3.1325 \(\int \frac{(1-2 x)^2 (2+3 x)^4}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=59 \[ \frac{81 x^4}{125}-\frac{72 x^3}{625}-\frac{4779 x^2}{6250}+\frac{1419 x}{3125}-\frac{1408}{78125 (5 x+3)}-\frac{121}{156250 (5 x+3)^2}+\frac{1202 \log (5 x+3)}{15625} \]

[Out]

(1419*x)/3125 - (4779*x^2)/6250 - (72*x^3)/625 + (81*x^4)/125 - 121/(156250*(3 + 5*x)^2) - 1408/(78125*(3 + 5*
x)) + (1202*Log[3 + 5*x])/15625

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Rubi [A]  time = 0.0283423, antiderivative size = 59, 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} \[ \frac{81 x^4}{125}-\frac{72 x^3}{625}-\frac{4779 x^2}{6250}+\frac{1419 x}{3125}-\frac{1408}{78125 (5 x+3)}-\frac{121}{156250 (5 x+3)^2}+\frac{1202 \log (5 x+3)}{15625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1419*x)/3125 - (4779*x^2)/6250 - (72*x^3)/625 + (81*x^4)/125 - 121/(156250*(3 + 5*x)^2) - 1408/(78125*(3 + 5*
x)) + (1202*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)^4}{(3+5 x)^3} \, dx &=\int \left (\frac{1419}{3125}-\frac{4779 x}{3125}-\frac{216 x^2}{625}+\frac{324 x^3}{125}+\frac{121}{15625 (3+5 x)^3}+\frac{1408}{15625 (3+5 x)^2}+\frac{1202}{3125 (3+5 x)}\right ) \, dx\\ &=\frac{1419 x}{3125}-\frac{4779 x^2}{6250}-\frac{72 x^3}{625}+\frac{81 x^4}{125}-\frac{121}{156250 (3+5 x)^2}-\frac{1408}{78125 (3+5 x)}+\frac{1202 \log (3+5 x)}{15625}\\ \end{align*}

Mathematica [A]  time = 0.0317847, size = 58, normalized size = 0.98 \[ \frac{506250 x^6+517500 x^5-523125 x^4-394500 x^3+553500 x^2+536320 x+2404 (5 x+3)^2 \log (6 (5 x+3))+121714}{31250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(121714 + 536320*x + 553500*x^2 - 394500*x^3 - 523125*x^4 + 517500*x^5 + 506250*x^6 + 2404*(3 + 5*x)^2*Log[6*(
3 + 5*x)])/(31250*(3 + 5*x)^2)

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \begin{align*}{\frac{1419\,x}{3125}}-{\frac{4779\,{x}^{2}}{6250}}-{\frac{72\,{x}^{3}}{625}}+{\frac{81\,{x}^{4}}{125}}-{\frac{121}{156250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{1408}{234375+390625\,x}}+{\frac{1202\,\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)^4/(3+5*x)^3,x)

[Out]

1419/3125*x-4779/6250*x^2-72/625*x^3+81/125*x^4-121/156250/(3+5*x)^2-1408/78125/(3+5*x)+1202/15625*ln(3+5*x)

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Maxima [A]  time = 1.64782, size = 62, normalized size = 1.05 \begin{align*} \frac{81}{125} \, x^{4} - \frac{72}{625} \, x^{3} - \frac{4779}{6250} \, x^{2} + \frac{1419}{3125} \, x - \frac{11 \,{\left (1280 \, x + 779\right )}}{156250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{1202}{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)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

81/125*x^4 - 72/625*x^3 - 4779/6250*x^2 + 1419/3125*x - 11/156250*(1280*x + 779)/(25*x^2 + 30*x + 9) + 1202/15
625*log(5*x + 3)

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Fricas [A]  time = 1.3018, size = 215, normalized size = 3.64 \begin{align*} \frac{2531250 \, x^{6} + 2587500 \, x^{5} - 2615625 \, x^{4} - 1972500 \, x^{3} + 1053225 \, x^{2} + 12020 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 624470 \, x - 8569}{156250 \,{\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)^2*(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/156250*(2531250*x^6 + 2587500*x^5 - 2615625*x^4 - 1972500*x^3 + 1053225*x^2 + 12020*(25*x^2 + 30*x + 9)*log(
5*x + 3) + 624470*x - 8569)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.122422, size = 49, normalized size = 0.83 \begin{align*} \frac{81 x^{4}}{125} - \frac{72 x^{3}}{625} - \frac{4779 x^{2}}{6250} + \frac{1419 x}{3125} - \frac{14080 x + 8569}{3906250 x^{2} + 4687500 x + 1406250} + \frac{1202 \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)**4/(3+5*x)**3,x)

[Out]

81*x**4/125 - 72*x**3/625 - 4779*x**2/6250 + 1419*x/3125 - (14080*x + 8569)/(3906250*x**2 + 4687500*x + 140625
0) + 1202*log(5*x + 3)/15625

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Giac [A]  time = 1.7011, size = 57, normalized size = 0.97 \begin{align*} \frac{81}{125} \, x^{4} - \frac{72}{625} \, x^{3} - \frac{4779}{6250} \, x^{2} + \frac{1419}{3125} \, x - \frac{11 \,{\left (1280 \, x + 779\right )}}{156250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{1202}{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)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

81/125*x^4 - 72/625*x^3 - 4779/6250*x^2 + 1419/3125*x - 11/156250*(1280*x + 779)/(5*x + 3)^2 + 1202/15625*log(
abs(5*x + 3))