3.1553 \(\int \frac{3+5 x}{(1-2 x)^2 (2+3 x)^7} \, dx\)

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

[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

________________________________________________________________________________________

Rubi [A]  time = 0.0531247, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ \frac{352}{823543 (1-2 x)}-\frac{2608}{823543 (3 x+2)}-\frac{520}{117649 (3 x+2)^2}-\frac{388}{50421 (3 x+2)^3}-\frac{32}{2401 (3 x+2)^4}-\frac{31}{1715 (3 x+2)^5}+\frac{1}{294 (3 x+2)^6}-\frac{128 \log (1-2 x)}{117649}+\frac{128 \log (3 x+2)}{117649} \]

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*}

Mathematica [A]  time = 0.037875, size = 67, normalized size = 0.68 \[ \frac{-\frac{7 \left (311040 x^6+1062720 x^5+1398240 x^4+807480 x^3+84708 x^2-132772 x-49131\right )}{(2 x-1) (3 x+2)^6}-1280 \log (1-2 x)+1280 \log (6 x+4)}{1176490} \]

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

________________________________________________________________________________________

Maple [A]  time = 0.01, size = 81, normalized size = 0.8 \begin{align*} -{\frac{352}{1647086\,x-823543}}-{\frac{128\,\ln \left ( 2\,x-1 \right ) }{117649}}+{\frac{1}{294\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{31}{1715\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{32}{2401\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{388}{50421\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{520}{117649\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{2608}{1647086+2470629\,x}}+{\frac{128\,\ln \left ( 2+3\,x \right ) }{117649}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-352/823543/(2*x-1)-128/117649*ln(2*x-1)+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)+128/117649*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 1.12631, size = 109, normalized size = 1.11 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.54728, size = 466, normalized size = 4.76 \begin{align*} -\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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.198172, size = 80, normalized size = 0.82 \begin{align*} - \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{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.88734, size = 117, normalized size = 1.19 \begin{align*} -\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{align*}

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))