\(\int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx\) [1207]

Optimal result

Integrand size = 20, antiderivative size = 69 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {5555478 x}{390625}+\frac {5740767 x^2}{156250}+\frac {92592 x^3}{3125}-\frac {513783 x^4}{12500}-\frac {336798 x^5}{3125}-\frac {21627 x^6}{250}-\frac {4374 x^7}{175}-\frac {11}{1953125 (3+5 x)}+\frac {229 \log (3+5 x)}{1953125} \]

[Out]

5555478/390625*x+5740767/156250*x^2+92592/3125*x^3-513783/12500*x^4-336798/3125*x^5-21627/250*x^6-4374/175*x^7
-11/1953125/(3+5*x)+229/1953125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 69, 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} \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=-\frac {4374 x^7}{175}-\frac {21627 x^6}{250}-\frac {336798 x^5}{3125}-\frac {513783 x^4}{12500}+\frac {92592 x^3}{3125}+\frac {5740767 x^2}{156250}+\frac {5555478 x}{390625}-\frac {11}{1953125 (5 x+3)}+\frac {229 \log (5 x+3)}{1953125} \]

[In]

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

[Out]

(5555478*x)/390625 + (5740767*x^2)/156250 + (92592*x^3)/3125 - (513783*x^4)/12500 - (336798*x^5)/3125 - (21627
*x^6)/250 - (4374*x^7)/175 - 11/(1953125*(3 + 5*x)) + (229*Log[3 + 5*x])/1953125

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 {5555478}{390625}+\frac {5740767 x}{78125}+\frac {277776 x^2}{3125}-\frac {513783 x^3}{3125}-\frac {336798 x^4}{625}-\frac {64881 x^5}{125}-\frac {4374 x^6}{25}+\frac {11}{390625 (3+5 x)^2}+\frac {229}{390625 (3+5 x)}\right ) \, dx \\ & = \frac {5555478 x}{390625}+\frac {5740767 x^2}{156250}+\frac {92592 x^3}{3125}-\frac {513783 x^4}{12500}-\frac {336798 x^5}{3125}-\frac {21627 x^6}{250}-\frac {4374 x^7}{175}-\frac {11}{1953125 (3+5 x)}+\frac {229 \log (3+5 x)}{1953125} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.23 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {82320 (2+3 x)+171150 (2+3 x)^2+455000 (2+3 x)^3+1273125 (2+3 x)^4+3360000 (2+3 x)^5+6781250 (2+3 x)^6-1875000 (2+3 x)^7-\frac {924}{3+5 x}+19236 \log (-3 (3+5 x))}{164062500} \]

[In]

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

[Out]

(82320*(2 + 3*x) + 171150*(2 + 3*x)^2 + 455000*(2 + 3*x)^3 + 1273125*(2 + 3*x)^4 + 3360000*(2 + 3*x)^5 + 67812
50*(2 + 3*x)^6 - 1875000*(2 + 3*x)^7 - 924/(3 + 5*x) + 19236*Log[-3*(3 + 5*x)])/164062500

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.72

method	result	size
risch	\(-\frac {4374 x^{7}}{175}-\frac {21627 x^{6}}{250}-\frac {336798 x^{5}}{3125}-\frac {513783 x^{4}}{12500}+\frac {92592 x^{3}}{3125}+\frac {5740767 x^{2}}{156250}+\frac {5555478 x}{390625}-\frac {11}{9765625 \left (x +\frac {3}{5}\right )}+\frac {229 \ln \left (3+5 x \right )}{1953125}\)	\(50\)
default	\(\frac {5555478 x}{390625}+\frac {5740767 x^{2}}{156250}+\frac {92592 x^{3}}{3125}-\frac {513783 x^{4}}{12500}-\frac {336798 x^{5}}{3125}-\frac {21627 x^{6}}{250}-\frac {4374 x^{7}}{175}-\frac {11}{1953125 \left (3+5 x \right )}+\frac {229 \ln \left (3+5 x \right )}{1953125}\)	\(52\)
norman	\(\frac {\frac {49999313}{1171875} x +\frac {28333257}{156250} x^{2}+\frac {8518527}{31250} x^{3}+\frac {310491}{12500} x^{4}-\frac {6610491}{12500} x^{5}-\frac {998001}{1250} x^{6}-\frac {177633}{350} x^{7}-\frac {4374}{35} x^{8}}{3+5 x}+\frac {229 \ln \left (3+5 x \right )}{1953125}\)	\(57\)
parallelrisch	\(\frac {-20503125000 x^{8}-83265468750 x^{7}-130987631250 x^{6}-86762694375 x^{5}[phone] x^{4}[phone] x^{3}+96180 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}+57708 \ln \left (x +\frac {3}{5}\right )[phone] x}{492187500+820312500 x}\)	\(62\)
meijerg	\(-\frac {2624 x}{45 \left (1+\frac {5 x}{3}\right )}+\frac {229 \ln \left (1+\frac {5 x}{3}\right )}{1953125}+\frac {224 x \left (5 x +6\right )}{5 \left (1+\frac {5 x}{3}\right )}-\frac {2268 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}-\frac {4536 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{625 \left (1+\frac {5 x}{3}\right )}+\frac {56133 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}-\frac {177147 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{78125 \left (1+\frac {5 x}{3}\right )}+\frac {59049 x \left (-\frac {312500}{729} x^{6}+\frac {87500}{243} x^{5}-\frac {8750}{27} x^{4}+\frac {8750}{27} x^{3}-\frac {3500}{9} x^{2}+700 x +840\right )}{125000 \left (1+\frac {5 x}{3}\right )}-\frac {354294 x \left (\frac {390625}{243} x^{7}-\frac {312500}{243} x^{6}+\frac {87500}{81} x^{5}-\frac {8750}{9} x^{4}+\frac {8750}{9} x^{3}-\frac {3500}{3} x^{2}+2100 x +2520\right )}{13671875 \left (1+\frac {5 x}{3}\right )}\)	\(230\)

[In]

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

[Out]

-4374/175*x^7-21627/250*x^6-336798/3125*x^5-513783/12500*x^4+92592/3125*x^3+5740767/156250*x^2+5555478/390625*
x-11/9765625/(x+3/5)+229/1953125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=-\frac {6834375000 \, x^{8} + 27755156250 \, x^{7} + 43662543750 \, x^{6} + 28920898125 \, x^{5} - 1358398125 \, x^{4} - 14907422250 \, x^{3} - 9916639950 \, x^{2} - 6412 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 2333300760 \, x + 308}{54687500 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

-1/54687500*(6834375000*x^8 + 27755156250*x^7 + 43662543750*x^6 + 28920898125*x^5 - 1358398125*x^4 - 149074222
50*x^3 - 9916639950*x^2 - 6412*(5*x + 3)*log(5*x + 3) - 2333300760*x + 308)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.88 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=- \frac {4374 x^{7}}{175} - \frac {21627 x^{6}}{250} - \frac {336798 x^{5}}{3125} - \frac {513783 x^{4}}{12500} + \frac {92592 x^{3}}{3125} + \frac {5740767 x^{2}}{156250} + \frac {5555478 x}{390625} + \frac {229 \log {\left (5 x + 3 \right )}}{1953125} - \frac {11}{9765625 x + 5859375} \]

[In]

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

[Out]

-4374*x**7/175 - 21627*x**6/250 - 336798*x**5/3125 - 513783*x**4/12500 + 92592*x**3/3125 + 5740767*x**2/156250
+ 5555478*x/390625 + 229*log(5*x + 3)/1953125 - 11/(9765625*x + 5859375)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=-\frac {4374}{175} \, x^{7} - \frac {21627}{250} \, x^{6} - \frac {336798}{3125} \, x^{5} - \frac {513783}{12500} \, x^{4} + \frac {92592}{3125} \, x^{3} + \frac {5740767}{156250} \, x^{2} + \frac {5555478}{390625} \, x - \frac {11}{1953125 \, {\left (5 \, x + 3\right )}} + \frac {229}{1953125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-4374/175*x^7 - 21627/250*x^6 - 336798/3125*x^5 - 513783/12500*x^4 + 92592/3125*x^3 + 5740767/156250*x^2 + 555
5478/390625*x - 11/1953125/(5*x + 3) + 229/1953125*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.35 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {3}{273437500} \, {\left (5 \, x + 3\right )}^{7} {\left (\frac {107730}{5 \, x + 3} + \frac {428652}{{\left (5 \, x + 3\right )}^{2}} + \frac {588735}{{\left (5 \, x + 3\right )}^{3}} + \frac {455700}{{\left (5 \, x + 3\right )}^{4}} + \frac {233730}{{\left (5 \, x + 3\right )}^{5}} + \frac {95060}{{\left (5 \, x + 3\right )}^{6}} - 29160\right )} - \frac {11}{1953125 \, {\left (5 \, x + 3\right )}} - \frac {229}{1953125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

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

[Out]

3/273437500*(5*x + 3)^7*(107730/(5*x + 3) + 428652/(5*x + 3)^2 + 588735/(5*x + 3)^3 + 455700/(5*x + 3)^4 + 233
730/(5*x + 3)^5 + 95060/(5*x + 3)^6 - 29160) - 11/1953125/(5*x + 3) - 229/1953125*log(1/5*abs(5*x + 3)/(5*x +
3)^2)

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x) (2+3 x)^7}{(3+5 x)^2} \, dx=\frac {5555478\,x}{390625}+\frac {229\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {11}{9765625\,\left (x+\frac {3}{5}\right )}+\frac {5740767\,x^2}{156250}+\frac {92592\,x^3}{3125}-\frac {513783\,x^4}{12500}-\frac {336798\,x^5}{3125}-\frac {21627\,x^6}{250}-\frac {4374\,x^7}{175} \]

[In]

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

[Out]

(5555478*x)/390625 + (229*log(x + 3/5))/1953125 - 11/(9765625*(x + 3/5)) + (5740767*x^2)/156250 + (92592*x^3)/
3125 - (513783*x^4)/12500 - (336798*x^5)/3125 - (21627*x^6)/250 - (4374*x^7)/175