3.1618 \(\int \frac{(2+3 x)^7}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=73 \[ \frac{729 x^3}{500}+\frac{21141 x^2}{2500}+\frac{1467477 x}{50000}+\frac{823543}{42592 (1-2 x)}-\frac{47}{4159375 (5 x+3)}-\frac{1}{3781250 (5 x+3)^2}+\frac{7411887 \log (1-2 x)}{234256}+\frac{4761 \log (5 x+3)}{45753125} \]

[Out]

823543/(42592*(1 - 2*x)) + (1467477*x)/50000 + (21141*x^2)/2500 + (729*x^3)/500 - 1/(3781250*(3 + 5*x)^2) - 47
/(4159375*(3 + 5*x)) + (7411887*Log[1 - 2*x])/234256 + (4761*Log[3 + 5*x])/45753125

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Rubi [A]  time = 0.0387854, antiderivative size = 73, 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} \[ \frac{729 x^3}{500}+\frac{21141 x^2}{2500}+\frac{1467477 x}{50000}+\frac{823543}{42592 (1-2 x)}-\frac{47}{4159375 (5 x+3)}-\frac{1}{3781250 (5 x+3)^2}+\frac{7411887 \log (1-2 x)}{234256}+\frac{4761 \log (5 x+3)}{45753125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

823543/(42592*(1 - 2*x)) + (1467477*x)/50000 + (21141*x^2)/2500 + (729*x^3)/500 - 1/(3781250*(3 + 5*x)^2) - 47
/(4159375*(3 + 5*x)) + (7411887*Log[1 - 2*x])/234256 + (4761*Log[3 + 5*x])/45753125

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)^2 (3+5 x)^3} \, dx &=\int \left (\frac{1467477}{50000}+\frac{21141 x}{1250}+\frac{2187 x^2}{500}+\frac{823543}{21296 (-1+2 x)^2}+\frac{7411887}{117128 (-1+2 x)}+\frac{1}{378125 (3+5 x)^3}+\frac{47}{831875 (3+5 x)^2}+\frac{4761}{9150625 (3+5 x)}\right ) \, dx\\ &=\frac{823543}{42592 (1-2 x)}+\frac{1467477 x}{50000}+\frac{21141 x^2}{2500}+\frac{729 x^3}{500}-\frac{1}{3781250 (3+5 x)^2}-\frac{47}{4159375 (3+5 x)}+\frac{7411887 \log (1-2 x)}{234256}+\frac{4761 \log (3+5 x)}{45753125}\\ \end{align*}

Mathematica [A]  time = 0.0404867, size = 67, normalized size = 0.92 \[ \frac{\frac{11 \left (48514950000 x^6+315347175000 x^5+1161933052500 x^4+42644641050 x^3-1002031406415 x^2-426293494632 x-14162188399\right )}{(2 x-1) (5 x+3)^2}[phone] \log (1-2 x)+761760 \log (10 x+6)}{7320500000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(-14162188399 - 426293494632*x - 1002031406415*x^2 + 42644641050*x^3 + 1161933052500*x^4 + 315347175000*x
^5 + 48514950000*x^6))/((-1 + 2*x)*(3 + 5*x)^2) + 231621468750*Log[1 - 2*x] + 761760*Log[6 + 10*x])/7320500000

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Maple [A]  time = 0.007, size = 58, normalized size = 0.8 \begin{align*}{\frac{729\,{x}^{3}}{500}}+{\frac{21141\,{x}^{2}}{2500}}+{\frac{1467477\,x}{50000}}-{\frac{823543}{85184\,x-42592}}+{\frac{7411887\,\ln \left ( 2\,x-1 \right ) }{234256}}-{\frac{1}{3781250\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{47}{12478125+20796875\,x}}+{\frac{4761\,\ln \left ( 3+5\,x \right ) }{45753125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/500*x^3+21141/2500*x^2+1467477/50000*x-823543/42592/(2*x-1)+7411887/234256*ln(2*x-1)-1/3781250/(3+5*x)^2-4
7/4159375/(3+5*x)+4761/45753125*ln(3+5*x)

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Maxima [A]  time = 1.05511, size = 80, normalized size = 1.1 \begin{align*} \frac{729}{500} \, x^{3} + \frac{21141}{2500} \, x^{2} + \frac{1467477}{50000} \, x - \frac{321696559575 \, x^{2} + 386035789122 \, x + 115810711639}{665500000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{4761}{45753125} \, \log \left (5 \, x + 3\right ) + \frac{7411887}{234256} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729/500*x^3 + 21141/2500*x^2 + 1467477/50000*x - 1/665500000*(321696559575*x^2 + 386035789122*x + 115810711639
)/(50*x^3 + 35*x^2 - 12*x - 9) + 4761/45753125*log(5*x + 3) + 7411887/234256*log(2*x - 1)

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Fricas [A]  time = 1.49145, size = 385, normalized size = 5.27 \begin{align*} \frac{533664450000 \, x^{6} + 3468818925000 \, x^{5} + 12781263577500 \, x^{4} + 6680945249550 \, x^{3} - 6674047531965 \, x^{2} + 761760 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 231621468750 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 6180073448472 \, x - 1273917828029}{7320500000 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7320500000*(533664450000*x^6 + 3468818925000*x^5 + 12781263577500*x^4 + 6680945249550*x^3 - 6674047531965*x^
2 + 761760*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 231621468750*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1)
- 6180073448472*x - 1273917828029)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 0.178777, size = 63, normalized size = 0.86 \begin{align*} \frac{729 x^{3}}{500} + \frac{21141 x^{2}}{2500} + \frac{1467477 x}{50000} - \frac{321696559575 x^{2} + 386035789122 x + 115810711639}{33275000000 x^{3} + 23292500000 x^{2} - 7986000000 x - 5989500000} + \frac{7411887 \log{\left (x - \frac{1}{2} \right )}}{234256} + \frac{4761 \log{\left (x + \frac{3}{5} \right )}}{45753125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

729*x**3/500 + 21141*x**2/2500 + 1467477*x/50000 - (321696559575*x**2 + 386035789122*x + 115810711639)/(332750
00000*x**3 + 23292500000*x**2 - 7986000000*x - 5989500000) + 7411887*log(x - 1/2)/234256 + 4761*log(x + 3/5)/4
5753125

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Giac [A]  time = 1.80633, size = 139, normalized size = 1.9 \begin{align*} \frac{{\left (2 \, x - 1\right )}^{3}{\left (\frac{25349061375}{2 \, x - 1} + \frac{234545525775}{{\left (2 \, x - 1\right )}^{2}} + \frac{720756547985}{{\left (2 \, x - 1\right )}^{3}} + \frac{689127341628}{{\left (2 \, x - 1\right )}^{4}} + 1334161125\right )}}{292820000 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} - \frac{823543}{42592 \,{\left (2 \, x - 1\right )}} - \frac{1582011}{50000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{4761}{45753125} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/292820000*(2*x - 1)^3*(25349061375/(2*x - 1) + 234545525775/(2*x - 1)^2 + 720756547985/(2*x - 1)^3 + 6891273
41628/(2*x - 1)^4 + 1334161125)/(11/(2*x - 1) + 5)^2 - 823543/42592/(2*x - 1) - 1582011/50000*log(1/2*abs(2*x
- 1)/(2*x - 1)^2) + 4761/45753125*log(abs(-11/(2*x - 1) - 5))