3.1620 \(\int \frac{(2+3 x)^5}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=59 \[ \frac{243 x}{500}+\frac{16807}{10648 (1-2 x)}-\frac{169}{831875 (5 x+3)}-\frac{1}{151250 (5 x+3)^2}+\frac{36015 \log (1-2 x)}{29282}+\frac{11562 \log (5 x+3)}{9150625} \]

[Out]

16807/(10648*(1 - 2*x)) + (243*x)/500 - 1/(151250*(3 + 5*x)^2) - 169/(831875*(3 + 5*x)) + (36015*Log[1 - 2*x])
/29282 + (11562*Log[3 + 5*x])/9150625

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Rubi [A]  time = 0.0271595, 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} \[ \frac{243 x}{500}+\frac{16807}{10648 (1-2 x)}-\frac{169}{831875 (5 x+3)}-\frac{1}{151250 (5 x+3)^2}+\frac{36015 \log (1-2 x)}{29282}+\frac{11562 \log (5 x+3)}{9150625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(10648*(1 - 2*x)) + (243*x)/500 - 1/(151250*(3 + 5*x)^2) - 169/(831875*(3 + 5*x)) + (36015*Log[1 - 2*x])
/29282 + (11562*Log[3 + 5*x])/9150625

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)^2 (3+5 x)^3} \, dx &=\int \left (\frac{243}{500}+\frac{16807}{5324 (-1+2 x)^2}+\frac{36015}{14641 (-1+2 x)}+\frac{1}{15125 (3+5 x)^3}+\frac{169}{166375 (3+5 x)^2}+\frac{11562}{1830125 (3+5 x)}\right ) \, dx\\ &=\frac{16807}{10648 (1-2 x)}+\frac{243 x}{500}-\frac{1}{151250 (3+5 x)^2}-\frac{169}{831875 (3+5 x)}+\frac{36015 \log (1-2 x)}{29282}+\frac{11562 \log (3+5 x)}{9150625}\\ \end{align*}

Mathematica [A]  time = 0.0418083, size = 55, normalized size = 0.93 \[ \frac{17788815 (2 x-1)+\frac{115548125}{1-2 x}-\frac{14872}{5 x+3}-\frac{484}{(5 x+3)^2}[phone] \log (1-2 x)+92496 \log (10 x+6)}{73205000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(115548125/(1 - 2*x) + 17788815*(-1 + 2*x) - 484/(3 + 5*x)^2 - 14872/(3 + 5*x) + 90037500*Log[1 - 2*x] + 92496
*Log[6 + 10*x])/73205000

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Maple [A]  time = 0.008, size = 48, normalized size = 0.8 \begin{align*}{\frac{243\,x}{500}}-{\frac{16807}{21296\,x-10648}}+{\frac{36015\,\ln \left ( 2\,x-1 \right ) }{29282}}-{\frac{1}{151250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{169}{2495625+4159375\,x}}+{\frac{11562\,\ln \left ( 3+5\,x \right ) }{9150625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/500*x-16807/10648/(2*x-1)+36015/29282*ln(2*x-1)-1/151250/(3+5*x)^2-169/831875/(3+5*x)+11562/9150625*ln(3+5
*x)

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Maxima [A]  time = 1.10145, size = 66, normalized size = 1.12 \begin{align*} \frac{243}{500} \, x - \frac{52524579 \, x^{2} + 63026538 \, x + 18907055}{1331000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{11562}{9150625} \, \log \left (5 \, x + 3\right ) + \frac{36015}{29282} \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

243/500*x - 1/1331000*(52524579*x^2 + 63026538*x + 18907055)/(50*x^3 + 35*x^2 - 12*x - 9) + 11562/9150625*log(
5*x + 3) + 36015/29282*log(2*x - 1)

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Fricas [A]  time = 1.24844, size = 301, normalized size = 5.1 \begin{align*} \frac{1778881500 \, x^{4} + 1245217050 \, x^{3} - 3315783405 \, x^{2} + 92496 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 90037500 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 3786658260 \, x - 1039888025}{73205000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/73205000*(1778881500*x^4 + 1245217050*x^3 - 3315783405*x^2 + 92496*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3)
+ 90037500*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) - 3786658260*x - 1039888025)/(50*x^3 + 35*x^2 - 12*x - 9
)

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Sympy [A]  time = 0.172143, size = 49, normalized size = 0.83 \begin{align*} \frac{243 x}{500} - \frac{52524579 x^{2} + 63026538 x + 18907055}{66550000 x^{3} + 46585000 x^{2} - 15972000 x - 11979000} + \frac{36015 \log{\left (x - \frac{1}{2} \right )}}{29282} + \frac{11562 \log{\left (x + \frac{3}{5} \right )}}{9150625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243*x/500 - (52524579*x**2 + 63026538*x + 18907055)/(66550000*x**3 + 46585000*x**2 - 15972000*x - 11979000) +
36015*log(x - 1/2)/29282 + 11562*log(x + 3/5)/9150625

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Giac [A]  time = 2.35726, size = 112, normalized size = 1.9 \begin{align*} \frac{{\left (2 \, x - 1\right )}{\left (\frac{391367530}{2 \, x - 1} + \frac{430519419}{{\left (2 \, x - 1\right )}^{2}} + 88944075\right )}}{14641000 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} - \frac{16807}{10648 \,{\left (2 \, x - 1\right )}} - \frac{1539}{1250} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{11562}{9150625} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14641000*(2*x - 1)*(391367530/(2*x - 1) + 430519419/(2*x - 1)^2 + 88944075)/(11/(2*x - 1) + 5)^2 - 16807/106
48/(2*x - 1) - 1539/1250*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 11562/9150625*log(abs(-11/(2*x - 1) - 5))