3.16.59 \(\int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=84 \[ -\frac {2187 x^3}{1000}-\frac {330237 x^2}{20000}-\frac {242028 x}{3125}-\frac {130943337}{937024 (1-2 x)}-\frac {54}{45753125 (5 x+3)}+\frac {5764801}{170368 (1-2 x)^2}-\frac {1}{41593750 (5 x+3)^2}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (5 x+3)}{100656875} \]

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Rubi [A]  time = 0.05, antiderivative size = 84, 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^3}{1000}-\frac {330237 x^2}{20000}-\frac {242028 x}{3125}-\frac {130943337}{937024 (1-2 x)}-\frac {54}{45753125 (5 x+3)}+\frac {5764801}{170368 (1-2 x)^2}-\frac {1}{41593750 (5 x+3)^2}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (5 x+3)}{100656875} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^8/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

5764801/(170368*(1 - 2*x)^2) - 130943337/(937024*(1 - 2*x)) - (242028*x)/3125 - (330237*x^2)/20000 - (2187*x^3
)/1000 - 1/(41593750*(3 + 5*x)^2) - 54/(45753125*(3 + 5*x)) - (595421589*Log[1 - 2*x])/5153632 + (1284*Log[3 +
5*x])/100656875

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)^8}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac {242028}{3125}-\frac {330237 x}{10000}-\frac {6561 x^2}{1000}-\frac {5764801}{42592 (-1+2 x)^3}-\frac {130943337}{468512 (-1+2 x)^2}-\frac {595421589}{2576816 (-1+2 x)}+\frac {1}{4159375 (3+5 x)^3}+\frac {54}{9150625 (3+5 x)^2}+\frac {1284}{20131375 (3+5 x)}\right ) \, dx\\ &=\frac {5764801}{170368 (1-2 x)^2}-\frac {130943337}{937024 (1-2 x)}-\frac {242028 x}{3125}-\frac {330237 x^2}{20000}-\frac {2187 x^3}{1000}-\frac {1}{41593750 (3+5 x)^2}-\frac {54}{45753125 (3+5 x)}-\frac {595421589 \log (1-2 x)}{5153632}+\frac {1284 \log (3+5 x)}{100656875}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 70, normalized size = 0.83 \begin {gather*} \frac {-\frac {11 \left (6403973400000 x^7+49630793850000 x^6+232677700200000 x^5+148045752548100 x^4-314407515766380 x^3-254889143270829 x^2+31893783102814 x+39754322426279\right )}{\left (10 x^2+x-3\right )^2}-37213849312500 \log (3-6 x)+4108800 \log (-3 (5 x+3))}{322102000000} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^8/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

((-11*(39754322426279 + 31893783102814*x - 254889143270829*x^2 - 314407515766380*x^3 + 148045752548100*x^4 + 2
32677700200000*x^5 + 49630793850000*x^6 + 6403973400000*x^7))/(-3 + x + 10*x^2)^2 - 37213849312500*Log[3 - 6*x
] + 4108800*Log[-3*(3 + 5*x)])/322102000000

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^8}{(1-2 x)^3 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(2 + 3*x)^8/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

IntegrateAlgebraic[(2 + 3*x)^8/((1 - 2*x)^3*(3 + 5*x)^3), x]

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fricas [A]  time = 1.20, size = 115, normalized size = 1.37 \begin {gather*} -\frac {70443707400000 \, x^{7} + 545938732350000 \, x^{6} + 2559454702200000 \, x^{5} + 180911181221100 \, x^{4} - 3748001092791780 \, x^{3} - 1949701238862399 \, x^{2} - 4108800 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 37213849312500 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 437687139939434 \, x + 307014257976349}{322102000000 \, {\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)^8/(1-2*x)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/322102000000*(70443707400000*x^7 + 545938732350000*x^6 + 2559454702200000*x^5 + 180911181221100*x^4 - 37480
01092791780*x^3 - 1949701238862399*x^2 - 4108800*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 37213849
312500*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 437687139939434*x + 307014257976349)/(100*x^4 + 20
*x^3 - 59*x^2 - 6*x + 9)

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giac [A]  time = 1.17, size = 63, normalized size = 0.75 \begin {gather*} -\frac {2187}{1000} \, x^{3} - \frac {330237}{20000} \, x^{2} - \frac {242028}{3125} \, x + \frac {204598963371300 \, x^{3} + 167989904414289 \, x^{2} - 19378995974014 \, x - 27910387088759}{29282000000 \, {\left (5 \, x + 3\right )}^{2} {\left (2 \, x - 1\right )}^{2}} + \frac {1284}{100656875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {595421589}{5153632} \, \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)^8/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-2187/1000*x^3 - 330237/20000*x^2 - 242028/3125*x + 1/29282000000*(204598963371300*x^3 + 167989904414289*x^2 -
19378995974014*x - 27910387088759)/((5*x + 3)^2*(2*x - 1)^2) + 1284/100656875*log(abs(5*x + 3)) - 595421589/5
153632*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 67, normalized size = 0.80 \begin {gather*} -\frac {2187 x^{3}}{1000}-\frac {330237 x^{2}}{20000}-\frac {242028 x}{3125}-\frac {595421589 \ln \left (2 x -1\right )}{5153632}+\frac {1284 \ln \left (5 x +3\right )}{100656875}-\frac {1}{41593750 \left (5 x +3\right )^{2}}-\frac {54}{45753125 \left (5 x +3\right )}+\frac {5764801}{170368 \left (2 x -1\right )^{2}}+\frac {130943337}{937024 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^8/(1-2*x)^3/(5*x+3)^3,x)

[Out]

-2187/1000*x^3-330237/20000*x^2-242028/3125*x-1/41593750/(5*x+3)^2-54/45753125/(5*x+3)+1284/100656875*ln(5*x+3
)+5764801/170368/(2*x-1)^2+130943337/937024/(2*x-1)-595421589/5153632*ln(2*x-1)

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maxima [A]  time = 0.53, size = 69, normalized size = 0.82 \begin {gather*} -\frac {2187}{1000} \, x^{3} - \frac {330237}{20000} \, x^{2} - \frac {242028}{3125} \, x + \frac {204598963371300 \, x^{3} + 167989904414289 \, x^{2} - 19378995974014 \, x - 27910387088759}{29282000000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {1284}{100656875} \, \log \left (5 \, x + 3\right ) - \frac {595421589}{5153632} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^8/(1-2*x)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

-2187/1000*x^3 - 330237/20000*x^2 - 242028/3125*x + 1/29282000000*(204598963371300*x^3 + 167989904414289*x^2 -
19378995974014*x - 27910387088759)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 1284/100656875*log(5*x + 3) - 5954
21589/5153632*log(2*x - 1)

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mupad [B]  time = 1.08, size = 63, normalized size = 0.75 \begin {gather*} \frac {1284\,\ln \left (x+\frac {3}{5}\right )}{100656875}-\frac {595421589\,\ln \left (x-\frac {1}{2}\right )}{5153632}-\frac {242028\,x}{3125}-\frac {-\frac {2045989633713\,x^3}{29282000000}-\frac {167989904414289\,x^2}{2928200000000}+\frac {9689497987007\,x}{1464100000000}+\frac {27910387088759}{2928200000000}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}}-\frac {330237\,x^2}{20000}-\frac {2187\,x^3}{1000} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(3*x + 2)^8/((2*x - 1)^3*(5*x + 3)^3),x)

[Out]

(1284*log(x + 3/5))/100656875 - (595421589*log(x - 1/2))/5153632 - (242028*x)/3125 - ((9689497987007*x)/146410
0000000 - (167989904414289*x^2)/2928200000000 - (2045989633713*x^3)/29282000000 + 27910387088759/2928200000000
)/(x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100) - (330237*x^2)/20000 - (2187*x^3)/1000

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sympy [A]  time = 0.23, size = 73, normalized size = 0.87 \begin {gather*} - \frac {2187 x^{3}}{1000} - \frac {330237 x^{2}}{20000} - \frac {242028 x}{3125} - \frac {- 204598963371300 x^{3} - 167989904414289 x^{2} + 19378995974014 x + 27910387088759}{2928200000000 x^{4} + 585640000000 x^{3} - 1727638000000 x^{2} - 175692000000 x + 263538000000} - \frac {595421589 \log {\left (x - \frac {1}{2} \right )}}{5153632} + \frac {1284 \log {\left (x + \frac {3}{5} \right )}}{100656875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**8/(1-2*x)**3/(3+5*x)**3,x)

[Out]

-2187*x**3/1000 - 330237*x**2/20000 - 242028*x/3125 - (-204598963371300*x**3 - 167989904414289*x**2 + 19378995
974014*x + 27910387088759)/(2928200000000*x**4 + 585640000000*x**3 - 1727638000000*x**2 - 175692000000*x + 263
538000000) - 595421589*log(x - 1/2)/5153632 + 1284*log(x + 3/5)/100656875

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