3.15.14 \(\int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^7} \, dx\)

Optimal. Leaf size=98 \[ \frac {352}{823543 (1-2 x)}-\frac {2608}{823543 (3 x+2)}-\frac {520}{117649 (3 x+2)^2}-\frac {388}{50421 (3 x+2)^3}-\frac {32}{2401 (3 x+2)^4}-\frac {31}{1715 (3 x+2)^5}+\frac {1}{294 (3 x+2)^6}-\frac {128 \log (1-2 x)}{117649}+\frac {128 \log (3 x+2)}{117649} \]

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Rubi [A]  time = 0.05, antiderivative size = 98, 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 {352}{823543 (1-2 x)}-\frac {2608}{823543 (3 x+2)}-\frac {520}{117649 (3 x+2)^2}-\frac {388}{50421 (3 x+2)^3}-\frac {32}{2401 (3 x+2)^4}-\frac {31}{1715 (3 x+2)^5}+\frac {1}{294 (3 x+2)^6}-\frac {128 \log (1-2 x)}{117649}+\frac {128 \log (3 x+2)}{117649} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^7),x]

[Out]

352/(823543*(1 - 2*x)) + 1/(294*(2 + 3*x)^6) - 31/(1715*(2 + 3*x)^5) - 32/(2401*(2 + 3*x)^4) - 388/(50421*(2 +
3*x)^3) - 520/(117649*(2 + 3*x)^2) - 2608/(823543*(2 + 3*x)) - (128*Log[1 - 2*x])/117649 + (128*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)^7} \, dx &=\int \left (\frac {704}{823543 (-1+2 x)^2}-\frac {256}{117649 (-1+2 x)}-\frac {3}{49 (2+3 x)^7}+\frac {93}{343 (2+3 x)^6}+\frac {384}{2401 (2+3 x)^5}+\frac {1164}{16807 (2+3 x)^4}+\frac {3120}{117649 (2+3 x)^3}+\frac {7824}{823543 (2+3 x)^2}+\frac {384}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {352}{823543 (1-2 x)}+\frac {1}{294 (2+3 x)^6}-\frac {31}{1715 (2+3 x)^5}-\frac {32}{2401 (2+3 x)^4}-\frac {388}{50421 (2+3 x)^3}-\frac {520}{117649 (2+3 x)^2}-\frac {2608}{823543 (2+3 x)}-\frac {128 \log (1-2 x)}{117649}+\frac {128 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 67, normalized size = 0.68 \begin {gather*} \frac {-\frac {7 \left (311040 x^6+1062720 x^5+1398240 x^4+807480 x^3+84708 x^2-132772 x-49131\right )}{(2 x-1) (3 x+2)^6}-1280 \log (1-2 x)+1280 \log (6 x+4)}{1176490} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^7),x]

[Out]

((-7*(-49131 - 132772*x + 84708*x^2 + 807480*x^3 + 1398240*x^4 + 1062720*x^5 + 311040*x^6))/((-1 + 2*x)*(2 + 3
*x)^6) - 1280*Log[1 - 2*x] + 1280*Log[4 + 6*x])/1176490

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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)^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^7),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^7), x]

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fricas [A]  time = 1.41, size = 140, normalized size = 1.43 \begin {gather*} -\frac {2177280 \, x^{6} + 7439040 \, x^{5} + 9787680 \, x^{4} + 5652360 \, x^{3} + 592956 \, x^{2} - 1280 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (3 \, x + 2\right ) + 1280 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )} \log \left (2 \, x - 1\right ) - 929404 \, x - 343917}{1176490 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\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)^7,x, algorithm="fricas")

[Out]

-1/1176490*(2177280*x^6 + 7439040*x^5 + 9787680*x^4 + 5652360*x^3 + 592956*x^2 - 1280*(1458*x^7 + 5103*x^6 + 6
804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64)*log(3*x + 2) + 1280*(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1
008*x^2 - 448*x - 64)*log(2*x - 1) - 929404*x - 343917)/(1458*x^7 + 5103*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2
- 448*x - 64)

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giac [A]  time = 1.10, size = 87, normalized size = 0.89 \begin {gather*} -\frac {352}{823543 \, {\left (2 \, x - 1\right )}} + \frac {288 \, {\left (\frac {1446039}{2 \, x - 1} + \frac {7393365}{{\left (2 \, x - 1\right )}^{2}} + \frac {19147975}{{\left (2 \, x - 1\right )}^{3}} + \frac {25210500}{{\left (2 \, x - 1\right )}^{4}} + \frac {13529635}{{\left (2 \, x - 1\right )}^{5}} + 114291\right )}}{28824005 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{6}} + \frac {128}{117649} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \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)^7,x, algorithm="giac")

[Out]

-352/823543/(2*x - 1) + 288/28824005*(1446039/(2*x - 1) + 7393365/(2*x - 1)^2 + 19147975/(2*x - 1)^3 + 2521050
0/(2*x - 1)^4 + 13529635/(2*x - 1)^5 + 114291)/(7/(2*x - 1) + 3)^6 + 128/117649*log(abs(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 81, normalized size = 0.83 \begin {gather*} -\frac {128 \ln \left (2 x -1\right )}{117649}+\frac {128 \ln \left (3 x +2\right )}{117649}+\frac {1}{294 \left (3 x +2\right )^{6}}-\frac {31}{1715 \left (3 x +2\right )^{5}}-\frac {32}{2401 \left (3 x +2\right )^{4}}-\frac {388}{50421 \left (3 x +2\right )^{3}}-\frac {520}{117649 \left (3 x +2\right )^{2}}-\frac {2608}{823543 \left (3 x +2\right )}-\frac {352}{823543 \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)^7,x)

[Out]

1/294/(3*x+2)^6-31/1715/(3*x+2)^5-32/2401/(3*x+2)^4-388/50421/(3*x+2)^3-520/117649/(3*x+2)^2-2608/823543/(3*x+
2)+128/117649*ln(3*x+2)-352/823543/(2*x-1)-128/117649*ln(2*x-1)

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maxima [A]  time = 0.51, size = 81, normalized size = 0.83 \begin {gather*} -\frac {311040 \, x^{6} + 1062720 \, x^{5} + 1398240 \, x^{4} + 807480 \, x^{3} + 84708 \, x^{2} - 132772 \, x - 49131}{168070 \, {\left (1458 \, x^{7} + 5103 \, x^{6} + 6804 \, x^{5} + 3780 \, x^{4} - 1008 \, x^{2} - 448 \, x - 64\right )}} + \frac {128}{117649} \, \log \left (3 \, x + 2\right ) - \frac {128}{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)^7,x, algorithm="maxima")

[Out]

-1/168070*(311040*x^6 + 1062720*x^5 + 1398240*x^4 + 807480*x^3 + 84708*x^2 - 132772*x - 49131)/(1458*x^7 + 510
3*x^6 + 6804*x^5 + 3780*x^4 - 1008*x^2 - 448*x - 64) + 128/117649*log(3*x + 2) - 128/117649*log(2*x - 1)

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mupad [B]  time = 1.07, size = 71, normalized size = 0.72 \begin {gather*} \frac {256\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}-\frac {\frac {64\,x^6}{50421}+\frac {656\,x^5}{151263}+\frac {7768\,x^4}{1361367}+\frac {4486\,x^3}{1361367}+\frac {2353\,x^2}{6806835}-\frac {33193\,x}{61261515}-\frac {5459}{27227340}}{x^7+\frac {7\,x^6}{2}+\frac {14\,x^5}{3}+\frac {70\,x^4}{27}-\frac {56\,x^2}{81}-\frac {224\,x}{729}-\frac {32}{729}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)/((2*x - 1)^2*(3*x + 2)^7),x)

[Out]

(256*atanh((12*x)/7 + 1/7))/117649 - ((2353*x^2)/6806835 - (33193*x)/61261515 + (4486*x^3)/1361367 + (7768*x^4
)/1361367 + (656*x^5)/151263 + (64*x^6)/50421 - 5459/27227340)/((70*x^4)/27 - (56*x^2)/81 - (224*x)/729 + (14*
x^5)/3 + (7*x^6)/2 + x^7 - 32/729)

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sympy [A]  time = 0.21, size = 80, normalized size = 0.82 \begin {gather*} \frac {- 311040 x^{6} - 1062720 x^{5} - 1398240 x^{4} - 807480 x^{3} - 84708 x^{2} + 132772 x + 49131}{245046060 x^{7} + 857661210 x^{6} + 1143548280 x^{5} + 635304600 x^{4} - 169414560 x^{2} - 75295360 x - 10756480} - \frac {128 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {128 \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)**7,x)

[Out]

(-311040*x**6 - 1062720*x**5 - 1398240*x**4 - 807480*x**3 - 84708*x**2 + 132772*x + 49131)/(245046060*x**7 + 8
57661210*x**6 + 1143548280*x**5 + 635304600*x**4 - 169414560*x**2 - 75295360*x - 10756480) - 128*log(x - 1/2)/
117649 + 128*log(x + 2/3)/117649

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