3.1636 \(\int \frac{(2+3 x)^2 (3+5 x)}{(1-2 x)^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac{45 x}{8}-\frac{707}{16 (1-2 x)}+\frac{539}{32 (1-2 x)^2}-\frac{309}{16} \log (1-2 x) \]

[Out]

539/(32*(1 - 2*x)^2) - 707/(16*(1 - 2*x)) - (45*x)/8 - (309*Log[1 - 2*x])/16

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Rubi [A]  time = 0.0277792, antiderivative size = 38, 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} \[ -\frac{45 x}{8}-\frac{707}{16 (1-2 x)}+\frac{539}{32 (1-2 x)^2}-\frac{309}{16} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

539/(32*(1 - 2*x)^2) - 707/(16*(1 - 2*x)) - (45*x)/8 - (309*Log[1 - 2*x])/16

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{(2+3 x)^2 (3+5 x)}{(1-2 x)^3} \, dx &=\int \left (-\frac{45}{8}-\frac{539}{8 (-1+2 x)^3}-\frac{707}{8 (-1+2 x)^2}-\frac{309}{8 (-1+2 x)}\right ) \, dx\\ &=\frac{539}{32 (1-2 x)^2}-\frac{707}{16 (1-2 x)}-\frac{45 x}{8}-\frac{309}{16} \log (1-2 x)\\ \end{align*}

Mathematica [A]  time = 0.0215737, size = 34, normalized size = 0.89 \[ \frac{1}{32} \left (\frac{360 x^2+2468 x-785}{(1-2 x)^2}-180 x-618 \log (1-2 x)\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-180*x + (-785 + 2468*x + 360*x^2)/(1 - 2*x)^2 - 618*Log[1 - 2*x])/32

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Maple [A]  time = 0.004, size = 31, normalized size = 0.8 \begin{align*} -{\frac{45\,x}{8}}-{\frac{309\,\ln \left ( 2\,x-1 \right ) }{16}}+{\frac{539}{32\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{707}{32\,x-16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/8*x-309/16*ln(2*x-1)+539/32/(2*x-1)^2+707/16/(2*x-1)

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Maxima [A]  time = 1.07712, size = 42, normalized size = 1.11 \begin{align*} -\frac{45}{8} \, x + \frac{7 \,{\left (404 \, x - 125\right )}}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{309}{16} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/8*x + 7/32*(404*x - 125)/(4*x^2 - 4*x + 1) - 309/16*log(2*x - 1)

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Fricas [A]  time = 1.33014, size = 131, normalized size = 3.45 \begin{align*} -\frac{720 \, x^{3} - 720 \, x^{2} + 618 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 2648 \, x + 875}{32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(720*x^3 - 720*x^2 + 618*(4*x^2 - 4*x + 1)*log(2*x - 1) - 2648*x + 875)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.116651, size = 29, normalized size = 0.76 \begin{align*} - \frac{45 x}{8} + \frac{2828 x - 875}{128 x^{2} - 128 x + 32} - \frac{309 \log{\left (2 x - 1 \right )}}{16} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45*x/8 + (2828*x - 875)/(128*x**2 - 128*x + 32) - 309*log(2*x - 1)/16

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Giac [A]  time = 5.90088, size = 36, normalized size = 0.95 \begin{align*} -\frac{45}{8} \, x + \frac{7 \,{\left (404 \, x - 125\right )}}{32 \,{\left (2 \, x - 1\right )}^{2}} - \frac{309}{16} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-45/8*x + 7/32*(404*x - 125)/(2*x - 1)^2 - 309/16*log(abs(2*x - 1))