\(\int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx\) [1405]

Optimal result

Integrand size = 22, antiderivative size = 62 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {133659 x}{78125}-\frac {1816 x^2}{625}-\frac {4217 x^3}{3125}+\frac {7317 x^4}{1250}+\frac {108 x^5}{625}-\frac {108 x^6}{25}-\frac {1331}{390625 (3+5 x)}+\frac {15246 \log (3+5 x)}{390625} \]

[Out]

133659/78125*x-1816/625*x^2-4217/3125*x^3+7317/1250*x^4+108/625*x^5-108/25*x^6-1331/390625/(3+5*x)+15246/39062
5*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 62, 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} \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {108 x^6}{25}+\frac {108 x^5}{625}+\frac {7317 x^4}{1250}-\frac {4217 x^3}{3125}-\frac {1816 x^2}{625}+\frac {133659 x}{78125}-\frac {1331}{390625 (5 x+3)}+\frac {15246 \log (5 x+3)}{390625} \]

[In]

Int[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

(133659*x)/78125 - (1816*x^2)/625 - (4217*x^3)/3125 + (7317*x^4)/1250 + (108*x^5)/625 - (108*x^6)/25 - 1331/(3
90625*(3 + 5*x)) + (15246*Log[3 + 5*x])/390625

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 {133659}{78125}-\frac {3632 x}{625}-\frac {12651 x^2}{3125}+\frac {14634 x^3}{625}+\frac {108 x^4}{125}-\frac {648 x^5}{25}+\frac {1331}{78125 (3+5 x)^2}+\frac {15246}{78125 (3+5 x)}\right ) \, dx \\ & = \frac {133659 x}{78125}-\frac {1816 x^2}{625}-\frac {4217 x^3}{3125}+\frac {7317 x^4}{1250}+\frac {108 x^5}{625}-\frac {108 x^6}{25}-\frac {1331}{390625 (3+5 x)}+\frac {15246 \log (3+5 x)}{390625} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.98 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {14487499+44216865 x-635250 x^2-72563750 x^3+42240625 x^4+116353125 x^5-47250000 x^6-84375000 x^7+152460 (3+5 x) \log (6 (3+5 x))}{3906250 (3+5 x)} \]

[In]

Integrate[((1 - 2*x)^3*(2 + 3*x)^4)/(3 + 5*x)^2,x]

[Out]

(14487499 + 44216865*x - 635250*x^2 - 72563750*x^3 + 42240625*x^4 + 116353125*x^5 - 47250000*x^6 - 84375000*x^
7 + 152460*(3 + 5*x)*Log[6*(3 + 5*x)])/(3906250*(3 + 5*x))

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {108 x^{6}}{25}+\frac {108 x^{5}}{625}+\frac {7317 x^{4}}{1250}-\frac {4217 x^{3}}{3125}-\frac {1816 x^{2}}{625}+\frac {133659 x}{78125}-\frac {1331}{1953125 \left (x +\frac {3}{5}\right )}+\frac {15246 \ln \left (3+5 x \right )}{390625}\)	\(45\)
default	\(\frac {133659 x}{78125}-\frac {1816 x^{2}}{625}-\frac {4217 x^{3}}{3125}+\frac {7317 x^{4}}{1250}+\frac {108 x^{5}}{625}-\frac {108 x^{6}}{25}-\frac {1331}{390625 \left (3+5 x \right )}+\frac {15246 \ln \left (3+5 x \right )}{390625}\)	\(47\)
norman	\(\frac {\frac {1204262}{234375} x -\frac {2541}{15625} x^{2}-\frac {58051}{3125} x^{3}+\frac {13517}{1250} x^{4}+\frac {37233}{1250} x^{5}-\frac {1512}{125} x^{6}-\frac {108}{5} x^{7}}{3+5 x}+\frac {15246 \ln \left (3+5 x \right )}{390625}\)	\(52\)
parallelrisch	\(\frac {-50625000 x^{7}-28350000 x^{6}[phone] x^{5}[phone] x^{4}-43538250 x^{3}+457380 \ln \left (x +\frac {3}{5}\right ) x -381150 x^{2}+274428 \ln \left (x +\frac {3}{5}\right )[phone] x}{7031250+11718750 x}\)	\(57\)
meijerg	\(\frac {16 x}{9 \left (1+\frac {5 x}{3}\right )}-\frac {56 x \left (5 x +6\right )}{25 \left (1+\frac {5 x}{3}\right )}+\frac {15246 \ln \left (1+\frac {5 x}{3}\right )}{390625}+\frac {42 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )}+\frac {1827 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{3125 \left (1+\frac {5 x}{3}\right )}-\frac {1701 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{6250 \left (1+\frac {5 x}{3}\right )}-\frac {4374 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{78125 \left (1+\frac {5 x}{3}\right )}+\frac {6561 x \left (-\frac {312500}{729} x^{6}+\frac {87500}{243} x^{5}-\frac {8750}{27} x^{4}+\frac {8750}{27} x^{3}-\frac {3500}{9} x^{2}+700 x +840\right )}{390625 \left (1+\frac {5 x}{3}\right )}\)	\(185\)

[In]

int((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x,method=_RETURNVERBOSE)

[Out]

-108/25*x^6+108/625*x^5+7317/1250*x^4-4217/3125*x^3-1816/625*x^2+133659/78125*x-1331/1953125/(x+3/5)+15246/390
625*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.92 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {16875000 \, x^{7} + 9450000 \, x^{6} - 23270625 \, x^{5} - 8448125 \, x^{4} + 14512750 \, x^{3} + 127050 \, x^{2} - 30492 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 4009770 \, x + 2662}{781250 \, {\left (5 \, x + 3\right )}} \]

[In]

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/781250*(16875000*x^7 + 9450000*x^6 - 23270625*x^5 - 8448125*x^4 + 14512750*x^3 + 127050*x^2 - 30492*(5*x +
3)*log(5*x + 3) - 4009770*x + 2662)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=- \frac {108 x^{6}}{25} + \frac {108 x^{5}}{625} + \frac {7317 x^{4}}{1250} - \frac {4217 x^{3}}{3125} - \frac {1816 x^{2}}{625} + \frac {133659 x}{78125} + \frac {15246 \log {\left (5 x + 3 \right )}}{390625} - \frac {1331}{1953125 x + 1171875} \]

[In]

integrate((1-2*x)**3*(2+3*x)**4/(3+5*x)**2,x)

[Out]

-108*x**6/25 + 108*x**5/625 + 7317*x**4/1250 - 4217*x**3/3125 - 1816*x**2/625 + 133659*x/78125 + 15246*log(5*x
+ 3)/390625 - 1331/(1953125*x + 1171875)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=-\frac {108}{25} \, x^{6} + \frac {108}{625} \, x^{5} + \frac {7317}{1250} \, x^{4} - \frac {4217}{3125} \, x^{3} - \frac {1816}{625} \, x^{2} + \frac {133659}{78125} \, x - \frac {1331}{390625 \, {\left (5 \, x + 3\right )}} + \frac {15246}{390625} \, \log \left (5 \, x + 3\right ) \]

[In]

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-108/25*x^6 + 108/625*x^5 + 7317/1250*x^4 - 4217/3125*x^3 - 1816/625*x^2 + 133659/78125*x - 1331/390625/(5*x +
3) + 15246/390625*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.35 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {1}{3906250} \, {\left (5 \, x + 3\right )}^{6} {\left (\frac {19656}{5 \, x + 3} - \frac {112455}{{\left (5 \, x + 3\right )}^{2}} + \frac {121450}{{\left (5 \, x + 3\right )}^{3}} + \frac {530600}{{\left (5 \, x + 3\right )}^{4}} + \frac {632940}{{\left (5 \, x + 3\right )}^{5}} - 1080\right )} - \frac {1331}{390625 \, {\left (5 \, x + 3\right )}} - \frac {15246}{390625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

integrate((1-2*x)^3*(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

1/3906250*(5*x + 3)^6*(19656/(5*x + 3) - 112455/(5*x + 3)^2 + 121450/(5*x + 3)^3 + 530600/(5*x + 3)^4 + 632940
/(5*x + 3)^5 - 1080) - 1331/390625/(5*x + 3) - 15246/390625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^3 (2+3 x)^4}{(3+5 x)^2} \, dx=\frac {133659\,x}{78125}+\frac {15246\,\ln \left (x+\frac {3}{5}\right )}{390625}-\frac {1331}{1953125\,\left (x+\frac {3}{5}\right )}-\frac {1816\,x^2}{625}-\frac {4217\,x^3}{3125}+\frac {7317\,x^4}{1250}+\frac {108\,x^5}{625}-\frac {108\,x^6}{25} \]

[In]

int(-((2*x - 1)^3*(3*x + 2)^4)/(5*x + 3)^2,x)

[Out]

(133659*x)/78125 + (15246*log(x + 3/5))/390625 - 1331/(1953125*(x + 3/5)) - (1816*x^2)/625 - (4217*x^3)/3125 +
(7317*x^4)/1250 + (108*x^5)/625 - (108*x^6)/25