3.1424 \(\int \frac{(1-2 x)^3 (2+3 x)}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac{12 x^2}{125}+\frac{316 x}{625}-\frac{3267}{3125 (5 x+3)}-\frac{1331}{6250 (5 x+3)^2}-\frac{2046 \log (5 x+3)}{3125} \]

[Out]

(316*x)/625 - (12*x^2)/125 - 1331/(6250*(3 + 5*x)^2) - 3267/(3125*(3 + 5*x)) - (2046*Log[3 + 5*x])/3125

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Rubi [A]  time = 0.0210411, antiderivative size = 45, 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} \[ -\frac{12 x^2}{125}+\frac{316 x}{625}-\frac{3267}{3125 (5 x+3)}-\frac{1331}{6250 (5 x+3)^2}-\frac{2046 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(316*x)/625 - (12*x^2)/125 - 1331/(6250*(3 + 5*x)^2) - 3267/(3125*(3 + 5*x)) - (2046*Log[3 + 5*x])/3125

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)^3 (2+3 x)}{(3+5 x)^3} \, dx &=\int \left (\frac{316}{625}-\frac{24 x}{125}+\frac{1331}{625 (3+5 x)^3}+\frac{3267}{625 (3+5 x)^2}-\frac{2046}{625 (3+5 x)}\right ) \, dx\\ &=\frac{316 x}{625}-\frac{12 x^2}{125}-\frac{1331}{6250 (3+5 x)^2}-\frac{3267}{3125 (3+5 x)}-\frac{2046 \log (3+5 x)}{3125}\\ \end{align*}

Mathematica [A]  time = 0.0187059, size = 46, normalized size = 1.02 \[ -\frac{15000 x^4-61000 x^3-53650 x^2+47130 x+4092 (5 x+3)^2 \log (10 x+6)+33803}{6250 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(33803 + 47130*x - 53650*x^2 - 61000*x^3 + 15000*x^4 + 4092*(3 + 5*x)^2*Log[6 + 10*x])/(6250*(3 + 5*x)^2)

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{316\,x}{625}}-{\frac{12\,{x}^{2}}{125}}-{\frac{1331}{6250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{3267}{9375+15625\,x}}-{\frac{2046\,\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)/(3+5*x)^3,x)

[Out]

316/625*x-12/125*x^2-1331/6250/(3+5*x)^2-3267/3125/(3+5*x)-2046/3125*ln(3+5*x)

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Maxima [A]  time = 1.01657, size = 49, normalized size = 1.09 \begin{align*} -\frac{12}{125} \, x^{2} + \frac{316}{625} \, x - \frac{121 \,{\left (270 \, x + 173\right )}}{6250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{2046}{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)/(3+5*x)^3,x, algorithm="maxima")

[Out]

-12/125*x^2 + 316/625*x - 121/6250*(270*x + 173)/(25*x^2 + 30*x + 9) - 2046/3125*log(5*x + 3)

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Fricas [A]  time = 1.33372, size = 165, normalized size = 3.67 \begin{align*} -\frac{15000 \, x^{4} - 61000 \, x^{3} - 89400 \, x^{2} + 4092 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 4230 \, x + 20933}{6250 \,{\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)/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/6250*(15000*x^4 - 61000*x^3 - 89400*x^2 + 4092*(25*x^2 + 30*x + 9)*log(5*x + 3) + 4230*x + 20933)/(25*x^2 +
30*x + 9)

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Sympy [A]  time = 0.117702, size = 36, normalized size = 0.8 \begin{align*} - \frac{12 x^{2}}{125} + \frac{316 x}{625} - \frac{32670 x + 20933}{156250 x^{2} + 187500 x + 56250} - \frac{2046 \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)/(3+5*x)**3,x)

[Out]

-12*x**2/125 + 316*x/625 - (32670*x + 20933)/(156250*x**2 + 187500*x + 56250) - 2046*log(5*x + 3)/3125

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Giac [A]  time = 2.07504, size = 43, normalized size = 0.96 \begin{align*} -\frac{12}{125} \, x^{2} + \frac{316}{625} \, x - \frac{121 \,{\left (270 \, x + 173\right )}}{6250 \,{\left (5 \, x + 3\right )}^{2}} - \frac{2046}{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)/(3+5*x)^3,x, algorithm="giac")

[Out]

-12/125*x^2 + 316/625*x - 121/6250*(270*x + 173)/(5*x + 3)^2 - 2046/3125*log(abs(5*x + 3))