3.1209 \(\int \frac{(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=55 \[ -\frac{486 x^5}{125}-\frac{3969 x^4}{500}-\frac{1854 x^3}{625}+\frac{24093 x^2}{6250}+\frac{444 x}{125}-\frac{11}{78125 (5 x+3)}+\frac{163 \log (5 x+3)}{78125} \]

[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

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Rubi [A]  time = 0.025099, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{486 x^5}{125}-\frac{3969 x^4}{500}-\frac{1854 x^3}{625}+\frac{24093 x^2}{6250}+\frac{444 x}{125}-\frac{11}{78125 (5 x+3)}+\frac{163 \log (5 x+3)}{78125} \]

Antiderivative was successfully verified.

[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 77

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*} \int \frac{(1-2 x) (2+3 x)^5}{(3+5 x)^2} \, dx &=\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]  time = 0.018817, size = 48, normalized size = 0.87 \[ \frac{-3645000 x^5-7441875 x^4-2781000 x^3+3613950 x^2+3330000 x-\frac{132}{5 x+3}+1956 \log (-3 (5 x+3))+779800}{937500} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.005, size = 42, normalized size = 0.8 \begin{align*}{\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}{234375+390625\,x}}+{\frac{163\,\ln \left ( 3+5\,x \right ) }{78125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]  time = 2.863, size = 55, normalized size = 1. \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 1.46809, size = 188, normalized size = 3.42 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [A]  time = 0.106166, size = 48, normalized size = 0.87 \begin{align*} - \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 3.07442, size = 101, normalized size = 1.84 \begin{align*} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)