3.1293 \(\int \frac{(1-2 x)^2 (2+3 x)^3}{3+5 x} \, dx\)

Optimal. Leaf size=44 \[ \frac{108 x^5}{25}+\frac{54 x^4}{25}-\frac{591 x^3}{125}-\frac{1931 x^2}{1250}+\frac{8293 x}{3125}+\frac{121 \log (5 x+3)}{15625} \]

[Out]

(8293*x)/3125 - (1931*x^2)/1250 - (591*x^3)/125 + (54*x^4)/25 + (108*x^5)/25 + (121*Log[3 + 5*x])/15625

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Rubi [A]  time = 0.0189513, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{108 x^5}{25}+\frac{54 x^4}{25}-\frac{591 x^3}{125}-\frac{1931 x^2}{1250}+\frac{8293 x}{3125}+\frac{121 \log (5 x+3)}{15625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8293*x)/3125 - (1931*x^2)/1250 - (591*x^3)/125 + (54*x^4)/25 + (108*x^5)/25 + (121*Log[3 + 5*x])/15625

Rule 88

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*} \int \frac{(1-2 x)^2 (2+3 x)^3}{3+5 x} \, dx &=\int \left (\frac{8293}{3125}-\frac{1931 x}{625}-\frac{1773 x^2}{125}+\frac{216 x^3}{25}+\frac{108 x^4}{5}+\frac{121}{3125 (3+5 x)}\right ) \, dx\\ &=\frac{8293 x}{3125}-\frac{1931 x^2}{1250}-\frac{591 x^3}{125}+\frac{54 x^4}{25}+\frac{108 x^5}{25}+\frac{121 \log (3+5 x)}{15625}\\ \end{align*}

Mathematica [A]  time = 0.0107388, size = 37, normalized size = 0.84 \[ \frac{675000 x^5+337500 x^4-738750 x^3-241375 x^2+414650 x+1210 \log (5 x+3)+184863}{156250} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(184863 + 414650*x - 241375*x^2 - 738750*x^3 + 337500*x^4 + 675000*x^5 + 1210*Log[3 + 5*x])/156250

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Maple [A]  time = 0.002, size = 33, normalized size = 0.8 \begin{align*}{\frac{8293\,x}{3125}}-{\frac{1931\,{x}^{2}}{1250}}-{\frac{591\,{x}^{3}}{125}}+{\frac{54\,{x}^{4}}{25}}+{\frac{108\,{x}^{5}}{25}}+{\frac{121\,\ln \left ( 3+5\,x \right ) }{15625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8293/3125*x-1931/1250*x^2-591/125*x^3+54/25*x^4+108/25*x^5+121/15625*ln(3+5*x)

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Maxima [A]  time = 1.28447, size = 43, normalized size = 0.98 \begin{align*} \frac{108}{25} \, x^{5} + \frac{54}{25} \, x^{4} - \frac{591}{125} \, x^{3} - \frac{1931}{1250} \, x^{2} + \frac{8293}{3125} \, x + \frac{121}{15625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108/25*x^5 + 54/25*x^4 - 591/125*x^3 - 1931/1250*x^2 + 8293/3125*x + 121/15625*log(5*x + 3)

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Fricas [A]  time = 1.48363, size = 126, normalized size = 2.86 \begin{align*} \frac{108}{25} \, x^{5} + \frac{54}{25} \, x^{4} - \frac{591}{125} \, x^{3} - \frac{1931}{1250} \, x^{2} + \frac{8293}{3125} \, x + \frac{121}{15625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108/25*x^5 + 54/25*x^4 - 591/125*x^3 - 1931/1250*x^2 + 8293/3125*x + 121/15625*log(5*x + 3)

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Sympy [A]  time = 0.086615, size = 41, normalized size = 0.93 \begin{align*} \frac{108 x^{5}}{25} + \frac{54 x^{4}}{25} - \frac{591 x^{3}}{125} - \frac{1931 x^{2}}{1250} + \frac{8293 x}{3125} + \frac{121 \log{\left (5 x + 3 \right )}}{15625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108*x**5/25 + 54*x**4/25 - 591*x**3/125 - 1931*x**2/1250 + 8293*x/3125 + 121*log(5*x + 3)/15625

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Giac [A]  time = 2.90204, size = 45, normalized size = 1.02 \begin{align*} \frac{108}{25} \, x^{5} + \frac{54}{25} \, x^{4} - \frac{591}{125} \, x^{3} - \frac{1931}{1250} \, x^{2} + \frac{8293}{3125} \, x + \frac{121}{15625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108/25*x^5 + 54/25*x^4 - 591/125*x^3 - 1931/1250*x^2 + 8293/3125*x + 121/15625*log(abs(5*x + 3))