3.1669 \(\int \frac {(3+5 x)^3}{(1-2 x)^3 (2+3 x)^4} \, dx\)

Optimal. Leaf size=76 \[ \frac {3267}{16807 (1-2 x)}+\frac {1023}{16807 (3 x+2)}+\frac {1331}{4802 (1-2 x)^2}-\frac {33}{4802 (3 x+2)^2}+\frac {1}{3087 (3 x+2)^3}-\frac {7755 \log (1-2 x)}{117649}+\frac {7755 \log (3 x+2)}{117649} \]

[Out]

1331/4802/(1-2*x)^2+3267/16807/(1-2*x)+1/3087/(2+3*x)^3-33/4802/(2+3*x)^2+1023/16807/(2+3*x)-7755/117649*ln(1-
2*x)+7755/117649*ln(2+3*x)

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Rubi [A]  time = 0.04, antiderivative size = 76, 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} \[ \frac {3267}{16807 (1-2 x)}+\frac {1023}{16807 (3 x+2)}+\frac {1331}{4802 (1-2 x)^2}-\frac {33}{4802 (3 x+2)^2}+\frac {1}{3087 (3 x+2)^3}-\frac {7755 \log (1-2 x)}{117649}+\frac {7755 \log (3 x+2)}{117649} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(4802*(1 - 2*x)^2) + 3267/(16807*(1 - 2*x)) + 1/(3087*(2 + 3*x)^3) - 33/(4802*(2 + 3*x)^2) + 1023/(16807*
(2 + 3*x)) - (7755*Log[1 - 2*x])/117649 + (7755*Log[2 + 3*x])/117649

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)^4} \, dx &=\int \left (-\frac {2662}{2401 (-1+2 x)^3}+\frac {6534}{16807 (-1+2 x)^2}-\frac {15510}{117649 (-1+2 x)}-\frac {1}{343 (2+3 x)^4}+\frac {99}{2401 (2+3 x)^3}-\frac {3069}{16807 (2+3 x)^2}+\frac {23265}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {1331}{4802 (1-2 x)^2}+\frac {3267}{16807 (1-2 x)}+\frac {1}{3087 (2+3 x)^3}-\frac {33}{4802 (2+3 x)^2}+\frac {1023}{16807 (2+3 x)}-\frac {7755 \log (1-2 x)}{117649}+\frac {7755 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 57, normalized size = 0.75 \[ \frac {\frac {7 \left (-2512620 x^4-2303235 x^3+3054740 x^2+4131175 x+1210868\right )}{(1-2 x)^2 (3 x+2)^3}-139590 \log (1-2 x)+139590 \log (6 x+4)}{2117682} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(1210868 + 4131175*x + 3054740*x^2 - 2303235*x^3 - 2512620*x^4))/((1 - 2*x)^2*(2 + 3*x)^3) - 139590*Log[1
- 2*x] + 139590*Log[4 + 6*x])/2117682

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fricas [A]  time = 0.60, size = 115, normalized size = 1.51 \[ -\frac {17588340 \, x^{4} + 16122645 \, x^{3} - 21383180 \, x^{2} - 139590 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 139590 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \log \left (2 \, x - 1\right ) - 28918225 \, x - 8476076}{2117682 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2117682*(17588340*x^4 + 16122645*x^3 - 21383180*x^2 - 139590*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8
)*log(3*x + 2) + 139590*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*log(2*x - 1) - 28918225*x - 8476076)/(
108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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giac [A]  time = 1.20, size = 55, normalized size = 0.72 \[ -\frac {2512620 \, x^{4} + 2303235 \, x^{3} - 3054740 \, x^{2} - 4131175 \, x - 1210868}{302526 \, {\left (3 \, x + 2\right )}^{3} {\left (2 \, x - 1\right )}^{2}} + \frac {7755}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {7755}{117649} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/302526*(2512620*x^4 + 2303235*x^3 - 3054740*x^2 - 4131175*x - 1210868)/((3*x + 2)^3*(2*x - 1)^2) + 7755/117
649*log(abs(3*x + 2)) - 7755/117649*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 63, normalized size = 0.83 \[ -\frac {7755 \ln \left (2 x -1\right )}{117649}+\frac {7755 \ln \left (3 x +2\right )}{117649}+\frac {1}{3087 \left (3 x +2\right )^{3}}-\frac {33}{4802 \left (3 x +2\right )^{2}}+\frac {1023}{16807 \left (3 x +2\right )}+\frac {1331}{4802 \left (2 x -1\right )^{2}}-\frac {3267}{16807 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3087/(3*x+2)^3-33/4802/(3*x+2)^2+1023/16807/(3*x+2)+7755/117649*ln(3*x+2)+1331/4802/(2*x-1)^2-3267/16807/(2*
x-1)-7755/117649*ln(2*x-1)

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maxima [A]  time = 0.56, size = 66, normalized size = 0.87 \[ -\frac {2512620 \, x^{4} + 2303235 \, x^{3} - 3054740 \, x^{2} - 4131175 \, x - 1210868}{302526 \, {\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} + \frac {7755}{117649} \, \log \left (3 \, x + 2\right ) - \frac {7755}{117649} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/302526*(2512620*x^4 + 2303235*x^3 - 3054740*x^2 - 4131175*x - 1210868)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2
+ 4*x + 8) + 7755/117649*log(3*x + 2) - 7755/117649*log(2*x - 1)

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mupad [B]  time = 1.07, size = 53, normalized size = 0.70 \[ \frac {15510\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {-\frac {2585\,x^4}{33614}-\frac {28435\,x^3}{403368}+\frac {763685\,x^2}{8168202}+\frac {4131175\,x}{32672808}+\frac {302717}{8168202}}{x^5+x^4-\frac {5\,x^3}{12}-\frac {29\,x^2}{54}+\frac {x}{27}+\frac {2}{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(15510*atanh((12*x)/7 + 1/7))/117649 + ((4131175*x)/32672808 + (763685*x^2)/8168202 - (28435*x^3)/403368 - (25
85*x^4)/33614 + 302717/8168202)/(x/27 - (29*x^2)/54 - (5*x^3)/12 + x^4 + x^5 + 2/27)

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sympy [A]  time = 0.20, size = 65, normalized size = 0.86 \[ - \frac {2512620 x^{4} + 2303235 x^{3} - 3054740 x^{2} - 4131175 x - 1210868}{32672808 x^{5} + 32672808 x^{4} - 13613670 x^{3} - 17546508 x^{2} + 1210104 x + 2420208} - \frac {7755 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {7755 \log {\left (x + \frac {2}{3} \right )}}{117649} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2512620*x**4 + 2303235*x**3 - 3054740*x**2 - 4131175*x - 1210868)/(32672808*x**5 + 32672808*x**4 - 13613670*
x**3 - 17546508*x**2 + 1210104*x + 2420208) - 7755*log(x - 1/2)/117649 + 7755*log(x + 2/3)/117649

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