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)-313845/1288408*ln(1-2*x)+1190
4/20131375*ln(3+5*x)

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Rubi [A]  time = 0.03, antiderivative size = 65, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.70, size = 95, normalized size = 1.46 \[ \frac {19807648080 \, x^{3} + 20221158771 \, x^{2} + 190464 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 78461250 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 2872410794 \, x - 1277723381}{322102000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

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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giac [A]  time = 1.14, size = 50, normalized size = 0.77 \[ \frac {1800695280 \, x^{3} + 1838287161 \, x^{2} + 261128254 \, x - 116156671}{29282000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {11904}{20131375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {313845}{1288408} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \[ -\frac {313845 \ln \left (2 x -1\right )}{1288408}+\frac {11904 \ln \left (5 x +3\right )}{20131375}-\frac {1}{332750 \left (5 x +3\right )^{2}}-\frac {171}{1830125 \left (5 x +3\right )}+\frac {16807}{21296 \left (2 x -1\right )^{2}}+\frac {36015}{29282 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/332750/(5*x+3)^2-171/1830125/(5*x+3)+11904/20131375*ln(5*x+3)+16807/21296/(2*x-1)^2+36015/29282/(2*x-1)-313
845/1288408*ln(2*x-1)

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maxima [A]  time = 0.52, size = 56, normalized size = 0.86 \[ \frac {1800695280 \, x^{3} + 1838287161 \, x^{2} + 261128254 \, x - 116156671}{29282000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {11904}{20131375} \, \log \left (5 \, x + 3\right ) - \frac {313845}{1288408} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 0.10, size = 49, normalized size = 0.75 \[ \frac {11904\,\ln \left (x+\frac {3}{5}\right )}{20131375}-\frac {313845\,\ln \left (x-\frac {1}{2}\right )}{1288408}+\frac {\frac {22508691\,x^3}{36602500}+\frac {1838287161\,x^2}{2928200000}+\frac {130564127\,x}{1464100000}-\frac {116156671}{2928200000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(11904*log(x + 3/5))/20131375 - (313845*log(x - 1/2))/1288408 + ((130564127*x)/1464100000 + (1838287161*x^2)/2
928200000 + (22508691*x^3)/36602500 - 116156671/2928200000)/(x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100)

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sympy [A]  time = 0.21, size = 54, normalized size = 0.83 \[ - \frac {- 1800695280 x^{3} - 1838287161 x^{2} - 261128254 x + 116156671}{2928200000 x^{4} + 585640000 x^{3} - 1727638000 x^{2} - 175692000 x + 263538000} - \frac {313845 \log {\left (x - \frac {1}{2} \right )}}{1288408} + \frac {11904 \log {\left (x + \frac {3}{5} \right )}}{20131375} \]

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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