\(\int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx\) [1193]

Optimal result

Integrand size = 20, antiderivative size = 58 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=\frac {1666663 x}{78125}+\frac {1777779 x^2}{31250}+\frac {152469 x^3}{3125}-\frac {152469 x^4}{2500}-\frac {106677 x^5}{625}-\frac {7047 x^6}{50}-\frac {1458 x^7}{35}+\frac {11 \log (3+5 x)}{390625} \]

[Out]

1666663/78125*x+1777779/31250*x^2+152469/3125*x^3-152469/2500*x^4-106677/625*x^5-7047/50*x^6-1458/35*x^7+11/39
0625*ln(3+5*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=-\frac {1458 x^7}{35}-\frac {7047 x^6}{50}-\frac {106677 x^5}{625}-\frac {152469 x^4}{2500}+\frac {152469 x^3}{3125}+\frac {1777779 x^2}{31250}+\frac {1666663 x}{78125}+\frac {11 \log (5 x+3)}{390625} \]

[In]

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

[Out]

(1666663*x)/78125 + (1777779*x^2)/31250 + (152469*x^3)/3125 - (152469*x^4)/2500 - (106677*x^5)/625 - (7047*x^6
)/50 - (1458*x^7)/35 + (11*Log[3 + 5*x])/390625

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1666663}{78125}+\frac {1777779 x}{15625}+\frac {457407 x^2}{3125}-\frac {152469 x^3}{625}-\frac {106677 x^4}{125}-\frac {21141 x^5}{25}-\frac {1458 x^6}{5}+\frac {11}{78125 (3+5 x)}\right ) \, dx \\ & = \frac {1666663 x}{78125}+\frac {1777779 x^2}{31250}+\frac {152469 x^3}{3125}-\frac {152469 x^4}{2500}-\frac {106677 x^5}{625}-\frac {7047 x^6}{50}-\frac {1458 x^7}{35}+\frac {11 \log (3+5 x)}{390625} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.81 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=\frac {158585307+1166664100 x+3111113250 x^2+2668207500 x^3-3335259375 x^4-9334237500 x^5-7707656250 x^6-2278125000 x^7+1540 \log (3+5 x)}{54687500} \]

[In]

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

[Out]

(158585307 + 1166664100*x + 3111113250*x^2 + 2668207500*x^3 - 3335259375*x^4 - 9334237500*x^5 - 7707656250*x^6
- 2278125000*x^7 + 1540*Log[3 + 5*x])/54687500

Maple [A] (verified)

Time = 2.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.71

method	result	size
parallelrisch	\(-\frac {1458 x^{7}}{35}-\frac {7047 x^{6}}{50}-\frac {106677 x^{5}}{625}-\frac {152469 x^{4}}{2500}+\frac {152469 x^{3}}{3125}+\frac {1777779 x^{2}}{31250}+\frac {1666663 x}{78125}+\frac {11 \ln \left (x +\frac {3}{5}\right )}{390625}\)	\(41\)
default	\(\frac {1666663 x}{78125}+\frac {1777779 x^{2}}{31250}+\frac {152469 x^{3}}{3125}-\frac {152469 x^{4}}{2500}-\frac {106677 x^{5}}{625}-\frac {7047 x^{6}}{50}-\frac {1458 x^{7}}{35}+\frac {11 \ln \left (3+5 x \right )}{390625}\)	\(43\)
norman	\(\frac {1666663 x}{78125}+\frac {1777779 x^{2}}{31250}+\frac {152469 x^{3}}{3125}-\frac {152469 x^{4}}{2500}-\frac {106677 x^{5}}{625}-\frac {7047 x^{6}}{50}-\frac {1458 x^{7}}{35}+\frac {11 \ln \left (3+5 x \right )}{390625}\)	\(43\)
risch	\(\frac {1666663 x}{78125}+\frac {1777779 x^{2}}{31250}+\frac {152469 x^{3}}{3125}-\frac {152469 x^{4}}{2500}-\frac {106677 x^{5}}{625}-\frac {7047 x^{6}}{50}-\frac {1458 x^{7}}{35}+\frac {11 \ln \left (3+5 x \right )}{390625}\)	\(43\)
meijerg	\(\frac {11 \ln \left (1+\frac {5 x}{3}\right )}{390625}+\frac {448 x}{5}-\frac {504 x \left (-5 x +6\right )}{25}+\frac {1701 x \left (-\frac {625}{9} x^{3}+\frac {500}{9} x^{2}-50 x +60\right )}{625}-\frac {45927 x \left (\frac {2500}{27} x^{4}-\frac {625}{9} x^{3}+\frac {500}{9} x^{2}-50 x +60\right )}{15625}+\frac {59049 x \left (-\frac {218750}{243} x^{5}+\frac {17500}{27} x^{4}-\frac {4375}{9} x^{3}+\frac {3500}{9} x^{2}-350 x +420\right )}{312500}-\frac {177147 x \left (\frac {625000}{243} x^{6}-\frac {437500}{243} x^{5}+\frac {35000}{27} x^{4}-\frac {8750}{9} x^{3}+\frac {7000}{9} x^{2}-700 x +840\right )}{10937500}\)	\(123\)

[In]

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

[Out]

-1458/35*x^7-7047/50*x^6-106677/625*x^5-152469/2500*x^4+152469/3125*x^3+1777779/31250*x^2+1666663/78125*x+11/3
90625*ln(x+3/5)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=-\frac {1458}{35} \, x^{7} - \frac {7047}{50} \, x^{6} - \frac {106677}{625} \, x^{5} - \frac {152469}{2500} \, x^{4} + \frac {152469}{3125} \, x^{3} + \frac {1777779}{31250} \, x^{2} + \frac {1666663}{78125} \, x + \frac {11}{390625} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-1458/35*x^7 - 7047/50*x^6 - 106677/625*x^5 - 152469/2500*x^4 + 152469/3125*x^3 + 1777779/31250*x^2 + 1666663/
78125*x + 11/390625*log(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.93 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=- \frac {1458 x^{7}}{35} - \frac {7047 x^{6}}{50} - \frac {106677 x^{5}}{625} - \frac {152469 x^{4}}{2500} + \frac {152469 x^{3}}{3125} + \frac {1777779 x^{2}}{31250} + \frac {1666663 x}{78125} + \frac {11 \log {\left (5 x + 3 \right )}}{390625} \]

[In]

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

[Out]

-1458*x**7/35 - 7047*x**6/50 - 106677*x**5/625 - 152469*x**4/2500 + 152469*x**3/3125 + 1777779*x**2/31250 + 16
66663*x/78125 + 11*log(5*x + 3)/390625

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=-\frac {1458}{35} \, x^{7} - \frac {7047}{50} \, x^{6} - \frac {106677}{625} \, x^{5} - \frac {152469}{2500} \, x^{4} + \frac {152469}{3125} \, x^{3} + \frac {1777779}{31250} \, x^{2} + \frac {1666663}{78125} \, x + \frac {11}{390625} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-1458/35*x^7 - 7047/50*x^6 - 106677/625*x^5 - 152469/2500*x^4 + 152469/3125*x^3 + 1777779/31250*x^2 + 1666663/
78125*x + 11/390625*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=-\frac {1458}{35} \, x^{7} - \frac {7047}{50} \, x^{6} - \frac {106677}{625} \, x^{5} - \frac {152469}{2500} \, x^{4} + \frac {152469}{3125} \, x^{3} + \frac {1777779}{31250} \, x^{2} + \frac {1666663}{78125} \, x + \frac {11}{390625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

[In]

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

[Out]

-1458/35*x^7 - 7047/50*x^6 - 106677/625*x^5 - 152469/2500*x^4 + 152469/3125*x^3 + 1777779/31250*x^2 + 1666663/
78125*x + 11/390625*log(abs(5*x + 3))

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.69 \[ \int \frac {(1-2 x) (2+3 x)^6}{3+5 x} \, dx=\frac {1666663\,x}{78125}+\frac {11\,\ln \left (x+\frac {3}{5}\right )}{390625}+\frac {1777779\,x^2}{31250}+\frac {152469\,x^3}{3125}-\frac {152469\,x^4}{2500}-\frac {106677\,x^5}{625}-\frac {7047\,x^6}{50}-\frac {1458\,x^7}{35} \]

[In]

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

[Out]

(1666663*x)/78125 + (11*log(x + 3/5))/390625 + (1777779*x^2)/31250 + (152469*x^3)/3125 - (152469*x^4)/2500 - (
106677*x^5)/625 - (7047*x^6)/50 - (1458*x^7)/35