3.1643 \(\int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx\)

Optimal. Leaf size=87 \[ \frac {1040}{117649 (1-2 x)}-\frac {2280}{117649 (3 x+2)}+\frac {44}{16807 (1-2 x)^2}-\frac {279}{16807 (3 x+2)^2}-\frac {29}{2401 (3 x+2)^3}+\frac {3}{1372 (3 x+2)^4}-\frac {7680 \log (1-2 x)}{823543}+\frac {7680 \log (3 x+2)}{823543} \]

[Out]

44/16807/(1-2*x)^2+1040/117649/(1-2*x)+3/1372/(2+3*x)^4-29/2401/(2+3*x)^3-279/16807/(2+3*x)^2-2280/117649/(2+3
*x)-7680/823543*ln(1-2*x)+7680/823543*ln(2+3*x)

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Rubi [A]  time = 0.05, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {1040}{117649 (1-2 x)}-\frac {2280}{117649 (3 x+2)}+\frac {44}{16807 (1-2 x)^2}-\frac {279}{16807 (3 x+2)^2}-\frac {29}{2401 (3 x+2)^3}+\frac {3}{1372 (3 x+2)^4}-\frac {7680 \log (1-2 x)}{823543}+\frac {7680 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

44/(16807*(1 - 2*x)^2) + 1040/(117649*(1 - 2*x)) + 3/(1372*(2 + 3*x)^4) - 29/(2401*(2 + 3*x)^3) - 279/(16807*(
2 + 3*x)^2) - 2280/(117649*(2 + 3*x)) - (7680*Log[1 - 2*x])/823543 + (7680*Log[2 + 3*x])/823543

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^5} \, dx &=\int \left (-\frac {176}{16807 (-1+2 x)^3}+\frac {2080}{117649 (-1+2 x)^2}-\frac {15360}{823543 (-1+2 x)}-\frac {9}{343 (2+3 x)^5}+\frac {261}{2401 (2+3 x)^4}+\frac {1674}{16807 (2+3 x)^3}+\frac {6840}{117649 (2+3 x)^2}+\frac {23040}{823543 (2+3 x)}\right ) \, dx\\ &=\frac {44}{16807 (1-2 x)^2}+\frac {1040}{117649 (1-2 x)}+\frac {3}{1372 (2+3 x)^4}-\frac {29}{2401 (2+3 x)^3}-\frac {279}{16807 (2+3 x)^2}-\frac {2280}{117649 (2+3 x)}-\frac {7680 \log (1-2 x)}{823543}+\frac {7680 \log (2+3 x)}{823543}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 64, normalized size = 0.74 \[ \frac {4 \left (-\frac {7 \left (1658880 x^5+2626560 x^4+384000 x^3-1101440 x^2-403584 x+28275\right )}{16 (1-2 x)^2 (3 x+2)^4}-1920 \log (1-2 x)+1920 \log (6 x+4)\right )}{823543} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(28275 - 403584*x - 1101440*x^2 + 384000*x^3 + 2626560*x^4 + 1658880*x^5))/(16*(1 - 2*x)^2*(2 + 3*x)^4
) - 1920*Log[1 - 2*x] + 1920*Log[4 + 6*x]))/823543

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fricas [A]  time = 0.45, size = 135, normalized size = 1.55 \[ -\frac {11612160 \, x^{5} + 18385920 \, x^{4} + 2688000 \, x^{3} - 7710080 \, x^{2} - 30720 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (3 \, x + 2\right ) + 30720 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} \log \left (2 \, x - 1\right ) - 2825088 \, x + 197925}{3294172 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3294172*(11612160*x^5 + 18385920*x^4 + 2688000*x^3 - 7710080*x^2 - 30720*(324*x^6 + 540*x^5 + 81*x^4 - 264*
x^3 - 104*x^2 + 32*x + 16)*log(3*x + 2) + 30720*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*l
og(2*x - 1) - 2825088*x + 197925)/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)

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giac [A]  time = 1.20, size = 78, normalized size = 0.90 \[ -\frac {2280}{117649 \, {\left (3 \, x + 2\right )}} + \frac {48 \, {\left (\frac {1141}{3 \, x + 2} - 293\right )}}{823543 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}^{2}} - \frac {279}{16807 \, {\left (3 \, x + 2\right )}^{2}} - \frac {29}{2401 \, {\left (3 \, x + 2\right )}^{3}} + \frac {3}{1372 \, {\left (3 \, x + 2\right )}^{4}} - \frac {7680}{823543} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2280/117649/(3*x + 2) + 48/823543*(1141/(3*x + 2) - 293)/(7/(3*x + 2) - 2)^2 - 279/16807/(3*x + 2)^2 - 29/240
1/(3*x + 2)^3 + 3/1372/(3*x + 2)^4 - 7680/823543*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 72, normalized size = 0.83 \[ -\frac {7680 \ln \left (2 x -1\right )}{823543}+\frac {7680 \ln \left (3 x +2\right )}{823543}+\frac {3}{1372 \left (3 x +2\right )^{4}}-\frac {29}{2401 \left (3 x +2\right )^{3}}-\frac {279}{16807 \left (3 x +2\right )^{2}}-\frac {2280}{117649 \left (3 x +2\right )}+\frac {44}{16807 \left (2 x -1\right )^{2}}-\frac {1040}{117649 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/1372/(3*x+2)^4-29/2401/(3*x+2)^3-279/16807/(3*x+2)^2-2280/117649/(3*x+2)+7680/823543*ln(3*x+2)+44/16807/(2*x
-1)^2-1040/117649/(2*x-1)-7680/823543*ln(2*x-1)

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maxima [A]  time = 0.53, size = 76, normalized size = 0.87 \[ -\frac {1658880 \, x^{5} + 2626560 \, x^{4} + 384000 \, x^{3} - 1101440 \, x^{2} - 403584 \, x + 28275}{470596 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} + \frac {7680}{823543} \, \log \left (3 \, x + 2\right ) - \frac {7680}{823543} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/470596*(1658880*x^5 + 2626560*x^4 + 384000*x^3 - 1101440*x^2 - 403584*x + 28275)/(324*x^6 + 540*x^5 + 81*x^
4 - 264*x^3 - 104*x^2 + 32*x + 16) + 7680/823543*log(3*x + 2) - 7680/823543*log(2*x - 1)

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mupad [B]  time = 0.04, size = 66, normalized size = 0.76 \[ \frac {15360\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {1280\,x^5}{117649}+\frac {6080\,x^4}{352947}+\frac {8000\,x^3}{3176523}-\frac {68840\,x^2}{9529569}-\frac {8408\,x}{3176523}+\frac {9425}{50824368}}{x^6+\frac {5\,x^5}{3}+\frac {x^4}{4}-\frac {22\,x^3}{27}-\frac {26\,x^2}{81}+\frac {8\,x}{81}+\frac {4}{81}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(15360*atanh((12*x)/7 + 1/7))/823543 - ((8000*x^3)/3176523 - (68840*x^2)/9529569 - (8408*x)/3176523 + (6080*x^
4)/352947 + (1280*x^5)/117649 + 9425/50824368)/((8*x)/81 - (26*x^2)/81 - (22*x^3)/27 + x^4/4 + (5*x^5)/3 + x^6
+ 4/81)

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sympy [A]  time = 0.20, size = 75, normalized size = 0.86 \[ - \frac {1658880 x^{5} + 2626560 x^{4} + 384000 x^{3} - 1101440 x^{2} - 403584 x + 28275}{152473104 x^{6} + 254121840 x^{5} + 38118276 x^{4} - 124237344 x^{3} - 48941984 x^{2} + 15059072 x + 7529536} - \frac {7680 \log {\left (x - \frac {1}{2} \right )}}{823543} + \frac {7680 \log {\left (x + \frac {2}{3} \right )}}{823543} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1658880*x**5 + 2626560*x**4 + 384000*x**3 - 1101440*x**2 - 403584*x + 28275)/(152473104*x**6 + 254121840*x**
5 + 38118276*x**4 - 124237344*x**3 - 48941984*x**2 + 15059072*x + 7529536) - 7680*log(x - 1/2)/823543 + 7680*l
og(x + 2/3)/823543

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