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

Optimal. Leaf size=65 \[ \frac {128}{2401 (1-2 x)}-\frac {87}{2401 (3 x+2)}+\frac {11}{343 (1-2 x)^2}+\frac {3}{686 (3 x+2)^2}-\frac {558 \log (1-2 x)}{16807}+\frac {558 \log (3 x+2)}{16807} \]

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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {128}{2401 (1-2 x)}-\frac {87}{2401 (3 x+2)}+\frac {11}{343 (1-2 x)^2}+\frac {3}{686 (3 x+2)^2}-\frac {558 \log (1-2 x)}{16807}+\frac {558 \log (3 x+2)}{16807} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

11/(343*(1 - 2*x)^2) + 128/(2401*(1 - 2*x)) + 3/(686*(2 + 3*x)^2) - 87/(2401*(2 + 3*x)) - (558*Log[1 - 2*x])/1
6807 + (558*Log[2 + 3*x])/16807

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)^3} \, dx &=\int \left (-\frac {44}{343 (-1+2 x)^3}+\frac {256}{2401 (-1+2 x)^2}-\frac {1116}{16807 (-1+2 x)}-\frac {9}{343 (2+3 x)^3}+\frac {261}{2401 (2+3 x)^2}+\frac {1674}{16807 (2+3 x)}\right ) \, dx\\ &=\frac {11}{343 (1-2 x)^2}+\frac {128}{2401 (1-2 x)}+\frac {3}{686 (2+3 x)^2}-\frac {87}{2401 (2+3 x)}-\frac {558 \log (1-2 x)}{16807}+\frac {558 \log (2+3 x)}{16807}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.74 \begin {gather*} \frac {\frac {7 \left (-6696 x^3-1674 x^2+3658 x+1313\right )}{\left (6 x^2+x-2\right )^2}-1116 \log (1-2 x)+1116 \log (3 x+2)}{33614} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(1313 + 3658*x - 1674*x^2 - 6696*x^3))/(-2 + x + 6*x^2)^2 - 1116*Log[1 - 2*x] + 1116*Log[2 + 3*x])/33614

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3+5 x}{(1-2 x)^3 (2+3 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.82, size = 95, normalized size = 1.46 \begin {gather*} -\frac {46872 \, x^{3} + 11718 \, x^{2} - 1116 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 1116 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )} \log \left (2 \, x - 1\right ) - 25606 \, x - 9191}{33614 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/33614*(46872*x^3 + 11718*x^2 - 1116*(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)*log(3*x + 2) + 1116*(36*x^4 + 12*x
^3 - 23*x^2 - 4*x + 4)*log(2*x - 1) - 25606*x - 9191)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4)

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giac [A]  time = 1.25, size = 46, normalized size = 0.71 \begin {gather*} -\frac {6696 \, x^{3} + 1674 \, x^{2} - 3658 \, x - 1313}{4802 \, {\left (6 \, x^{2} + x - 2\right )}^{2}} + \frac {558}{16807} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {558}{16807} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4802*(6696*x^3 + 1674*x^2 - 3658*x - 1313)/(6*x^2 + x - 2)^2 + 558/16807*log(abs(3*x + 2)) - 558/16807*log(
abs(2*x - 1))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \begin {gather*} -\frac {558 \ln \left (2 x -1\right )}{16807}+\frac {558 \ln \left (3 x +2\right )}{16807}+\frac {3}{686 \left (3 x +2\right )^{2}}-\frac {87}{2401 \left (3 x +2\right )}+\frac {11}{343 \left (2 x -1\right )^{2}}-\frac {128}{2401 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/686/(3*x+2)^2-87/2401/(3*x+2)+558/16807*ln(3*x+2)+11/343/(2*x-1)^2-128/2401/(2*x-1)-558/16807*ln(2*x-1)

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maxima [A]  time = 0.56, size = 56, normalized size = 0.86 \begin {gather*} -\frac {6696 \, x^{3} + 1674 \, x^{2} - 3658 \, x - 1313}{4802 \, {\left (36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4\right )}} + \frac {558}{16807} \, \log \left (3 \, x + 2\right ) - \frac {558}{16807} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4802*(6696*x^3 + 1674*x^2 - 3658*x - 1313)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4) + 558/16807*log(3*x + 2) -
558/16807*log(2*x - 1)

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mupad [B]  time = 1.07, size = 45, normalized size = 0.69 \begin {gather*} \frac {1116\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}+\frac {-\frac {93\,x^3}{2401}-\frac {93\,x^2}{9604}+\frac {1829\,x}{86436}+\frac {1313}{172872}}{x^4+\frac {x^3}{3}-\frac {23\,x^2}{36}-\frac {x}{9}+\frac {1}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1116*atanh((12*x)/7 + 1/7))/16807 + ((1829*x)/86436 - (93*x^2)/9604 - (93*x^3)/2401 + 1313/172872)/(x^3/3 - (
23*x^2)/36 - x/9 + x^4 + 1/9)

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sympy [A]  time = 0.17, size = 54, normalized size = 0.83 \begin {gather*} - \frac {6696 x^{3} + 1674 x^{2} - 3658 x - 1313}{172872 x^{4} + 57624 x^{3} - 110446 x^{2} - 19208 x + 19208} - \frac {558 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {558 \log {\left (x + \frac {2}{3} \right )}}{16807} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(6696*x**3 + 1674*x**2 - 3658*x - 1313)/(172872*x**4 + 57624*x**3 - 110446*x**2 - 19208*x + 19208) - 558*log(
x - 1/2)/16807 + 558*log(x + 2/3)/16807

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