\(\int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx\) [1209]

Optimal result

Integrand size = 20, antiderivative size = 55 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {444 x}{125}+\frac {24093 x^2}{6250}-\frac {1854 x^3}{625}-\frac {3969 x^4}{500}-\frac {486 x^5}{125}-\frac {11}{78125 (3+5 x)}+\frac {163 \log (3+5 x)}{78125} \]

[Out]

444/125*x+24093/6250*x^2-1854/625*x^3-3969/500*x^4-486/125*x^5-11/78125/(3+5*x)+163/78125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, 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} \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {486 x^5}{125}-\frac {3969 x^4}{500}-\frac {1854 x^3}{625}+\frac {24093 x^2}{6250}+\frac {444 x}{125}-\frac {11}{78125 (5 x+3)}+\frac {163 \log (5 x+3)}{78125} \]

[In]

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

[Out]

(444*x)/125 + (24093*x^2)/6250 - (1854*x^3)/625 - (3969*x^4)/500 - (486*x^5)/125 - 11/(78125*(3 + 5*x)) + (163
*Log[3 + 5*x])/78125

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 {444}{125}+\frac {24093 x}{3125}-\frac {5562 x^2}{625}-\frac {3969 x^3}{125}-\frac {486 x^4}{25}+\frac {11}{15625 (3+5 x)^2}+\frac {163}{15625 (3+5 x)}\right ) \, dx \\ & = \frac {444 x}{125}+\frac {24093 x^2}{6250}-\frac {1854 x^3}{625}-\frac {3969 x^4}{500}-\frac {486 x^5}{125}-\frac {11}{78125 (3+5 x)}+\frac {163 \log (3+5 x)}{78125} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {779800+3330000 x+3613950 x^2-2781000 x^3-7441875 x^4-3645000 x^5-\frac {132}{3+5 x}+1956 \log (-3 (3+5 x))}{937500} \]

[In]

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

[Out]

(779800 + 3330000*x + 3613950*x^2 - 2781000*x^3 - 7441875*x^4 - 3645000*x^5 - 132/(3 + 5*x) + 1956*Log[-3*(3 +
5*x)])/937500

Maple [A] (verified)

Time = 2.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.73

method	result	size
risch	\(-\frac {486 x^{5}}{125}-\frac {3969 x^{4}}{500}-\frac {1854 x^{3}}{625}+\frac {24093 x^{2}}{6250}+\frac {444 x}{125}-\frac {11}{390625 \left (x +\frac {3}{5}\right )}+\frac {163 \ln \left (3+5 x \right )}{78125}\)	\(40\)
default	\(\frac {444 x}{125}+\frac {24093 x^{2}}{6250}-\frac {1854 x^{3}}{625}-\frac {3969 x^{4}}{500}-\frac {486 x^{5}}{125}-\frac {11}{78125 \left (3+5 x \right )}+\frac {163 \ln \left (3+5 x \right )}{78125}\)	\(42\)
norman	\(\frac {\frac {499511}{46875} x +\frac {183279}{6250} x^{2}+\frac {12969}{1250} x^{3}-\frac {19323}{500} x^{4}-\frac {25677}{500} x^{5}-\frac {486}{25} x^{6}}{3+5 x}+\frac {163 \ln \left (3+5 x \right )}{78125}\)	\(47\)
parallelrisch	\(\frac {-18225000 x^{6}-48144375 x^{5}-36230625 x^{4}+9726750 x^{3}+9780 \ln \left (x +\frac {3}{5}\right ) x [phone] x^{2}+5868 \ln \left (x +\frac {3}{5}\right )+9990220 x}{2812500+4687500 x}\)	\(52\)
meijerg	\(-\frac {368 x}{45 \left (1+\frac {5 x}{3}\right )}+\frac {163 \ln \left (1+\frac {5 x}{3}\right )}{78125}+\frac {16 x \left (5 x +6\right )}{5 \left (1+\frac {5 x}{3}\right )}+\frac {54 x \left (-\frac {50}{9} x^{2}+10 x +12\right )}{25 \left (1+\frac {5 x}{3}\right )}-\frac {162 x \left (\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{125 \left (1+\frac {5 x}{3}\right )}+\frac {12393 x \left (-\frac {625}{27} x^{4}+\frac {625}{27} x^{3}-\frac {250}{9} x^{2}+50 x +60\right )}{12500 \left (1+\frac {5 x}{3}\right )}-\frac {19683 x \left (\frac {43750}{243} x^{5}-\frac {4375}{27} x^{4}+\frac {4375}{27} x^{3}-\frac {1750}{9} x^{2}+350 x +420\right )}{546875 \left (1+\frac {5 x}{3}\right )}\)	\(145\)

[In]

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

[Out]

-486/125*x^5-3969/500*x^4-1854/625*x^3+24093/6250*x^2+444/125*x-11/390625/(x+3/5)+163/78125*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {6075000 \, x^{6} + 16048125 \, x^{5} + 12076875 \, x^{4} - 3242250 \, x^{3} - 9163950 \, x^{2} - 652 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 3330000 \, x + 44}{312500 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

-1/312500*(6075000*x^6 + 16048125*x^5 + 12076875*x^4 - 3242250*x^3 - 9163950*x^2 - 652*(5*x + 3)*log(5*x + 3)
- 3330000*x + 44)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=- \frac {486 x^{5}}{125} - \frac {3969 x^{4}}{500} - \frac {1854 x^{3}}{625} + \frac {24093 x^{2}}{6250} + \frac {444 x}{125} + \frac {163 \log {\left (5 x + 3 \right )}}{78125} - \frac {11}{390625 x + 234375} \]

[In]

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

[Out]

-486*x**5/125 - 3969*x**4/500 - 1854*x**3/625 + 24093*x**2/6250 + 444*x/125 + 163*log(5*x + 3)/78125 - 11/(390
625*x + 234375)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.75 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=-\frac {486}{125} \, x^{5} - \frac {3969}{500} \, x^{4} - \frac {1854}{625} \, x^{3} + \frac {24093}{6250} \, x^{2} + \frac {444}{125} \, x - \frac {11}{78125 \, {\left (5 \, x + 3\right )}} + \frac {163}{78125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-486/125*x^5 - 3969/500*x^4 - 1854/625*x^3 + 24093/6250*x^2 + 444/125*x - 11/78125/(5*x + 3) + 163/78125*log(5
*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.36 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {3}{1562500} \, {\left (5 \, x + 3\right )}^{5} {\left (\frac {3105}{5 \, x + 3} + \frac {8700}{{\left (5 \, x + 3\right )}^{2}} + \frac {9300}{{\left (5 \, x + 3\right )}^{3}} + \frac {6400}{{\left (5 \, x + 3\right )}^{4}} - 648\right )} - \frac {11}{78125 \, {\left (5 \, x + 3\right )}} - \frac {163}{78125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

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

[Out]

3/1562500*(5*x + 3)^5*(3105/(5*x + 3) + 8700/(5*x + 3)^2 + 9300/(5*x + 3)^3 + 6400/(5*x + 3)^4 - 648) - 11/781
25/(5*x + 3) - 163/78125*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx=\frac {444\,x}{125}+\frac {163\,\ln \left (x+\frac {3}{5}\right )}{78125}-\frac {11}{390625\,\left (x+\frac {3}{5}\right )}+\frac {24093\,x^2}{6250}-\frac {1854\,x^3}{625}-\frac {3969\,x^4}{500}-\frac {486\,x^5}{125} \]

[In]

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

[Out]

(444*x)/125 + (163*log(x + 3/5))/78125 - 11/(390625*(x + 3/5)) + (24093*x^2)/6250 - (1854*x^3)/625 - (3969*x^4
)/500 - (486*x^5)/125