3.1702 \(\int \frac {(2+3 x)^4}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac {9261}{58564 (1-2 x)}-\frac {138}{366025 (5 x+3)}+\frac {2401}{10648 (1-2 x)^2}-\frac {1}{66550 (5 x+3)^2}-\frac {294 \log (1-2 x)}{161051}+\frac {294 \log (5 x+3)}{161051} \]

[Out]

2401/10648/(1-2*x)^2-9261/58564/(1-2*x)-1/66550/(3+5*x)^2-138/366025/(3+5*x)-294/161051*ln(1-2*x)+294/161051*l
n(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} \[ -\frac {9261}{58564 (1-2 x)}-\frac {138}{366025 (5 x+3)}+\frac {2401}{10648 (1-2 x)^2}-\frac {1}{66550 (5 x+3)^2}-\frac {294 \log (1-2 x)}{161051}+\frac {294 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

2401/(10648*(1 - 2*x)^2) - 9261/(58564*(1 - 2*x)) - 1/(66550*(3 + 5*x)^2) - 138/(366025*(3 + 5*x)) - (294*Log[
1 - 2*x])/161051 + (294*Log[3 + 5*x])/161051

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)^4}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {2401}{2662 (-1+2 x)^3}-\frac {9261}{29282 (-1+2 x)^2}-\frac {588}{161051 (-1+2 x)}+\frac {1}{6655 (3+5 x)^3}+\frac {138}{73205 (3+5 x)^2}+\frac {1470}{161051 (3+5 x)}\right ) \, dx\\ &=\frac {2401}{10648 (1-2 x)^2}-\frac {9261}{58564 (1-2 x)}-\frac {1}{66550 (3+5 x)^2}-\frac {138}{366025 (3+5 x)}-\frac {294 \log (1-2 x)}{161051}+\frac {294 \log (3+5 x)}{161051}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 48, normalized size = 0.74 \[ \frac {\frac {11 \left (23130420 x^3+32722281 x^2+14259554 x+1771669\right )}{\left (10 x^2+x-3\right )^2}+58800 \log (-5 x-3)-58800 \log (1-2 x)}{32210200} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

((11*(1771669 + 14259554*x + 32722281*x^2 + 23130420*x^3))/(-3 + x + 10*x^2)^2 + 58800*Log[-3 - 5*x] - 58800*L
og[1 - 2*x])/32210200

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fricas [A]  time = 0.59, size = 95, normalized size = 1.46 \[ \frac {254434620 \, x^{3} + 359945091 \, x^{2} + 58800 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 58800 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 156855094 \, x + 19488359}{32210200 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/32210200*(254434620*x^3 + 359945091*x^2 + 58800*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) - 58800*(
100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 156855094*x + 19488359)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x +
9)

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giac [A]  time = 1.19, size = 46, normalized size = 0.71 \[ \frac {23130420 \, x^{3} + 32722281 \, x^{2} + 14259554 \, x + 1771669}{2928200 \, {\left (10 \, x^{2} + x - 3\right )}^{2}} + \frac {294}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {294}{161051} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

1/2928200*(23130420*x^3 + 32722281*x^2 + 14259554*x + 1771669)/(10*x^2 + x - 3)^2 + 294/161051*log(abs(5*x + 3
)) - 294/161051*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \[ -\frac {294 \ln \left (2 x -1\right )}{161051}+\frac {294 \ln \left (5 x +3\right )}{161051}-\frac {1}{66550 \left (5 x +3\right )^{2}}-\frac {138}{366025 \left (5 x +3\right )}+\frac {2401}{10648 \left (2 x -1\right )^{2}}+\frac {9261}{58564 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^4/(1-2*x)^3/(5*x+3)^3,x)

[Out]

-1/66550/(5*x+3)^2-138/366025/(5*x+3)+294/161051*ln(5*x+3)+2401/10648/(2*x-1)^2+9261/58564/(2*x-1)-294/161051*
ln(2*x-1)

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maxima [A]  time = 0.64, size = 56, normalized size = 0.86 \[ \frac {23130420 \, x^{3} + 32722281 \, x^{2} + 14259554 \, x + 1771669}{2928200 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {294}{161051} \, \log \left (5 \, x + 3\right ) - \frac {294}{161051} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/2928200*(23130420*x^3 + 32722281*x^2 + 14259554*x + 1771669)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 294/161
051*log(5*x + 3) - 294/161051*log(2*x - 1)

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mupad [B]  time = 1.06, size = 45, normalized size = 0.69 \[ \frac {588\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{161051}+\frac {\frac {1156521\,x^3}{14641000}+\frac {32722281\,x^2}{292820000}+\frac {7129777\,x}{146410000}+\frac {1771669}{292820000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 2)^4/((2*x - 1)^3*(5*x + 3)^3),x)

[Out]

(588*atanh((20*x)/11 + 1/11))/161051 + ((7129777*x)/146410000 + (32722281*x^2)/292820000 + (1156521*x^3)/14641
000 + 1771669/292820000)/(x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100)

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sympy [A]  time = 0.19, size = 56, normalized size = 0.86 \[ - \frac {- 23130420 x^{3} - 32722281 x^{2} - 14259554 x - 1771669}{292820000 x^{4} + 58564000 x^{3} - 172763800 x^{2} - 17569200 x + 26353800} - \frac {294 \log {\left (x - \frac {1}{2} \right )}}{161051} + \frac {294 \log {\left (x + \frac {3}{5} \right )}}{161051} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**3/(3+5*x)**3,x)

[Out]

-(-23130420*x**3 - 32722281*x**2 - 14259554*x - 1771669)/(292820000*x**4 + 58564000*x**3 - 172763800*x**2 - 17
569200*x + 26353800) - 294*log(x - 1/2)/161051 + 294*log(x + 3/5)/161051

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