3.16.52 \(\int \frac {3+5 x}{(1-2 x)^2 (2+3 x)^6} \, dx\) [1552]

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

[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
)-2608/823543*ln(1-2*x)+2608/823543*ln(2+3*x)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 87, 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 = {78} \begin {gather*} \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} \end {gather*}

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 78

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 = 0.04, size = 62, normalized size = 0.71 \begin {gather*} \frac {-\frac {7 \left (-2104258-3488689 x+10410810 x^2+33741000 x^3+34855920 x^4+12674880 x^5\right )}{(-1+2 x) (2+3 x)^5}-156480 \log (3-6 x)+156480 \log (2+3 x)}{49412580} \end {gather*}

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 = 0.09, size = 72, normalized size = 0.83

method	result	size
risch	\(\frac {-\frac {211248}{117649} x^{5}-\frac {580932}{117649} x^{4}-\frac {562350}{117649} x^{3}-\frac {347027}{235298} x^{2}+\frac {3488689}{7058940} x +\frac {1052129}{3529470}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {2608 \ln \left (-1+2 x \right )}{823543}+\frac {2608 \ln \left (2+3 x \right )}{823543}\)	\(59\)
norman	\(\frac {\frac {415327851}{37647680} x^{5}+\frac {28755981}{3764768} x^{4}+\frac {85222449}{18823840} x^{6}-\frac {1342413}{941192} x^{3}-\frac {269491}{235298} x -\frac {1746183}{470596} x^{2}}{\left (-1+2 x \right ) \left (2+3 x \right )^{5}}-\frac {2608 \ln \left (-1+2 x \right )}{823543}+\frac {2608 \ln \left (2+3 x \right )}{823543}\)	\(62\)
default	\(-\frac {176}{117649 \left (-1+2 x \right )}-\frac {2608 \ln \left (-1+2 x \right )}{823543}+\frac {1}{245 \left (2+3 x \right )^{5}}-\frac {31}{1372 \left (2+3 x \right )^{4}}-\frac {128}{7203 \left (2+3 x \right )^{3}}-\frac {194}{16807 \left (2+3 x \right )^{2}}-\frac {1040}{117649 \left (2+3 x \right )}+\frac {2608 \ln \left (2+3 x \right )}{823543}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-176/117649/(-1+2*x)-2608/823543*ln(-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)+2608/823543*ln(2+3*x)

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 76, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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 = 1.03, size = 135, normalized size = 1.55 \begin {gather*} -\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 {gather*}

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 = 0.08, size = 75, normalized size = 0.86 \begin {gather*} \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 {gather*}

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 = 1.21, size = 78, normalized size = 0.90 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

Mupad [B]
time = 1.07, size = 66, normalized size = 0.76 \begin {gather*} \frac {5216\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{823543}-\frac {\frac {1304\,x^5}{352947}+\frac {3586\,x^4}{352947}+\frac {93725\,x^3}{9529569}+\frac {347027\,x^2}{114354828}-\frac {3488689\,x}{3430644840}-\frac {1052129}{1715322420}}{x^6+\frac {17\,x^5}{6}+\frac {25\,x^4}{9}+\frac {20\,x^3}{27}-\frac {40\,x^2}{81}-\frac {88\,x}{243}-\frac {16}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5216*atanh((12*x)/7 + 1/7))/823543 - ((347027*x^2)/114354828 - (3488689*x)/3430644840 + (93725*x^3)/9529569 +
(3586*x^4)/352947 + (1304*x^5)/352947 - 1052129/1715322420)/((20*x^3)/27 - (40*x^2)/81 - (88*x)/243 + (25*x^4
)/9 + (17*x^5)/6 + x^6 - 16/243)

________________________________________________________________________________________