3.1423 \(\int \frac{(1-2 x)^3 (2+3 x)^2}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=52 \[ -\frac{24 x^3}{125}+\frac{354 x^2}{625}-\frac{2978 x}{3125}-\frac{1452}{3125 (5 x+3)}-\frac{1331}{31250 (5 x+3)^2}+\frac{1551 \log (5 x+3)}{3125} \]

[Out]

(-2978*x)/3125 + (354*x^2)/625 - (24*x^3)/125 - 1331/(31250*(3 + 5*x)^2) - 1452/(3125*(3 + 5*x)) + (1551*Log[3
+ 5*x])/3125

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Rubi [A]  time = 0.0246695, antiderivative size = 52, 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} \[ -\frac{24 x^3}{125}+\frac{354 x^2}{625}-\frac{2978 x}{3125}-\frac{1452}{3125 (5 x+3)}-\frac{1331}{31250 (5 x+3)^2}+\frac{1551 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2978*x)/3125 + (354*x^2)/625 - (24*x^3)/125 - 1331/(31250*(3 + 5*x)^2) - 1452/(3125*(3 + 5*x)) + (1551*Log[3
+ 5*x])/3125

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)^3 (2+3 x)^2}{(3+5 x)^3} \, dx &=\int \left (-\frac{2978}{3125}+\frac{708 x}{625}-\frac{72 x^2}{125}+\frac{1331}{3125 (3+5 x)^3}+\frac{1452}{625 (3+5 x)^2}+\frac{1551}{625 (3+5 x)}\right ) \, dx\\ &=-\frac{2978 x}{3125}+\frac{354 x^2}{625}-\frac{24 x^3}{125}-\frac{1331}{31250 (3+5 x)^2}-\frac{1452}{3125 (3+5 x)}+\frac{1551 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0329455, size = 53, normalized size = 1.02 \[ -\frac{30000 x^5-52500 x^4+53500 x^3+274500 x^2+221340 x-3102 (5 x+3)^2 \log (6 (5 x+3))+54943}{6250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(54943 + 221340*x + 274500*x^2 + 53500*x^3 - 52500*x^4 + 30000*x^5 - 3102*(3 + 5*x)^2*Log[6*(3 + 5*x)])/(6250
*(3 + 5*x)^2)

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Maple [A]  time = 0.006, size = 41, normalized size = 0.8 \begin{align*} -{\frac{2978\,x}{3125}}+{\frac{354\,{x}^{2}}{625}}-{\frac{24\,{x}^{3}}{125}}-{\frac{1331}{31250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{1452}{9375+15625\,x}}+{\frac{1551\,\ln \left ( 3+5\,x \right ) }{3125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2978/3125*x+354/625*x^2-24/125*x^3-1331/31250/(3+5*x)^2-1452/3125/(3+5*x)+1551/3125*ln(3+5*x)

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Maxima [A]  time = 2.58778, size = 55, normalized size = 1.06 \begin{align*} -\frac{24}{125} \, x^{3} + \frac{354}{625} \, x^{2} - \frac{2978}{3125} \, x - \frac{121 \,{\left (600 \, x + 371\right )}}{31250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{1551}{3125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24/125*x^3 + 354/625*x^2 - 2978/3125*x - 121/31250*(600*x + 371)/(25*x^2 + 30*x + 9) + 1551/3125*log(5*x + 3)

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Fricas [A]  time = 1.35644, size = 192, normalized size = 3.69 \begin{align*} -\frac{150000 \, x^{5} - 262500 \, x^{4} + 267500 \, x^{3} + 734100 \, x^{2} - 15510 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 340620 \, x + 44891}{31250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/31250*(150000*x^5 - 262500*x^4 + 267500*x^3 + 734100*x^2 - 15510*(25*x^2 + 30*x + 9)*log(5*x + 3) + 340620*
x + 44891)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.122591, size = 42, normalized size = 0.81 \begin{align*} - \frac{24 x^{3}}{125} + \frac{354 x^{2}}{625} - \frac{2978 x}{3125} - \frac{72600 x + 44891}{781250 x^{2} + 937500 x + 281250} + \frac{1551 \log{\left (5 x + 3 \right )}}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-24*x**3/125 + 354*x**2/625 - 2978*x/3125 - (72600*x + 44891)/(781250*x**2 + 937500*x + 281250) + 1551*log(5*x
+ 3)/3125

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Giac [A]  time = 1.95947, size = 50, normalized size = 0.96 \begin{align*} -\frac{24}{125} \, x^{3} + \frac{354}{625} \, x^{2} - \frac{2978}{3125} \, x - \frac{121 \,{\left (600 \, x + 371\right )}}{31250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{1551}{3125} \, \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)^3*(2+3*x)^2/(3+5*x)^3,x, algorithm="giac")

[Out]

-24/125*x^3 + 354/625*x^2 - 2978/3125*x - 121/31250*(600*x + 371)/(5*x + 3)^2 + 1551/3125*log(abs(5*x + 3))