615625 \log (3+5 x) \]

[Out]

-49/18/(2+3*x)^6-154/5/(2+3*x)^5-1133/4/(2+3*x)^4-7480/3/(2+3*x)^3-46475/2/(2+3*x)^2-277750/(2+3*x)-75625/(3+5
*x)+1615625*ln(2+3*x)-1615625*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 90, 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} \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {277750}{3 x+2}-\frac {75625}{5 x+3}-\frac {46475}{2 (3 x+2)^2}-\frac {7480}{3 (3 x+2)^3}-\frac {1133}{4 (3 x+2)^4}-\frac {154}{5 (3 x+2)^5}-\frac {49}{18 (3 x+2)^6}+1615625 \log (3 x+2)-1615625 \log (5 x+3) \]

[In]

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

[Out]

-49/(18*(2 + 3*x)^6) - 154/(5*(2 + 3*x)^5) - 1133/(4*(2 + 3*x)^4) - 7480/(3*(2 + 3*x)^3) - 46475/(2*(2 + 3*x)^
2) - 277750/(2 + 3*x) - 75625/(3 + 5*x) + 1615625*Log[2 + 3*x] - 1615625*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 {49}{(2+3 x)^7}+\frac {462}{(2+3 x)^6}+\frac {3399}{(2+3 x)^5}+\frac {22440}{(2+3 x)^4}+\frac {139425}{(2+3 x)^3}+\frac {833250}{(2+3 x)^2}+\frac {4846875}{2+3 x}+\frac {378125}{(3+5 x)^2}-\frac {8078125}{3+5 x}\right ) \, dx \\ & = -\frac {49}{18 (2+3 x)^6}-\frac {154}{5 (2+3 x)^5}-\frac {1133}{4 (2+3 x)^4}-\frac {7480}{3 (2+3 x)^3}-\frac {46475}{2 (2+3 x)^2}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \log (2+3 x)-1615625 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {49}{18 (2+3 x)^6}-\frac {154}{5 (2+3 x)^5}-\frac {1133}{4 (2+3 x)^4}-\frac {7480}{3 (2+3 x)^3}-\frac {46475}{2 (2+3 x)^2}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \log (5 (2+3 x))-1615625 \log (3+5 x) \]

[In]

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

[Out]

-49/(18*(2 + 3*x)^6) - 154/(5*(2 + 3*x)^5) - 1133/(4*(2 + 3*x)^4) - 7480/(3*(2 + 3*x)^3) - 46475/(2*(2 + 3*x)^
2) - 277750/(2 + 3*x) - 75625/(3 + 5*x) + 1615625*Log[5*(2 + 3*x)] - 1615625*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70

method	result	size
norman	\(\frac {-392596875 x^{6}-\frac {26722518673}{90} x -\frac {9070209225}{4} x^{3}-\frac {5146799625}{2} x^{4}-\frac {4494718899}{4} x^{2}-\frac {3114601875}{2} x^{5}-\frac {980484959}{30}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+1615625 \ln \left (2+3 x \right )-1615625 \ln \left (3+5 x \right )\)	\(63\)
risch	\(\frac {-392596875 x^{6}-\frac {26722518673}{90} x -\frac {9070209225}{4} x^{3}-\frac {5146799625}{2} x^{4}-\frac {4494718899}{4} x^{2}-\frac {3114601875}{2} x^{5}-\frac {980484959}{30}}{\left (2+3 x \right )^{6} \left (3+5 x \right )}+1615625 \ln \left (2+3 x \right )-1615625 \ln \left (3+5 x \right )\)	\(64\)
default	\(-\frac {49}{18 \left (2+3 x \right )^{6}}-\frac {154}{5 \left (2+3 x \right )^{5}}-\frac {1133}{4 \left (2+3 x \right )^{4}}-\frac {7480}{3 \left (2+3 x \right )^{3}}-\frac {46475}{2 \left (2+3 x \right )^{2}}-\frac {277750}{2+3 x}-\frac {75625}{3+5 x}+1615625 \ln \left (2+3 x \right )-1615625 \ln \left (3+5 x \right )\)	\(81\)
parallelrisch	\(\frac {99264000320 x -29034720000000 \ln \left (x +\frac {3}{5}\right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-6352896000000 \ln \left (x +\frac {3}{5}\right ) x [phone] \ln \left (\frac {2}{3}+x \right ) x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x [phone] x^{5}[phone] x^{6}[phone] x^{7}[phone] x^{3}[phone] x^{4}[phone] x^{2}[phone] \ln \left (\frac {2}{3}+x \right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{7}-11306790000000 \ln \left (x +\frac {3}{5}\right ) x^{7}-595584000000 \ln \left (x +\frac {3}{5}\right )[phone] \ln \left (\frac {2}{3}+x \right ) x^{5}-73703520000000 \ln \left (x +\frac {3}{5}\right ) x^{3}-102514896000000 \ln \left (x +\frac {3}{5}\right ) x^{5}-112230360000000 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{6}-52011234000000 \ln \left (x +\frac {3}{5}\right ) x^{6}}{1920 \left (2+3 x \right )^{6} \left (3+5 x \right )}\)	\(185\)

[In]

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

[Out]

(-392596875*x^6-26722518673/90*x-9070209225/4*x^3-5146799625/2*x^4-4494718899/4*x^2-3114601875/2*x^5-980484959
/30)/(2+3*x)^6/(3+5*x)+1615625*ln(2+3*x)-1615625*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.72 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {70667437500 \, x^{6} + 280314168750 \, x^{5} + 463211966250 \, x^{4} + 408159415125 \, x^{3} + 202262350455 \, x^{2} + 290812500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 290812500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 53445037346 \, x + 5882909754}{180 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \]

[In]

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

[Out]

-1/180*(70667437500*x^6 + 280314168750*x^5 + 463211966250*x^4 + 408159415125*x^3 + 202262350455*x^2 + 29081250
0*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 29081250
0*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(3*x + 2) + 53445037
346*x + 5882909754)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=\frac {- 70667437500 x^{6} - 280314168750 x^{5} - 463211966250 x^{4} - 408159415125 x^{3} - 202262350455 x^{2} - 53445037346 x - 5882909754}{656100 x^{7} + 3018060 x^{6} + 5948640 x^{5} + 6512400 x^{4} + 4276800 x^{3} + 1684800 x^{2} + 368640 x + 34560} - 1615625 \log {\left (x + \frac {3}{5} \right )} + 1615625 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

(-70667437500*x**6 - 280314168750*x**5 - 463211966250*x**4 - 408159415125*x**3 - 202262350455*x**2 - 534450373
46*x - 5882909754)/(656100*x**7 + 3018060*x**6 + 5948640*x**5 + 6512400*x**4 + 4276800*x**3 + 1684800*x**2 + 3
68640*x + 34560) - 1615625*log(x + 3/5) + 1615625*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {70667437500 \, x^{6} + 280314168750 \, x^{5} + 463211966250 \, x^{4} + 408159415125 \, x^{3} + 202262350455 \, x^{2} + 53445037346 \, x + 5882909754}{180 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 1615625 \, \log \left (5 \, x + 3\right ) + 1615625 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/180*(70667437500*x^6 + 280314168750*x^5 + 463211966250*x^4 + 408159415125*x^3 + 202262350455*x^2 + 53445037
346*x + 5882909754)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192) - 161
5625*log(5*x + 3) + 1615625*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=-\frac {75625}{5 \, x + 3} + \frac {625 \, {\left (\frac {22074930}{5 \, x + 3} + \frac {16294797}{{\left (5 \, x + 3\right )}^{2}} + \frac {6120660}{{\left (5 \, x + 3\right )}^{3}} + \frac {1179210}{{\left (5 \, x + 3\right )}^{4}} + \frac {94660}{{\left (5 \, x + 3\right )}^{5}} + 12117357\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 1615625 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

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

[Out]

-75625/(5*x + 3) + 625/4*(22074930/(5*x + 3) + 16294797/(5*x + 3)^2 + 6120660/(5*x + 3)^3 + 1179210/(5*x + 3)^
4 + 94660/(5*x + 3)^5 + 12117357)/(1/(5*x + 3) + 3)^6 + 1615625*log(abs(-1/(5*x + 3) - 3))

Mupad [B] (verification not implemented)

Time = 1.18 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)^2} \, dx=3231250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {323125\,x^6}{3}+\frac {7690375\,x^5}{18}+\frac {114373325\,x^4}{162}+\frac {67186735\,x^3}{108}+\frac {499413211\,x^2}{1620}+\frac {26722518673\,x}{328050}+\frac {980484959}{109350}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \]

[In]

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

[Out]

3231250*atanh(30*x + 19) - ((26722518673*x)/328050 + (499413211*x^2)/1620 + (67186735*x^3)/108 + (114373325*x^
4)/162 + (7690375*x^5)/18 + (323125*x^6)/3 + 980484959/109350)/((2048*x)/3645 + (208*x^2)/81 + (176*x^3)/27 +
(268*x^4)/27 + (136*x^5)/15 + (23*x^6)/5 + x^7 + 64/1215)