3.1664 \(\int \frac{(2+3 x) (3+5 x)^3}{(1-2 x)^3} \, dx\)

Optimal. Leaf size=45 \[ -\frac{375 x^2}{16}-\frac{2975 x}{16}-\frac{8349}{16 (1-2 x)}+\frac{9317}{64 (1-2 x)^2}-\frac{2805}{8} \log (1-2 x) \]

[Out]

9317/(64*(1 - 2*x)^2) - 8349/(16*(1 - 2*x)) - (2975*x)/16 - (375*x^2)/16 - (2805*Log[1 - 2*x])/8

________________________________________________________________________________________

Rubi [A]  time = 0.0209593, 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} \[ -\frac{375 x^2}{16}-\frac{2975 x}{16}-\frac{8349}{16 (1-2 x)}+\frac{9317}{64 (1-2 x)^2}-\frac{2805}{8} \log (1-2 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

9317/(64*(1 - 2*x)^2) - 8349/(16*(1 - 2*x)) - (2975*x)/16 - (375*x^2)/16 - (2805*Log[1 - 2*x])/8

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) (3+5 x)^3}{(1-2 x)^3} \, dx &=\int \left (-\frac{2975}{16}-\frac{375 x}{8}-\frac{9317}{16 (-1+2 x)^3}-\frac{8349}{8 (-1+2 x)^2}-\frac{2805}{4 (-1+2 x)}\right ) \, dx\\ &=\frac{9317}{64 (1-2 x)^2}-\frac{8349}{16 (1-2 x)}-\frac{2975 x}{16}-\frac{375 x^2}{16}-\frac{2805}{8} \log (1-2 x)\\ \end{align*}

Mathematica [A]  time = 0.01306, size = 46, normalized size = 1.02 \[ -\frac{3000 x^4+20800 x^3-35700 x^2-14796 x+11220 (1-2 x)^2 \log (1-2 x)+8877}{32 (1-2 x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(8877 - 14796*x - 35700*x^2 + 20800*x^3 + 3000*x^4 + 11220*(1 - 2*x)^2*Log[1 - 2*x])/(32*(1 - 2*x)^2)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{375\,{x}^{2}}{16}}-{\frac{2975\,x}{16}}-{\frac{2805\,\ln \left ( 2\,x-1 \right ) }{8}}+{\frac{9317}{64\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{8349}{32\,x-16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375/16*x^2-2975/16*x-2805/8*ln(2*x-1)+9317/64/(2*x-1)^2+8349/16/(2*x-1)

________________________________________________________________________________________

Maxima [A]  time = 2.11124, size = 49, normalized size = 1.09 \begin{align*} -\frac{375}{16} \, x^{2} - \frac{2975}{16} \, x + \frac{121 \,{\left (552 \, x - 199\right )}}{64 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{2805}{8} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375/16*x^2 - 2975/16*x + 121/64*(552*x - 199)/(4*x^2 - 4*x + 1) - 2805/8*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.40517, size = 158, normalized size = 3.51 \begin{align*} -\frac{6000 \, x^{4} + 41600 \, x^{3} - 46100 \, x^{2} + 22440 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 54892 \, x + 24079}{64 \,{\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)*(3+5*x)^3/(1-2*x)^3,x, algorithm="fricas")

[Out]

-1/64*(6000*x^4 + 41600*x^3 - 46100*x^2 + 22440*(4*x^2 - 4*x + 1)*log(2*x - 1) - 54892*x + 24079)/(4*x^2 - 4*x
+ 1)

________________________________________________________________________________________

Sympy [A]  time = 0.123523, size = 36, normalized size = 0.8 \begin{align*} - \frac{375 x^{2}}{16} - \frac{2975 x}{16} + \frac{66792 x - 24079}{256 x^{2} - 256 x + 64} - \frac{2805 \log{\left (2 x - 1 \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-375*x**2/16 - 2975*x/16 + (66792*x - 24079)/(256*x**2 - 256*x + 64) - 2805*log(2*x - 1)/8

________________________________________________________________________________________

Giac [A]  time = 3.6649, size = 43, normalized size = 0.96 \begin{align*} -\frac{375}{16} \, x^{2} - \frac{2975}{16} \, x + \frac{121 \,{\left (552 \, x - 199\right )}}{64 \,{\left (2 \, x - 1\right )}^{2}} - \frac{2805}{8} \, \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)*(3+5*x)^3/(1-2*x)^3,x, algorithm="giac")

[Out]

-375/16*x^2 - 2975/16*x + 121/64*(552*x - 199)/(2*x - 1)^2 - 2805/8*log(abs(2*x - 1))