3.1700 \(\int \frac{(2+3 x)^6}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=70 \[ -\frac{729 x}{1000}-\frac{1563051}{234256 (1-2 x)}-\frac{204}{9150625 (5 x+3)}+\frac{117649}{42592 (1-2 x)^2}-\frac{1}{1663750 (5 x+3)^2}-\frac{6950895 \log (1-2 x)}{2576816}+\frac{17547 \log (5 x+3)}{100656875} \]

[Out]

117649/(42592*(1 - 2*x)^2) - 1563051/(234256*(1 - 2*x)) - (729*x)/1000 - 1/(1663750*(3 + 5*x)^2) - 204/(915062
5*(3 + 5*x)) - (6950895*Log[1 - 2*x])/2576816 + (17547*Log[3 + 5*x])/100656875

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Rubi [A]  time = 0.0337331, antiderivative size = 70, 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} \[ -\frac{729 x}{1000}-\frac{1563051}{234256 (1-2 x)}-\frac{204}{9150625 (5 x+3)}+\frac{117649}{42592 (1-2 x)^2}-\frac{1}{1663750 (5 x+3)^2}-\frac{6950895 \log (1-2 x)}{2576816}+\frac{17547 \log (5 x+3)}{100656875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

117649/(42592*(1 - 2*x)^2) - 1563051/(234256*(1 - 2*x)) - (729*x)/1000 - 1/(1663750*(3 + 5*x)^2) - 204/(915062
5*(3 + 5*x)) - (6950895*Log[1 - 2*x])/2576816 + (17547*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)^6}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac{729}{1000}-\frac{117649}{10648 (-1+2 x)^3}-\frac{1563051}{117128 (-1+2 x)^2}-\frac{6950895}{1288408 (-1+2 x)}+\frac{1}{166375 (3+5 x)^3}+\frac{204}{1830125 (3+5 x)^2}+\frac{17547}{20131375 (3+5 x)}\right ) \, dx\\ &=\frac{117649}{42592 (1-2 x)^2}-\frac{1563051}{234256 (1-2 x)}-\frac{729 x}{1000}-\frac{1}{1663750 (3+5 x)^2}-\frac{204}{9150625 (3+5 x)}-\frac{6950895 \log (1-2 x)}{2576816}+\frac{17547 \log (3+5 x)}{100656875}\\ \end{align*}

Mathematica [A]  time = 0.0344085, size = 60, normalized size = 0.86 \[ \frac{-\frac{55 \left (4269315600 x^5+3700073520 x^4-21487765512 x^3-19656314001 x^2+49588250 x+2317121263\right )}{\left (10 x^2+x-3\right )^2}-8688618750 \log (3-6 x)+561504 \log (-3 (5 x+3))}{3221020000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-55*(2317121263 + 49588250*x - 19656314001*x^2 - 21487765512*x^3 + 3700073520*x^4 + 4269315600*x^5))/(-3 + x
+ 10*x^2)^2 - 8688618750*Log[3 - 6*x] + 561504*Log[-3*(3 + 5*x)])/3221020000

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Maple [A]  time = 0.01, size = 57, normalized size = 0.8 \begin{align*} -{\frac{729\,x}{1000}}+{\frac{117649}{42592\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{1563051}{468512\,x-234256}}-{\frac{6950895\,\ln \left ( 2\,x-1 \right ) }{2576816}}-{\frac{1}{1663750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{204}{27451875+45753125\,x}}+{\frac{17547\,\ln \left ( 3+5\,x \right ) }{100656875}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/1000*x+117649/42592/(2*x-1)^2+1563051/234256/(2*x-1)-6950895/2576816*ln(2*x-1)-1/1663750/(3+5*x)^2-204/91
50625/(3+5*x)+17547/100656875*ln(3+5*x)

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Maxima [A]  time = 1.06181, size = 80, normalized size = 1.14 \begin{align*} -\frac{729}{1000} \, x + \frac{19538111388 \, x^{3} + 17720890929 \, x^{2} + 163877530 \, x - 2060962327}{58564000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac{17547}{100656875} \, \log \left (5 \, x + 3\right ) - \frac{6950895}{2576816} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/1000*x + 1/58564000*(19538111388*x^3 + 17720890929*x^2 + 163877530*x - 2060962327)/(100*x^4 + 20*x^3 - 59
*x^2 - 6*x + 9) + 17547/100656875*log(5*x + 3) - 6950895/2576816*log(2*x - 1)

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Fricas [A]  time = 1.5054, size = 383, normalized size = 5.47 \begin{align*} -\frac{234812358000 \, x^{5} + 46962471600 \, x^{4} - 1213135417560 \, x^{3} - 988737742575 \, x^{2} - 561504 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 8688618750 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 12119848070 \, x + 113352927985}{3221020000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3221020000*(234812358000*x^5 + 46962471600*x^4 - 1213135417560*x^3 - 988737742575*x^2 - 561504*(100*x^4 + 2
0*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 8688618750*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 12119
848070*x + 113352927985)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [A]  time = 0.199677, size = 60, normalized size = 0.86 \begin{align*} - \frac{729 x}{1000} + \frac{19538111388 x^{3} + 17720890929 x^{2} + 163877530 x - 2060962327}{5856400000 x^{4} + 1171280000 x^{3} - 3455276000 x^{2} - 351384000 x + 527076000} - \frac{6950895 \log{\left (x - \frac{1}{2} \right )}}{2576816} + \frac{17547 \log{\left (x + \frac{3}{5} \right )}}{100656875} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729*x/1000 + (19538111388*x**3 + 17720890929*x**2 + 163877530*x - 2060962327)/(5856400000*x**4 + 1171280000*x
**3 - 3455276000*x**2 - 351384000*x + 527076000) - 6950895*log(x - 1/2)/2576816 + 17547*log(x + 3/5)/100656875

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Giac [A]  time = 2.79653, size = 72, normalized size = 1.03 \begin{align*} -\frac{729}{1000} \, x + \frac{19538111388 \, x^{3} + 17720890929 \, x^{2} + 163877530 \, x - 2060962327}{58564000 \,{\left (5 \, x + 3\right )}^{2}{\left (2 \, x - 1\right )}^{2}} + \frac{17547}{100656875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{6950895}{2576816} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-729/1000*x + 1/58564000*(19538111388*x^3 + 17720890929*x^2 + 163877530*x - 2060962327)/((5*x + 3)^2*(2*x - 1)
^2) + 17547/100656875*log(abs(5*x + 3)) - 6950895/2576816*log(abs(2*x - 1))