Integrand size = 22, antiderivative size = 81 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {49}{54 (2+3 x)^6}+\frac {217}{45 (2+3 x)^5}+\frac {121}{4 (2+3 x)^4}+\frac {605}{3 (2+3 x)^3}+\frac {3025}{2 (2+3 x)^2}+\frac {15125}{2+3 x}-75625 \log (2+3 x)+75625 \log (3+5 x) \] Output:
49/54/(2+3*x)^6+217/45/(2+3*x)^5+121/4/(2+3*x)^4+605/3/(2+3*x)^3+3025/2/(2 +3*x)^2+15125/(2+3*x)-75625*ln(2+3*x)+75625*ln(3+5*x)
Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {275370238+2042732232 x+6063045615 x^2+9000258300 x^3+6681831750 x^4+1984702500 x^5}{540 (2+3 x)^6}-75625 \log (5 (2+3 x))+75625 \log (3+5 x) \] Input:
Integrate[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)),x]
Output:
(275370238 + 2042732232*x + 6063045615*x^2 + 9000258300*x^3 + 6681831750*x ^4 + 1984702500*x^5)/(540*(2 + 3*x)^6) - 75625*Log[5*(2 + 3*x)] + 75625*Lo g[3 + 5*x]
Time = 0.25 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {99, 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.
\(\displaystyle \int \frac {(1-2 x)^2}{(3 x+2)^7 (5 x+3)} \, dx\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (\frac {378125}{5 x+3}-\frac {226875}{3 x+2}-\frac {45375}{(3 x+2)^2}-\frac {9075}{(3 x+2)^3}-\frac {1815}{(3 x+2)^4}-\frac {363}{(3 x+2)^5}-\frac {217}{3 (3 x+2)^6}-\frac {49}{3 (3 x+2)^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {15125}{3 x+2}+\frac {3025}{2 (3 x+2)^2}+\frac {605}{3 (3 x+2)^3}+\frac {121}{4 (3 x+2)^4}+\frac {217}{45 (3 x+2)^5}+\frac {49}{54 (3 x+2)^6}-75625 \log (3 x+2)+75625 \log (5 x+3)\) |
Input:
Int[(1 - 2*x)^2/((2 + 3*x)^7*(3 + 5*x)),x]
Output:
49/(54*(2 + 3*x)^6) + 217/(45*(2 + 3*x)^5) + 121/(4*(2 + 3*x)^4) + 605/(3* (2 + 3*x)^3) + 3025/(2*(2 + 3*x)^2) + 15125/(2 + 3*x) - 75625*Log[2 + 3*x] + 75625*Log[3 + 5*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Time = 0.22 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63
method | result | size |
norman | \(\frac {3675375 x^{5}+16667145 x^{3}+\frac {24747525}{2} x^{4}+\frac {44911449}{4} x^{2}+\frac {56742562}{15} x +\frac {137685119}{270}}{\left (2+3 x \right )^{6}}-75625 \ln \left (2+3 x \right )+75625 \ln \left (3+5 x \right )\) | \(51\) |
risch | \(\frac {3675375 x^{5}+16667145 x^{3}+\frac {24747525}{2} x^{4}+\frac {44911449}{4} x^{2}+\frac {56742562}{15} x +\frac {137685119}{270}}{\left (2+3 x \right )^{6}}-75625 \ln \left (2+3 x \right )+75625 \ln \left (3+5 x \right )\) | \(52\) |
default | \(\frac {49}{54 \left (2+3 x \right )^{6}}+\frac {217}{45 \left (2+3 x \right )^{5}}+\frac {121}{4 \left (2+3 x \right )^{4}}+\frac {605}{3 \left (2+3 x \right )^{3}}+\frac {3025}{2 \left (2+3 x \right )^{2}}+\frac {15125}{2+3 x}-75625 \ln \left (2+3 x \right )+75625 \ln \left (3+5 x \right )\) | \(72\) |
parallelrisch | \(-\frac {516266560 x -104544000000 \ln \left (x +\frac {3}{5}\right ) x^{2}+11362646240 x^{3}+16864113420 x^{4}+3828977680 x^{2}+12517752852 x^{5}+3717498213 x^{6}+141134400000 \ln \left (\frac {2}{3}+x \right ) x^{5}-209088000000 \ln \left (x +\frac {3}{5}\right ) x^{3}+209088000000 \ln \left (\frac {2}{3}+x \right ) x^{3}+3097600000 \ln \left (\frac {2}{3}+x \right )+27878400000 \ln \left (\frac {2}{3}+x \right ) x +104544000000 \ln \left (\frac {2}{3}+x \right ) x^{2}+35283600000 \ln \left (\frac {2}{3}+x \right ) x^{6}-35283600000 \ln \left (x +\frac {3}{5}\right ) x^{6}-27878400000 \ln \left (x +\frac {3}{5}\right ) x +235224000000 \ln \left (\frac {2}{3}+x \right ) x^{4}-235224000000 \ln \left (x +\frac {3}{5}\right ) x^{4}-141134400000 \ln \left (x +\frac {3}{5}\right ) x^{5}-3097600000 \ln \left (x +\frac {3}{5}\right )}{640 \left (2+3 x \right )^{6}}\) | \(155\) |
Input:
int((1-2*x)^2/(2+3*x)^7/(3+5*x),x,method=_RETURNVERBOSE)
Output:
(3675375*x^5+16667145*x^3+24747525/2*x^4+44911449/4*x^2+56742562/15*x+1376 85119/270)/(2+3*x)^6-75625*ln(2+3*x)+75625*ln(3+5*x)
Time = 0.07 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.67 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {1984702500 \, x^{5} + 6681831750 \, x^{4} + 9000258300 \, x^{3} + 6063045615 \, x^{2} + 40837500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (5 \, x + 3\right ) - 40837500 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \log \left (3 \, x + 2\right ) + 2042732232 \, x + 275370238}{540 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \] Input:
integrate((1-2*x)^2/(2+3*x)^7/(3+5*x),x, algorithm="fricas")
Output:
1/540*(1984702500*x^5 + 6681831750*x^4 + 9000258300*x^3 + 6063045615*x^2 + 40837500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 6 4)*log(5*x + 3) - 40837500*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216 0*x^2 + 576*x + 64)*log(3*x + 2) + 2042732232*x + 275370238)/(729*x^6 + 29 16*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {1984702500 x^{5} + 6681831750 x^{4} + 9000258300 x^{3} + 6063045615 x^{2} + 2042732232 x + 275370238}{393660 x^{6} + 1574640 x^{5} + 2624400 x^{4} + 2332800 x^{3} + 1166400 x^{2} + 311040 x + 34560} + 75625 \log {\left (x + \frac {3}{5} \right )} - 75625 \log {\left (x + \frac {2}{3} \right )} \] Input:
integrate((1-2*x)**2/(2+3*x)**7/(3+5*x),x)
Output:
(1984702500*x**5 + 6681831750*x**4 + 9000258300*x**3 + 6063045615*x**2 + 2 042732232*x + 275370238)/(393660*x**6 + 1574640*x**5 + 2624400*x**4 + 2332 800*x**3 + 1166400*x**2 + 311040*x + 34560) + 75625*log(x + 3/5) - 75625*l og(x + 2/3)
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.94 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {1984702500 \, x^{5} + 6681831750 \, x^{4} + 9000258300 \, x^{3} + 6063045615 \, x^{2} + 2042732232 \, x + 275370238}{540 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + 75625 \, \log \left (5 \, x + 3\right ) - 75625 \, \log \left (3 \, x + 2\right ) \] Input:
integrate((1-2*x)^2/(2+3*x)^7/(3+5*x),x, algorithm="maxima")
Output:
1/540*(1984702500*x^5 + 6681831750*x^4 + 9000258300*x^3 + 6063045615*x^2 + 2042732232*x + 275370238)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 216 0*x^2 + 576*x + 64) + 75625*log(5*x + 3) - 75625*log(3*x + 2)
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {1984702500 \, x^{5} + 6681831750 \, x^{4} + 9000258300 \, x^{3} + 6063045615 \, x^{2} + 2042732232 \, x + 275370238}{540 \, {\left (3 \, x + 2\right )}^{6}} + 75625 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 75625 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \] Input:
integrate((1-2*x)^2/(2+3*x)^7/(3+5*x),x, algorithm="giac")
Output:
1/540*(1984702500*x^5 + 6681831750*x^4 + 9000258300*x^3 + 6063045615*x^2 + 2042732232*x + 275370238)/(3*x + 2)^6 + 75625*log(abs(5*x + 3)) - 75625*l og(abs(3*x + 2))
Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.80 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {\frac {15125\,x^5}{3}+\frac {305525\,x^4}{18}+\frac {1851905\,x^3}{81}+\frac {1663387\,x^2}{108}+\frac {56742562\,x}{10935}+\frac {137685119}{196830}}{x^6+4\,x^5+\frac {20\,x^4}{3}+\frac {160\,x^3}{27}+\frac {80\,x^2}{27}+\frac {64\,x}{81}+\frac {64}{729}}-151250\,\mathrm {atanh}\left (30\,x+19\right ) \] Input:
int((2*x - 1)^2/((3*x + 2)^7*(5*x + 3)),x)
Output:
((56742562*x)/10935 + (1663387*x^2)/108 + (1851905*x^3)/81 + (305525*x^4)/ 18 + (15125*x^5)/3 + 137685119/196830)/((64*x)/81 + (80*x^2)/27 + (160*x^3 )/27 + (20*x^4)/3 + 4*x^5 + x^6 + 64/729) - 151250*atanh(30*x + 19)
Time = 0.16 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.49 \[ \int \frac {(1-2 x)^2}{(2+3 x)^7 (3+5 x)} \, dx=\frac {29770537500 \,\mathrm {log}\left (5 x +3\right ) x^{6}+119082150000 \,\mathrm {log}\left (5 x +3\right ) x^{5}+198470250000 \,\mathrm {log}\left (5 x +3\right ) x^{4}+176418000000 \,\mathrm {log}\left (5 x +3\right ) x^{3}+88209000000 \,\mathrm {log}\left (5 x +3\right ) x^{2}+23522400000 \,\mathrm {log}\left (5 x +3\right ) x +2613600000 \,\mathrm {log}\left (5 x +3\right )-29770537500 \,\mathrm {log}\left (3 x +2\right ) x^{6}-119082150000 \,\mathrm {log}\left (3 x +2\right ) x^{5}-198470250000 \,\mathrm {log}\left (3 x +2\right ) x^{4}-176418000000 \,\mathrm {log}\left (3 x +2\right ) x^{3}-88209000000 \,\mathrm {log}\left (3 x +2\right ) x^{2}-23522400000 \,\mathrm {log}\left (3 x +2\right ) x -2613600000 \,\mathrm {log}\left (3 x +2\right )-496175625 x^{6}+3373994250 x^{4}+6059958300 x^{3}+4592895615 x^{2}+1650692232 x +231810238}{393660 x^{6}+1574640 x^{5}+2624400 x^{4}+2332800 x^{3}+1166400 x^{2}+311040 x +34560} \] Input:
int((1-2*x)^2/(2+3*x)^7/(3+5*x),x)
Output:
(29770537500*log(5*x + 3)*x**6 + 119082150000*log(5*x + 3)*x**5 + 19847025 0000*log(5*x + 3)*x**4 + 176418000000*log(5*x + 3)*x**3 + 88209000000*log( 5*x + 3)*x**2 + 23522400000*log(5*x + 3)*x + 2613600000*log(5*x + 3) - 297 70537500*log(3*x + 2)*x**6 - 119082150000*log(3*x + 2)*x**5 - 198470250000 *log(3*x + 2)*x**4 - 176418000000*log(3*x + 2)*x**3 - 88209000000*log(3*x + 2)*x**2 - 23522400000*log(3*x + 2)*x - 2613600000*log(3*x + 2) - 4961756 25*x**6 + 3373994250*x**4 + 6059958300*x**3 + 4592895615*x**2 + 1650692232 *x + 231810238)/(540*(729*x**6 + 2916*x**5 + 4860*x**4 + 4320*x**3 + 2160* x**2 + 576*x + 64))