3.1687 \(\int \frac{(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=59 \[ -\frac{243 x}{200}-\frac{228095}{21296 (1-2 x)}-\frac{1}{166375 (5 x+3)}+\frac{16807}{3872 (1-2 x)^2}-\frac{1034145 \log (1-2 x)}{234256}+\frac{171 \log (5 x+3)}{1830125} \]

[Out]

16807/(3872*(1 - 2*x)^2) - 228095/(21296*(1 - 2*x)) - (243*x)/200 - 1/(166375*(3 + 5*x)) - (1034145*Log[1 - 2*
x])/234256 + (171*Log[3 + 5*x])/1830125

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Rubi [A]  time = 0.0266856, 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} \[ -\frac{243 x}{200}-\frac{228095}{21296 (1-2 x)}-\frac{1}{166375 (5 x+3)}+\frac{16807}{3872 (1-2 x)^2}-\frac{1034145 \log (1-2 x)}{234256}+\frac{171 \log (5 x+3)}{1830125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(3872*(1 - 2*x)^2) - 228095/(21296*(1 - 2*x)) - (243*x)/200 - 1/(166375*(3 + 5*x)) - (1034145*Log[1 - 2*
x])/234256 + (171*Log[3 + 5*x])/1830125

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{(2+3 x)^5}{(1-2 x)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{243}{200}-\frac{16807}{968 (-1+2 x)^3}-\frac{228095}{10648 (-1+2 x)^2}-\frac{1034145}{117128 (-1+2 x)}+\frac{1}{33275 (3+5 x)^2}+\frac{171}{366025 (3+5 x)}\right ) \, dx\\ &=\frac{16807}{3872 (1-2 x)^2}-\frac{228095}{21296 (1-2 x)}-\frac{243 x}{200}-\frac{1}{166375 (3+5 x)}-\frac{1034145 \log (1-2 x)}{234256}+\frac{171 \log (3+5 x)}{1830125}\\ \end{align*}

Mathematica [A]  time = 0.0351674, size = 55, normalized size = 0.93 \[ \frac{35577630 (1-2 x)+\frac{627261250}{2 x-1}-\frac{352}{5 x+3}+\frac{254205875}{(1-2 x)^2}-258536250 \log (1-2 x)+5472 \log (10 x+6)}{58564000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(254205875/(1 - 2*x)^2 + 35577630*(1 - 2*x) + 627261250/(-1 + 2*x) - 352/(3 + 5*x) - 258536250*Log[1 - 2*x] +
5472*Log[6 + 10*x])/58564000

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Maple [A]  time = 0.009, size = 48, normalized size = 0.8 \begin{align*} -{\frac{243\,x}{200}}+{\frac{16807}{3872\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{228095}{42592\,x-21296}}-{\frac{1034145\,\ln \left ( 2\,x-1 \right ) }{234256}}-{\frac{1}{499125+831875\,x}}+{\frac{171\,\ln \left ( 3+5\,x \right ) }{1830125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/200*x+16807/3872/(2*x-1)^2+228095/21296/(2*x-1)-1034145/234256*ln(2*x-1)-1/166375/(3+5*x)+171/1830125*ln(
3+5*x)

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Maxima [A]  time = 1.06381, size = 66, normalized size = 1.12 \begin{align*} -\frac{243}{200} \, x + \frac{570237372 \, x^{2} + 172572003 \, x - 101742407}{5324000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{171}{1830125} \, \log \left (5 \, x + 3\right ) - \frac{1034145}{234256} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/200*x + 1/5324000*(570237372*x^2 + 172572003*x - 101742407)/(20*x^3 - 8*x^2 - 7*x + 3) + 171/1830125*log(
5*x + 3) - 1034145/234256*log(2*x - 1)

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Fricas [A]  time = 1.47731, size = 293, normalized size = 4.97 \begin{align*} -\frac{1423105200 \, x^{4} - 569242080 \, x^{3} - 6770697912 \, x^{2} - 5472 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 258536250 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) - 1684826253 \, x + 1119166477}{58564000 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/58564000*(1423105200*x^4 - 569242080*x^3 - 6770697912*x^2 - 5472*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) +
258536250*(20*x^3 - 8*x^2 - 7*x + 3)*log(2*x - 1) - 1684826253*x + 1119166477)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.183134, size = 49, normalized size = 0.83 \begin{align*} - \frac{243 x}{200} + \frac{570237372 x^{2} + 172572003 x - 101742407}{106480000 x^{3} - 42592000 x^{2} - 37268000 x + 15972000} - \frac{1034145 \log{\left (x - \frac{1}{2} \right )}}{234256} + \frac{171 \log{\left (x + \frac{3}{5} \right )}}{1830125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x/200 + (570237372*x**2 + 172572003*x - 101742407)/(106480000*x**3 - 42592000*x**2 - 37268000*x + 1597200
0) - 1034145*log(x - 1/2)/234256 + 171*log(x + 3/5)/1830125

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Giac [A]  time = 4.53909, size = 112, normalized size = 1.9 \begin{align*} \frac{{\left (5 \, x + 3\right )}{\left (\frac{389138447}{5 \, x + 3} - \frac{1420901823}{{\left (5 \, x + 3\right )}^{2}} - 14231052\right )}}{14641000 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} - \frac{1}{166375 \,{\left (5 \, x + 3\right )}} + \frac{8829}{2000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{1034145}{234256} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14641000*(5*x + 3)*(389138447/(5*x + 3) - 1420901823/(5*x + 3)^2 - 14231052)/(11/(5*x + 3) - 2)^2 - 1/166375
/(5*x + 3) + 8829/2000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 1034145/234256*log(abs(-11/(5*x + 3) + 2))