3.1552 \(\int \frac{3+5 x}{(1-2 x)^2 (2+3 x)^6} \, dx\)

Optimal. Leaf size=87 \[ \frac{176}{117649 (1-2 x)}-\frac{1040}{117649 (3 x+2)}-\frac{194}{16807 (3 x+2)^2}-\frac{128}{7203 (3 x+2)^3}-\frac{31}{1372 (3 x+2)^4}+\frac{1}{245 (3 x+2)^5}-\frac{2608 \log (1-2 x)}{823543}+\frac{2608 \log (3 x+2)}{823543} \]

[Out]

176/(117649*(1 - 2*x)) + 1/(245*(2 + 3*x)^5) - 31/(1372*(2 + 3*x)^4) - 128/(7203*(2 + 3*x)^3) - 194/(16807*(2
+ 3*x)^2) - 1040/(117649*(2 + 3*x)) - (2608*Log[1 - 2*x])/823543 + (2608*Log[2 + 3*x])/823543

________________________________________________________________________________________

Rubi [A]  time = 0.043672, antiderivative size = 87, 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} \[ \frac{176}{117649 (1-2 x)}-\frac{1040}{117649 (3 x+2)}-\frac{194}{16807 (3 x+2)^2}-\frac{128}{7203 (3 x+2)^3}-\frac{31}{1372 (3 x+2)^4}+\frac{1}{245 (3 x+2)^5}-\frac{2608 \log (1-2 x)}{823543}+\frac{2608 \log (3 x+2)}{823543} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^6),x]

[Out]

176/(117649*(1 - 2*x)) + 1/(245*(2 + 3*x)^5) - 31/(1372*(2 + 3*x)^4) - 128/(7203*(2 + 3*x)^3) - 194/(16807*(2
+ 3*x)^2) - 1040/(117649*(2 + 3*x)) - (2608*Log[1 - 2*x])/823543 + (2608*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)^2 (2+3 x)^6} \, dx &=\int \left (\frac{352}{117649 (-1+2 x)^2}-\frac{5216}{823543 (-1+2 x)}-\frac{3}{49 (2+3 x)^6}+\frac{93}{343 (2+3 x)^5}+\frac{384}{2401 (2+3 x)^4}+\frac{1164}{16807 (2+3 x)^3}+\frac{3120}{117649 (2+3 x)^2}+\frac{7824}{823543 (2+3 x)}\right ) \, dx\\ &=\frac{176}{117649 (1-2 x)}+\frac{1}{245 (2+3 x)^5}-\frac{31}{1372 (2+3 x)^4}-\frac{128}{7203 (2+3 x)^3}-\frac{194}{16807 (2+3 x)^2}-\frac{1040}{117649 (2+3 x)}-\frac{2608 \log (1-2 x)}{823543}+\frac{2608 \log (2+3 x)}{823543}\\ \end{align*}

Mathematica [A]  time = 0.0457832, size = 62, normalized size = 0.71 \[ \frac{-\frac{7 \left (12674880 x^5+34855920 x^4+33741000 x^3+10410810 x^2-3488689 x-2104258\right )}{(2 x-1) (3 x+2)^5}-156480 \log (3-6 x)+156480 \log (3 x+2)}{49412580} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/((1 - 2*x)^2*(2 + 3*x)^6),x]

[Out]

((-7*(-2104258 - 3488689*x + 10410810*x^2 + 33741000*x^3 + 34855920*x^4 + 12674880*x^5))/((-1 + 2*x)*(2 + 3*x)
^5) - 156480*Log[3 - 6*x] + 156480*Log[2 + 3*x])/49412580

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 72, normalized size = 0.8 \begin{align*} -{\frac{176}{235298\,x-117649}}-{\frac{2608\,\ln \left ( 2\,x-1 \right ) }{823543}}+{\frac{1}{245\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{31}{1372\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{128}{7203\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{194}{16807\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{1040}{235298+352947\,x}}+{\frac{2608\,\ln \left ( 2+3\,x \right ) }{823543}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)/(1-2*x)^2/(2+3*x)^6,x)

[Out]

-176/117649/(2*x-1)-2608/823543*ln(2*x-1)+1/245/(2+3*x)^5-31/1372/(2+3*x)^4-128/7203/(2+3*x)^3-194/16807/(2+3*
x)^2-1040/117649/(2+3*x)+2608/823543*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]  time = 2.64007, size = 103, normalized size = 1.18 \begin{align*} -\frac{12674880 \, x^{5} + 34855920 \, x^{4} + 33741000 \, x^{3} + 10410810 \, x^{2} - 3488689 \, x - 2104258}{7058940 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} + \frac{2608}{823543} \, \log \left (3 \, x + 2\right ) - \frac{2608}{823543} \, \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)^6,x, algorithm="maxima")

[Out]

-1/7058940*(12674880*x^5 + 34855920*x^4 + 33741000*x^3 + 10410810*x^2 - 3488689*x - 2104258)/(486*x^6 + 1377*x
^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32) + 2608/823543*log(3*x + 2) - 2608/823543*log(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.28924, size = 456, normalized size = 5.24 \begin{align*} -\frac{88724160 \, x^{5} + 243991440 \, x^{4} + 236187000 \, x^{3} + 72875670 \, x^{2} - 156480 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (3 \, x + 2\right ) + 156480 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} \log \left (2 \, x - 1\right ) - 24420823 \, x - 14729806}{49412580 \,{\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\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)^6,x, algorithm="fricas")

[Out]

-1/49412580*(88724160*x^5 + 243991440*x^4 + 236187000*x^3 + 72875670*x^2 - 156480*(486*x^6 + 1377*x^5 + 1350*x
^4 + 360*x^3 - 240*x^2 - 176*x - 32)*log(3*x + 2) + 156480*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2
- 176*x - 32)*log(2*x - 1) - 24420823*x - 14729806)/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x
- 32)

________________________________________________________________________________________

Sympy [A]  time = 0.184372, size = 75, normalized size = 0.86 \begin{align*} - \frac{12674880 x^{5} + 34855920 x^{4} + 33741000 x^{3} + 10410810 x^{2} - 3488689 x - 2104258}{3430644840 x^{6} + 9720160380 x^{5} + 9529569000 x^{4} + 2541218400 x^{3} - 1694145600 x^{2} - 1242373440 x - 225886080} - \frac{2608 \log{\left (x - \frac{1}{2} \right )}}{823543} + \frac{2608 \log{\left (x + \frac{2}{3} \right )}}{823543} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)/(1-2*x)**2/(2+3*x)**6,x)

[Out]

-(12674880*x**5 + 34855920*x**4 + 33741000*x**3 + 10410810*x**2 - 3488689*x - 2104258)/(3430644840*x**6 + 9720
160380*x**5 + 9529569000*x**4 + 2541218400*x**3 - 1694145600*x**2 - 1242373440*x - 225886080) - 2608*log(x - 1
/2)/823543 + 2608*log(x + 2/3)/823543

________________________________________________________________________________________

Giac [A]  time = 3.11993, size = 105, normalized size = 1.21 \begin{align*} -\frac{176}{117649 \,{\left (2 \, x - 1\right )}} + \frac{12 \,{\left (\frac{3424365}{2 \, x - 1} + \frac{13259400}{{\left (2 \, x - 1\right )}^{2}} + \frac{23152500}{{\left (2 \, x - 1\right )}^{3}} + \frac{15366400}{{\left (2 \, x - 1\right )}^{4}} + 335637\right )}}{4117715 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{5}} + \frac{2608}{823543} \, \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)^6,x, algorithm="giac")

[Out]

-176/117649/(2*x - 1) + 12/4117715*(3424365/(2*x - 1) + 13259400/(2*x - 1)^2 + 23152500/(2*x - 1)^3 + 15366400
/(2*x - 1)^4 + 335637)/(7/(2*x - 1) + 3)^5 + 2608/823543*log(abs(-7/(2*x - 1) - 3))