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

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

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Rubi [A]  time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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) \end {gather*}

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*}

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Mathematica [A]  time = 0.07, size = 95, normalized size = 0.98 \begin {gather*} -\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) \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x) (2+3 x)^6 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 1.42, size = 173, normalized size = 1.78 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.87, size = 91, normalized size = 0.94 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 80, normalized size = 0.82 \begin {gather*} -\frac {128 \ln \left (2 x -1\right )}{14235529}+\frac {2977686468 \ln \left (3 x +2\right )}{117649}-\frac {3062500 \ln \left (5 x +3\right )}{121}-\frac {15625}{11 \left (5 x +3\right )}-\frac {9}{35 \left (3 x +2\right )^{5}}-\frac {162}{49 \left (3 x +2\right )^{4}}-\frac {11457}{343 \left (3 x +2\right )^{3}}-\frac {806121}{2401 \left (3 x +2\right )^{2}}-\frac {70752609}{16807 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15625/11/(5*x+3)-3062500/121*ln(5*x+3)-9/35/(3*x+2)^5-162/49/(3*x+2)^4-11457/343/(3*x+2)^3-806121/2401/(3*x+2
)^2-70752609/16807/(3*x+2)[phone]/117649*ln(3*x+2)-128/14235529*ln(2*x-1)

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maxima [A]  time = 0.58, size = 84, normalized size = 0.87 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.05, size = 76, normalized size = 0.78 \begin {gather*} \frac {2977686468\,\ln \left (x+\frac {2}{3}\right )}{117649}-\frac {128\,\ln \left (x-\frac {1}{2}\right )}{14235529}-\frac {3062500\,\ln \left (x+\frac {3}{5}\right )}{121}-\frac {\frac {311948108\,x^5}{184877}+\frac {2205918766\,x^4}{396165}+\frac {61131431062\,x^3}{8319465}+\frac {120972437153\,x^2}{24958395}+\frac {119659011743\,x}{74875185}+\frac {236642515057}{1123127775}}{x^6+\frac {59\,x^5}{15}+\frac {58\,x^4}{9}+\frac {152\,x^3}{27}+\frac {224\,x^2}{81}+\frac {176\,x}{243}+\frac {32}{405}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2977686468*log(x + 2/3))/117649 - (128*log(x - 1/2))/14235529 - (3062500*log(x + 3/5))/121 - ((119659011743*x
)/74875185 + (120972437153*x^2)/24958395 + (61131431062*x^3)/8319465 + (2205918766*x^4)/396165 + (311948108*x^
5)/184877 + 236642515057/1123127775)/((176*x)/243 + (224*x^2)/81 + (152*x^3)/27 + (58*x^4)/9 + (59*x^5)/15 + x
^6 + 32/405)

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sympy [A]  time = 0.28, size = 85, normalized size = 0.88 \begin {gather*} - \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 {gather*}

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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