3.1515 \(\int \frac{1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{70752609}{16807 (3 x+2)}-\frac{15625}{11 (5 x+3)}-\frac{806121}{2401 (3 x+2)^2}-\frac{11457}{343 (3 x+2)^3}-\frac{162}{49 (3 x+2)^4}-\frac{9}{35 (3 x+2)^5}-\frac{128 \log (1-2 x)}{14235529}+\frac{2977686468 \log (3 x+2)}{117649}-\frac{3062500}{121} \log (5 x+3) \]

[Out]

-9/(35*(2 + 3*x)^5) - 162/(49*(2 + 3*x)^4) - 11457/(343*(2 + 3*x)^3) - 806121/(2401*(2 + 3*x)^2) - 70752609/(1
6807*(2 + 3*x)) - 15625/(11*(3 + 5*x)) - (128*Log[1 - 2*x])/14235529 + (2977686468*Log[2 + 3*x])/117649 - (306
2500*Log[3 + 5*x])/121

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Rubi [A]  time = 0.0509471, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ -\frac{70752609}{16807 (3 x+2)}-\frac{15625}{11 (5 x+3)}-\frac{806121}{2401 (3 x+2)^2}-\frac{11457}{343 (3 x+2)^3}-\frac{162}{49 (3 x+2)^4}-\frac{9}{35 (3 x+2)^5}-\frac{128 \log (1-2 x)}{14235529}+\frac{2977686468 \log (3 x+2)}{117649}-\frac{3062500}{121} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-9/(35*(2 + 3*x)^5) - 162/(49*(2 + 3*x)^4) - 11457/(343*(2 + 3*x)^3) - 806121/(2401*(2 + 3*x)^2) - 70752609/(1
6807*(2 + 3*x)) - 15625/(11*(3 + 5*x)) - (128*Log[1 - 2*x])/14235529 + (2977686468*Log[2 + 3*x])/117649 - (306
2500*Log[3 + 5*x])/121

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx &=\int \left (-\frac{256}{14235529 (-1+2 x)}+\frac{27}{7 (2+3 x)^6}+\frac{1944}{49 (2+3 x)^5}+\frac{103113}{343 (2+3 x)^4}+\frac{4836726}{2401 (2+3 x)^3}+\frac{212257827}{16807 (2+3 x)^2}+\frac{8933059404}{117649 (2+3 x)}+\frac{78125}{11 (3+5 x)^2}-\frac{15312500}{121 (3+5 x)}\right ) \, dx\\ &=-\frac{9}{35 (2+3 x)^5}-\frac{162}{49 (2+3 x)^4}-\frac{11457}{343 (2+3 x)^3}-\frac{806121}{2401 (2+3 x)^2}-\frac{70752609}{16807 (2+3 x)}-\frac{15625}{11 (3+5 x)}-\frac{128 \log (1-2 x)}{14235529}+\frac{2977686468 \log (2+3 x)}{117649}-\frac{3062500}{121} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0359911, size = 95, normalized size = 0.98 \[ -\frac{70752609}{16807 (3 x+2)}-\frac{15625}{55 x+33}-\frac{806121}{2401 (3 x+2)^2}-\frac{11457}{343 (3 x+2)^3}-\frac{162}{49 (3 x+2)^4}-\frac{9}{35 (3 x+2)^5}-\frac{128 \log (1-2 x)}{14235529}+\frac{2977686468 \log (6 x+4)}{117649}-\frac{3062500}{121} \log (10 x+6) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-9/(35*(2 + 3*x)^5) - 162/(49*(2 + 3*x)^4) - 11457/(343*(2 + 3*x)^3) - 806121/(2401*(2 + 3*x)^2) - 70752609/(1
6807*(2 + 3*x)) - 15625/(33 + 55*x) - (128*Log[1 - 2*x])/14235529 + (2977686468*Log[4 + 6*x])/117649 - (306250
0*Log[6 + 10*x])/121

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Maple [A]  time = 0.01, size = 80, normalized size = 0.8 \begin{align*} -{\frac{128\,\ln \left ( 2\,x-1 \right ) }{14235529}}-{\frac{9}{35\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{162}{49\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{11457}{343\, \left ( 2+3\,x \right ) ^{3}}}-{\frac{806121}{2401\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{70752609}{33614+50421\,x}}+{\frac{2977686468\,\ln \left ( 2+3\,x \right ) }{117649}}-{\frac{15625}{33+55\,x}}-{\frac{3062500\,\ln \left ( 3+5\,x \right ) }{121}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128/14235529*ln(2*x-1)-9/35/(2+3*x)^5-162/49/(2+3*x)^4-11457/343/(2+3*x)^3-806121/2401/(2+3*x)^2-70752609/168
07/(2+3*x)[phone]/117649*ln(2+3*x)-15625/11/(3+5*x)-3062500/121*ln(3+5*x)

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Maxima [A]  time = 1.12975, size = 113, normalized size = 1.16 \begin{align*} -\frac{1895084756100 \, x^{5} + 6253779701610 \, x^{4} + 8252743193370 \, x^{3} + 5443759671885 \, x^{2} + 1794885176145 \, x + 236642515057}{924385 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - \frac{3062500}{121} \, \log \left (5 \, x + 3\right ) + \frac{2977686468}{117649} \, \log \left (3 \, x + 2\right ) - \frac{128}{14235529} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/924385*(1895084756100*x^5 + 6253779701610*x^4 + 8252743193370*x^3 + 5443759671885*x^2 + 1794885176145*x + 2
36642515057)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 3062500/121*log(5*x + 3) +
2977686468/117649*log(3*x + 2) - 128/14235529*log(2*x - 1)

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Fricas [B]  time = 1.32452, size = 657, normalized size = 6.77 \begin{align*} -\frac{145921526219700 \, x^{5} + 481541037023970 \, x^{4} + 635461225889490 \, x^{3} + 419169494735145 \, x^{2} + 1801500312500 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 1801500313140 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 640 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (2 \, x - 1\right ) + 138206158563165 \, x + 18221473659389}{71177645 \,{\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/71177645*(145921526219700*x^5 + 481541037023970*x^4 + 635461225889490*x^3 + 419169494735145*x^2 + 180150031
2500*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 1801500313140*(1215*x^
6 + 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(3*x + 2) + 640*(1215*x^6 + 4779*x^5 + 7830*x^4
+ 6840*x^3 + 3360*x^2 + 880*x + 96)*log(2*x - 1) + 138206158563165*x + 18221473659389)/(1215*x^6 + 4779*x^5 +
7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)

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Sympy [A]  time = 0.260834, size = 85, normalized size = 0.88 \begin{align*} - \frac{1895084756100 x^{5} + 6253779701610 x^{4} + 8252743193370 x^{3} + 5443759671885 x^{2} + 1794885176145 x + 236642515057}{1123127775 x^{6} + 4417635915 x^{5} + 7237934550 x^{4} + 6322793400 x^{3} + 3105933600 x^{2} + 813458800 x + 88740960} - \frac{128 \log{\left (x - \frac{1}{2} \right )}}{14235529} - \frac{3062500 \log{\left (x + \frac{3}{5} \right )}}{121} + \frac{2977686468 \log{\left (x + \frac{2}{3} \right )}}{117649} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(1895084756100*x**5 + 6253779701610*x**4 + 8252743193370*x**3 + 5443759671885*x**2 + 1794885176145*x + 236642
515057)/(1123127775*x**6 + 4417635915*x**5 + 7237934550*x**4 + 6322793400*x**3 + 3105933600*x**2 + 813458800*x
+ 88740960) - 128*log(x - 1/2)/14235529 - 3062500*log(x + 3/5)/121 + 2977686468*log(x + 2/3)/117649

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Giac [A]  time = 2.93413, size = 123, normalized size = 1.27 \begin{align*} -\frac{15625}{11 \,{\left (5 \, x + 3\right )}} + \frac{135 \,{\left (\frac{1627470333}{5 \, x + 3} + \frac{915260769}{{\left (5 \, x + 3\right )}^{2}} + \frac{234430752}{{\left (5 \, x + 3\right )}^{3}} + \frac{23397131}{{\left (5 \, x + 3\right )}^{4}} + 1103836896\right )}}{16807 \,{\left (\frac{1}{5 \, x + 3} + 3\right )}^{5}} + \frac{2977686468}{117649} \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) - \frac{128}{14235529} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)/(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

-15625/11/(5*x + 3) + 135/16807*(1627470333/(5*x + 3) + 915260769/(5*x + 3)^2 + 234430752/(5*x + 3)^3 + 233971
31/(5*x + 3)^4 + 1103836896)/(1/(5*x + 3) + 3)^5 + 2977686468/117649*log(abs(-1/(5*x + 3) - 3)) - 128/14235529
*log(abs(-11/(5*x + 3) + 2))