3.1701 \(\int \frac{(2+3 x)^5}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac{36015}{29282 (1-2 x)}-\frac{171}{1830125 (5 x+3)}+\frac{16807}{21296 (1-2 x)^2}-\frac{1}{332750 (5 x+3)^2}-\frac{313845 \log (1-2 x)}{1288408}+\frac{11904 \log (5 x+3)}{20131375} \]

[Out]

16807/(21296*(1 - 2*x)^2) - 36015/(29282*(1 - 2*x)) - 1/(332750*(3 + 5*x)^2) - 171/(1830125*(3 + 5*x)) - (3138
45*Log[1 - 2*x])/1288408 + (11904*Log[3 + 5*x])/20131375

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Rubi [A]  time = 0.0299426, antiderivative size = 65, 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} \[ -\frac{36015}{29282 (1-2 x)}-\frac{171}{1830125 (5 x+3)}+\frac{16807}{21296 (1-2 x)^2}-\frac{1}{332750 (5 x+3)^2}-\frac{313845 \log (1-2 x)}{1288408}+\frac{11904 \log (5 x+3)}{20131375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

16807/(21296*(1 - 2*x)^2) - 36015/(29282*(1 - 2*x)) - 1/(332750*(3 + 5*x)^2) - 171/(1830125*(3 + 5*x)) - (3138
45*Log[1 - 2*x])/1288408 + (11904*Log[3 + 5*x])/20131375

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)^3} \, dx &=\int \left (-\frac{16807}{5324 (-1+2 x)^3}-\frac{36015}{14641 (-1+2 x)^2}-\frac{313845}{644204 (-1+2 x)}+\frac{1}{33275 (3+5 x)^3}+\frac{171}{366025 (3+5 x)^2}+\frac{11904}{4026275 (3+5 x)}\right ) \, dx\\ &=\frac{16807}{21296 (1-2 x)^2}-\frac{36015}{29282 (1-2 x)}-\frac{1}{332750 (3+5 x)^2}-\frac{171}{1830125 (3+5 x)}-\frac{313845 \log (1-2 x)}{1288408}+\frac{11904 \log (3+5 x)}{20131375}\\ \end{align*}

Mathematica [A]  time = 0.0283377, size = 50, normalized size = 0.77 \[ \frac{\frac{11 \left (1800695280 x^3+1838287161 x^2+261128254 x-116156671\right )}{\left (10 x^2+x-3\right )^2}-78461250 \log (3-6 x)+190464 \log (-3 (5 x+3))}{322102000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(-116156671 + 261128254*x + 1838287161*x^2 + 1800695280*x^3))/(-3 + x + 10*x^2)^2 - 78461250*Log[3 - 6*x]
+ 190464*Log[-3*(3 + 5*x)])/322102000

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \begin{align*}{\frac{16807}{21296\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{36015}{58564\,x-29282}}-{\frac{313845\,\ln \left ( 2\,x-1 \right ) }{1288408}}-{\frac{1}{332750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{171}{5490375+9150625\,x}}+{\frac{11904\,\ln \left ( 3+5\,x \right ) }{20131375}} \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)^3,x)

[Out]

16807/21296/(2*x-1)^2+36015/29282/(2*x-1)-313845/1288408*ln(2*x-1)-1/332750/(3+5*x)^2-171/1830125/(3+5*x)+1190
4/20131375*ln(3+5*x)

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Maxima [A]  time = 2.72467, size = 76, normalized size = 1.17 \begin{align*} \frac{1800695280 \, x^{3} + 1838287161 \, x^{2} + 261128254 \, x - 116156671}{29282000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac{11904}{20131375} \, \log \left (5 \, x + 3\right ) - \frac{313845}{1288408} \, \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)^3,x, algorithm="maxima")

[Out]

1/29282000*(1800695280*x^3 + 1838287161*x^2 + 261128254*x - 116156671)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) +
11904/20131375*log(5*x + 3) - 313845/1288408*log(2*x - 1)

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Fricas [A]  time = 1.50241, size = 320, normalized size = 4.92 \begin{align*} \frac{19807648080 \, x^{3} + 20221158771 \, x^{2} + 190464 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 78461250 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 2872410794 \, x - 1277723381}{322102000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

1/322102000*(19807648080*x^3 + 20221158771*x^2 + 190464*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) - 7
8461250*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 2872410794*x - 1277723381)/(100*x^4 + 20*x^3 - 59
*x^2 - 6*x + 9)

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Sympy [A]  time = 0.193623, size = 54, normalized size = 0.83 \begin{align*} \frac{1800695280 x^{3} + 1838287161 x^{2} + 261128254 x - 116156671}{2928200000 x^{4} + 585640000 x^{3} - 1727638000 x^{2} - 175692000 x + 263538000} - \frac{313845 \log{\left (x - \frac{1}{2} \right )}}{1288408} + \frac{11904 \log{\left (x + \frac{3}{5} \right )}}{20131375} \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)**3,x)

[Out]

(1800695280*x**3 + 1838287161*x**2 + 261128254*x - 116156671)/(2928200000*x**4 + 585640000*x**3 - 1727638000*x
**2 - 175692000*x + 263538000) - 313845*log(x - 1/2)/1288408 + 11904*log(x + 3/5)/20131375

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Giac [A]  time = 4.56542, size = 68, normalized size = 1.05 \begin{align*} \frac{1800695280 \, x^{3} + 1838287161 \, x^{2} + 261128254 \, x - 116156671}{29282000 \,{\left (5 \, x + 3\right )}^{2}{\left (2 \, x - 1\right )}^{2}} + \frac{11904}{20131375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{313845}{1288408} \, \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)^5/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

1/29282000*(1800695280*x^3 + 1838287161*x^2 + 261128254*x - 116156671)/((5*x + 3)^2*(2*x - 1)^2) + 11904/20131
375*log(abs(5*x + 3)) - 313845/1288408*log(abs(2*x - 1))