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

Optimal. Leaf size=76 \[ \frac {88}{16807 (1-2 x)}-\frac {388}{16807 (3 x+2)}-\frac {64}{2401 (3 x+2)^2}-\frac {31}{1029 (3 x+2)^3}+\frac {1}{196 (3 x+2)^4}-\frac {1040 \log (1-2 x)}{117649}+\frac {1040 \log (3 x+2)}{117649} \]

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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {88}{16807 (1-2 x)}-\frac {388}{16807 (3 x+2)}-\frac {64}{2401 (3 x+2)^2}-\frac {31}{1029 (3 x+2)^3}+\frac {1}{196 (3 x+2)^4}-\frac {1040 \log (1-2 x)}{117649}+\frac {1040 \log (3 x+2)}{117649} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

88/(16807*(1 - 2*x)) + 1/(196*(2 + 3*x)^4) - 31/(1029*(2 + 3*x)^3) - 64/(2401*(2 + 3*x)^2) - 388/(16807*(2 + 3
*x)) - (1040*Log[1 - 2*x])/117649 + (1040*Log[2 + 3*x])/117649

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)^2 (2+3 x)^5} \, dx &=\int \left (\frac {176}{16807 (-1+2 x)^2}-\frac {2080}{117649 (-1+2 x)}-\frac {3}{49 (2+3 x)^5}+\frac {93}{343 (2+3 x)^4}+\frac {384}{2401 (2+3 x)^3}+\frac {1164}{16807 (2+3 x)^2}+\frac {3120}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {88}{16807 (1-2 x)}+\frac {1}{196 (2+3 x)^4}-\frac {31}{1029 (2+3 x)^3}-\frac {64}{2401 (2+3 x)^2}-\frac {388}{16807 (2+3 x)}-\frac {1040 \log (1-2 x)}{117649}+\frac {1040 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 59, normalized size = 0.78 \begin {gather*} \frac {4 \left (-\frac {7 \left (336960 x^4+702000 x^3+429000 x^2-9230 x-52979\right )}{16 (2 x-1) (3 x+2)^4}-780 \log (1-2 x)+780 \log (6 x+4)\right )}{352947} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((-7*(-52979 - 9230*x + 429000*x^2 + 702000*x^3 + 336960*x^4))/(16*(-1 + 2*x)*(2 + 3*x)^4) - 780*Log[1 - 2*
x] + 780*Log[4 + 6*x]))/352947

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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)^2 (2+3 x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.35, size = 115, normalized size = 1.51 \begin {gather*} -\frac {2358720 \, x^{4} + 4914000 \, x^{3} + 3003000 \, x^{2} - 12480 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 12480 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) - 64610 \, x - 370853}{1411788 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1411788*(2358720*x^4 + 4914000*x^3 + 3003000*x^2 - 12480*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)
*log(3*x + 2) + 12480*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) - 64610*x - 370853)/(162
*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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giac [A]  time = 1.02, size = 67, normalized size = 0.88 \begin {gather*} -\frac {388}{16807 \, {\left (3 \, x + 2\right )}} + \frac {528}{117649 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} - \frac {64}{2401 \, {\left (3 \, x + 2\right )}^{2}} - \frac {31}{1029 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{196 \, {\left (3 \, x + 2\right )}^{4}} - \frac {1040}{117649} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-388/16807/(3*x + 2) + 528/117649/(7/(3*x + 2) - 2) - 64/2401/(3*x + 2)^2 - 31/1029/(3*x + 2)^3 + 1/196/(3*x +
2)^4 - 1040/117649*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.83 \begin {gather*} -\frac {1040 \ln \left (2 x -1\right )}{117649}+\frac {1040 \ln \left (3 x +2\right )}{117649}+\frac {1}{196 \left (3 x +2\right )^{4}}-\frac {31}{1029 \left (3 x +2\right )^{3}}-\frac {64}{2401 \left (3 x +2\right )^{2}}-\frac {388}{16807 \left (3 x +2\right )}-\frac {88}{16807 \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)^2/(3*x+2)^5,x)

[Out]

1/196/(3*x+2)^4-31/1029/(3*x+2)^3-64/2401/(3*x+2)^2-388/16807/(3*x+2)+1040/117649*ln(3*x+2)-88/16807/(2*x-1)-1
040/117649*ln(2*x-1)

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maxima [A]  time = 0.54, size = 66, normalized size = 0.87 \begin {gather*} -\frac {336960 \, x^{4} + 702000 \, x^{3} + 429000 \, x^{2} - 9230 \, x - 52979}{201684 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {1040}{117649} \, \log \left (3 \, x + 2\right ) - \frac {1040}{117649} \, \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)^2/(2+3*x)^5,x, algorithm="maxima")

[Out]

-1/201684*(336960*x^4 + 702000*x^3 + 429000*x^2 - 9230*x - 52979)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x
- 16) + 1040/117649*log(3*x + 2) - 1040/117649*log(2*x - 1)

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mupad [B]  time = 0.04, size = 57, normalized size = 0.75 \begin {gather*} \frac {2080\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {\frac {520\,x^4}{50421}+\frac {3250\,x^3}{151263}+\frac {17875\,x^2}{1361367}-\frac {4615\,x}{16336404}-\frac {52979}{32672808}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2080*atanh((12*x)/7 + 1/7))/117649 + ((17875*x^2)/1361367 - (4615*x)/16336404 + (3250*x^3)/151263 + (520*x^4)
/50421 - 52979/32672808)/((32*x)/81 + (4*x^2)/27 - (4*x^3)/3 - (13*x^4)/6 - x^5 + 8/81)

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sympy [A]  time = 0.18, size = 65, normalized size = 0.86 \begin {gather*} \frac {- 336960 x^{4} - 702000 x^{3} - 429000 x^{2} + 9230 x + 52979}{32672808 x^{5} + 70791084 x^{4} + 43563744 x^{3} - 4840416 x^{2} - 12907776 x - 3226944} - \frac {1040 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {1040 \log {\left (x + \frac {2}{3} \right )}}{117649} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-336960*x**4 - 702000*x**3 - 429000*x**2 + 9230*x + 52979)/(32672808*x**5 + 70791084*x**4 + 43563744*x**3 - 4
840416*x**2 - 12907776*x - 3226944) - 1040*log(x - 1/2)/117649 + 1040*log(x + 2/3)/117649

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