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

Optimal. Leaf size=77 \[ -\frac {2187 x^2}{2000}-\frac {95499 x}{10000}-\frac {7411887}{234256 (1-2 x)}-\frac {237}{45753125 (5 x+3)}+\frac {823543}{85184 (1-2 x)^2}-\frac {1}{8318750 (5 x+3)^2}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (5 x+3)}{503284375} \]

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Rubi [A]  time = 0.04, antiderivative size = 77, 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 {2187 x^2}{2000}-\frac {95499 x}{10000}-\frac {7411887}{234256 (1-2 x)}-\frac {237}{45753125 (5 x+3)}+\frac {823543}{85184 (1-2 x)^2}-\frac {1}{8318750 (5 x+3)^2}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (5 x+3)}{503284375} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

823543/(85184*(1 - 2*x)^2) - 7411887/(234256*(1 - 2*x)) - (95499*x)/10000 - (2187*x^2)/2000 - 1/(8318750*(3 +
5*x)^2) - 237/(45753125*(3 + 5*x)) - (25059237*Log[1 - 2*x])/1288408 + (24279*Log[3 + 5*x])/503284375

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 {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {95499}{10000}-\frac {2187 x}{1000}-\frac {823543}{21296 (-1+2 x)^3}-\frac {7411887}{117128 (-1+2 x)^2}-\frac {25059237}{644204 (-1+2 x)}+\frac {1}{831875 (3+5 x)^3}+\frac {237}{9150625 (3+5 x)^2}+\frac {24279}{100656875 (3+5 x)}\right ) \, dx\\ &=\frac {823543}{85184 (1-2 x)^2}-\frac {7411887}{234256 (1-2 x)}-\frac {95499 x}{10000}-\frac {2187 x^2}{2000}-\frac {1}{8318750 (3+5 x)^2}-\frac {237}{45753125 (3+5 x)}-\frac {25059237 \log (1-2 x)}{1288408}+\frac {24279 \log (3+5 x)}{503284375}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 65, normalized size = 0.84 \begin {gather*} \frac {-\frac {11 \left (320198670000 x^6+2860441452000 x^5+2092320420300 x^4-5957126547060 x^3-5105353973121 x^2+410862940766 x+734029874011\right )}{\left (10 x^2+x-3\right )^2}-626480925000 \log (3-6 x)+1553856 \log (-3 (5 x+3))}{32210200000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

((-11*(734029874011 + 410862940766*x - 5105353973121*x^2 - 5957126547060*x^3 + 2092320420300*x^4 + 28604414520
00*x^5 + 320198670000*x^6))/(-3 + x + 10*x^2)^2 - 626480925000*Log[3 - 6*x] + 1553856*Log[-3*(3 + 5*x)])/32210
200000

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^7}{(1-2 x)^3 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

IntegrateAlgebraic[(2 + 3*x)^7/((1 - 2*x)^3*(3 + 5*x)^3), x]

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fricas [A]  time = 1.38, size = 110, normalized size = 1.43 \begin {gather*} -\frac {3522185370000 \, x^{6} + 31464855972000 \, x^{5} + 4073994411300 \, x^{4} - 69316698060060 \, x^{3} - 44983390879251 \, x^{2} - 1553856 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 626480925000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 5655984161146 \, x + 6369590895041}{32210200000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/32210200000*(3522185370000*x^6 + 31464855972000*x^5 + 4073994411300*x^4 - 69316698060060*x^3 - 449833908792
51*x^2 - 1553856*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 626480925000*(100*x^4 + 20*x^3 - 59*x^2
- 6*x + 9)*log(2*x - 1) + 5655984161146*x + 6369590895041)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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giac [A]  time = 1.17, size = 58, normalized size = 0.75 \begin {gather*} -\frac {2187}{2000} \, x^{2} - \frac {95499}{10000} \, x + \frac {4632429071640 \, x^{3} + 3950432948061 \, x^{2} - 262504223666 \, x - 579053717731}{2928200000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {24279}{503284375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {25059237}{1288408} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-2187/2000*x^2 - 95499/10000*x + 1/2928200000*(4632429071640*x^3 + 3950432948061*x^2 - 262504223666*x - 579053
717731)/((5*x + 3)^2*(2*x - 1)^2) + 24279/503284375*log(abs(5*x + 3)) - 25059237/1288408*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 62, normalized size = 0.81 \begin {gather*} -\frac {2187 x^{2}}{2000}-\frac {95499 x}{10000}-\frac {25059237 \ln \left (2 x -1\right )}{1288408}+\frac {24279 \ln \left (5 x +3\right )}{503284375}-\frac {1}{8318750 \left (5 x +3\right )^{2}}-\frac {237}{45753125 \left (5 x +3\right )}+\frac {823543}{85184 \left (2 x -1\right )^{2}}+\frac {7411887}{234256 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187/2000*x^2-95499/10000*x-1/8318750/(5*x+3)^2-237/45753125/(5*x+3)+24279/503284375*ln(5*x+3)+823543/85184/(
2*x-1)^2+7411887/234256/(2*x-1)-25059237/1288408*ln(2*x-1)

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maxima [A]  time = 0.54, size = 64, normalized size = 0.83 \begin {gather*} -\frac {2187}{2000} \, x^{2} - \frac {95499}{10000} \, x + \frac {4632429071640 \, x^{3} + 3950432948061 \, x^{2} - 262504223666 \, x - 579053717731}{2928200000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {24279}{503284375} \, \log \left (5 \, x + 3\right ) - \frac {25059237}{1288408} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^7/(1-2*x)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

-2187/2000*x^2 - 95499/10000*x + 1/2928200000*(4632429071640*x^3 + 3950432948061*x^2 - 262504223666*x - 579053
717731)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 24279/503284375*log(5*x + 3) - 25059237/1288408*log(2*x - 1)

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mupad [B]  time = 1.09, size = 58, normalized size = 0.75 \begin {gather*} \frac {24279\,\ln \left (x+\frac {3}{5}\right )}{503284375}-\frac {25059237\,\ln \left (x-\frac {1}{2}\right )}{1288408}-\frac {95499\,x}{10000}-\frac {-\frac {115810726791\,x^3}{7320500000}-\frac {3950432948061\,x^2}{292820000000}+\frac {131252111833\,x}{146410000000}+\frac {579053717731}{292820000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {2187\,x^2}{2000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(24279*log(x + 3/5))/503284375 - (25059237*log(x - 1/2))/1288408 - (95499*x)/10000 - ((131252111833*x)/1464100
00000 - (3950432948061*x^2)/292820000000 - (115810726791*x^3)/7320500000 + 579053717731/292820000000)/(x^3/5 -
(59*x^2)/100 - (3*x)/50 + x^4 + 9/100) - (2187*x^2)/2000

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sympy [A]  time = 0.22, size = 66, normalized size = 0.86 \begin {gather*} - \frac {2187 x^{2}}{2000} - \frac {95499 x}{10000} - \frac {- 4632429071640 x^{3} - 3950432948061 x^{2} + 262504223666 x + 579053717731}{292820000000 x^{4} + 58564000000 x^{3} - 172763800000 x^{2} - 17569200000 x + 26353800000} - \frac {25059237 \log {\left (x - \frac {1}{2} \right )}}{1288408} + \frac {24279 \log {\left (x + \frac {3}{5} \right )}}{503284375} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2187*x**2/2000 - 95499*x/10000 - (-4632429071640*x**3 - 3950432948061*x**2 + 262504223666*x + 579053717731)/(
292820000000*x**4 + 58564000000*x**3 - 172763800000*x**2 - 17569200000*x + 26353800000) - 25059237*log(x - 1/2
)/1288408 + 24279*log(x + 3/5)/503284375

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