3.1668 \(\int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^3} \, dx\)

Optimal. Leaf size=65 \[ -\frac {363}{2401 (1-2 x)}-\frac {33}{2401 (3 x+2)}+\frac {1331}{1372 (1-2 x)^2}+\frac {1}{2058 (3 x+2)^2}+\frac {1023 \log (1-2 x)}{16807}-\frac {1023 \log (3 x+2)}{16807} \]

[Out]

1331/1372/(1-2*x)^2-363/2401/(1-2*x)+1/2058/(2+3*x)^2-33/2401/(2+3*x)+1023/16807*ln(1-2*x)-1023/16807*ln(2+3*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} \[ -\frac {363}{2401 (1-2 x)}-\frac {33}{2401 (3 x+2)}+\frac {1331}{1372 (1-2 x)^2}+\frac {1}{2058 (3 x+2)^2}+\frac {1023 \log (1-2 x)}{16807}-\frac {1023 \log (3 x+2)}{16807} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(1372*(1 - 2*x)^2) - 363/(2401*(1 - 2*x)) + 1/(2058*(2 + 3*x)^2) - 33/(2401*(2 + 3*x)) + (1023*Log[1 - 2*
x])/16807 - (1023*Log[2 + 3*x])/16807

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 {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^3} \, dx &=\int \left (-\frac {1331}{343 (-1+2 x)^3}-\frac {726}{2401 (-1+2 x)^2}+\frac {2046}{16807 (-1+2 x)}-\frac {1}{343 (2+3 x)^3}+\frac {99}{2401 (2+3 x)^2}-\frac {3069}{16807 (2+3 x)}\right ) \, dx\\ &=\frac {1331}{1372 (1-2 x)^2}-\frac {363}{2401 (1-2 x)}+\frac {1}{2058 (2+3 x)^2}-\frac {33}{2401 (2+3 x)}+\frac {1023 \log (1-2 x)}{16807}-\frac {1023 \log (2+3 x)}{16807}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 48, normalized size = 0.74 \[ \frac {\frac {7 \left (73656 x^3+318539 x^2+319912 x+93602\right )}{\left (6 x^2+x-2\right )^2}+12276 \log (1-2 x)-12276 \log (3 x+2)}{201684} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(93602 + 319912*x + 318539*x^2 + 73656*x^3))/(-2 + x + 6*x^2)^2 + 12276*Log[1 - 2*x] - 12276*Log[2 + 3*x])
/201684

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fricas [A]  time = 0.66, size = 95, normalized size = 1.46 \[ \frac {515592 \, x^{3} + 2229773 \, x^{2} - 12276 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 12276 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (2 \, x - 1\right ) + 2239384 \, x + 655214}{201684 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/201684*(515592*x^3 + 2229773*x^2 - 12276*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(3*x + 2) + 12276*(36*x^4 +
12*x^3 - 23*x^2 - 4*x + 4)*log(2*x - 1) + 2239384*x + 655214)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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giac [A]  time = 1.18, size = 46, normalized size = 0.71 \[ \frac {73656 \, x^{3} + 318539 \, x^{2} + 319912 \, x + 93602}{28812 \, {\left (6 \, x^{2} + x - 2\right )}^{2}} - \frac {1023}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1023}{16807} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28812*(73656*x^3 + 318539*x^2 + 319912*x + 93602)/(6*x^2 + x - 2)^2 - 1023/16807*log(abs(3*x + 2)) + 1023/16
807*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \[ \frac {1023 \ln \left (2 x -1\right )}{16807}-\frac {1023 \ln \left (3 x +2\right )}{16807}+\frac {1}{2058 \left (3 x +2\right )^{2}}-\frac {33}{2401 \left (3 x +2\right )}+\frac {1331}{1372 \left (2 x -1\right )^{2}}+\frac {363}{2401 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2058/(3*x+2)^2-33/2401/(3*x+2)-1023/16807*ln(3*x+2)+1331/1372/(2*x-1)^2+363/2401/(2*x-1)+1023/16807*ln(2*x-1
)

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maxima [A]  time = 0.55, size = 56, normalized size = 0.86 \[ \frac {73656 \, x^{3} + 318539 \, x^{2} + 319912 \, x + 93602}{28812 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} - \frac {1023}{16807} \, \log \left (3 \, x + 2\right ) + \frac {1023}{16807} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28812*(73656*x^3 + 318539*x^2 + 319912*x + 93602)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4) - 1023/16807*log(3*x
+ 2) + 1023/16807*log(2*x - 1)

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mupad [B]  time = 0.04, size = 45, normalized size = 0.69 \[ \frac {\frac {341\,x^3}{4802}+\frac {318539\,x^2}{1037232}+\frac {39989\,x}{129654}+\frac {46801}{518616}}{x^4+\frac {x^3}{3}-\frac {23\,x^2}{36}-\frac {x}{9}+\frac {1}{9}}-\frac {2046\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((39989*x)/129654 + (318539*x^2)/1037232 + (341*x^3)/4802 + 46801/518616)/(x^3/3 - (23*x^2)/36 - x/9 + x^4 + 1
/9) - (2046*atanh((12*x)/7 + 1/7))/16807

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sympy [A]  time = 0.19, size = 56, normalized size = 0.86 \[ - \frac {- 73656 x^{3} - 318539 x^{2} - 319912 x - 93602}{1037232 x^{4} + 345744 x^{3} - 662676 x^{2} - 115248 x + 115248} + \frac {1023 \log {\left (x - \frac {1}{2} \right )}}{16807} - \frac {1023 \log {\left (x + \frac {2}{3} \right )}}{16807} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-73656*x**3 - 318539*x**2 - 319912*x - 93602)/(1037232*x**4 + 345744*x**3 - 662676*x**2 - 115248*x + 115248)
+ 1023*log(x - 1/2)/16807 - 1023*log(x + 2/3)/16807

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