3.1604 \(\int \frac{(2+3 x)^7}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=69 \[ \frac{2187 x^4}{400}+\frac{16281 x^3}{500}+\frac{1974861 x^2}{20000}+\frac{6156243 x}{25000}+\frac{823543}{7744 (1-2 x)}-\frac{1}{1890625 (5 x+3)}+\frac{18941489 \log (1-2 x)}{85184}+\frac{47 \log (5 x+3)}{4159375} \]

[Out]

823543/(7744*(1 - 2*x)) + (6156243*x)/25000 + (1974861*x^2)/20000 + (16281*x^3)/500 + (2187*x^4)/400 - 1/(1890
625*(3 + 5*x)) + (18941489*Log[1 - 2*x])/85184 + (47*Log[3 + 5*x])/4159375

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Rubi [A]  time = 0.0395852, antiderivative size = 69, 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} \[ \frac{2187 x^4}{400}+\frac{16281 x^3}{500}+\frac{1974861 x^2}{20000}+\frac{6156243 x}{25000}+\frac{823543}{7744 (1-2 x)}-\frac{1}{1890625 (5 x+3)}+\frac{18941489 \log (1-2 x)}{85184}+\frac{47 \log (5 x+3)}{4159375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

823543/(7744*(1 - 2*x)) + (6156243*x)/25000 + (1974861*x^2)/20000 + (16281*x^3)/500 + (2187*x^4)/400 - 1/(1890
625*(3 + 5*x)) + (18941489*Log[1 - 2*x])/85184 + (47*Log[3 + 5*x])/4159375

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)^2} \, dx &=\int \left (\frac{6156243}{25000}+\frac{1974861 x}{10000}+\frac{48843 x^2}{500}+\frac{2187 x^3}{100}+\frac{823543}{3872 (-1+2 x)^2}+\frac{18941489}{42592 (-1+2 x)}+\frac{1}{378125 (3+5 x)^2}+\frac{47}{831875 (3+5 x)}\right ) \, dx\\ &=\frac{823543}{7744 (1-2 x)}+\frac{6156243 x}{25000}+\frac{1974861 x^2}{20000}+\frac{16281 x^3}{500}+\frac{2187 x^4}{400}-\frac{1}{1890625 (3+5 x)}+\frac{18941489 \log (1-2 x)}{85184}+\frac{47 \log (3+5 x)}{4159375}\\ \end{align*}

Mathematica [A]  time = 0.034418, size = 74, normalized size = 1.07 \[ \frac{-\frac{11 (64339297003 x+38603578061)}{10 x^2+x-3}[phone] (3 x+2)^4+886446000 (3 x+2)^3+7128103950 (3 x+2)^2+67228064640 (3 x+2)[phone] \log (3-6 x)+15040 \log (-3 (5 x+3))}{1331000000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(67228064640*(2 + 3*x) + 7128103950*(2 + 3*x)^2 + 886446000*(2 + 3*x)^3 + 89842500*(2 + 3*x)^4 - (11*(38603578
061 + 64339297003*x))/(-3 + x + 10*x^2) + 295960765625*Log[3 - 6*x] + 15040*Log[-3*(3 + 5*x)])/1331000000

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Maple [A]  time = 0.009, size = 54, normalized size = 0.8 \begin{align*}{\frac{2187\,{x}^{4}}{400}}+{\frac{16281\,{x}^{3}}{500}}+{\frac{1974861\,{x}^{2}}{20000}}+{\frac{6156243\,x}{25000}}-{\frac{823543}{15488\,x-7744}}+{\frac{18941489\,\ln \left ( 2\,x-1 \right ) }{85184}}-{\frac{1}{5671875+9453125\,x}}+{\frac{47\,\ln \left ( 3+5\,x \right ) }{4159375}} \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)^2,x)

[Out]

2187/400*x^4+16281/500*x^3+1974861/20000*x^2+6156243/25000*x-823543/7744/(2*x-1)[phone]/85184*ln(2*x-1)-1/18
90625/(3+5*x)+47/4159375*ln(3+5*x)

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Maxima [A]  time = 1.05327, size = 70, normalized size = 1.01 \begin{align*} \frac{2187}{400} \, x^{4} + \frac{16281}{500} \, x^{3} + \frac{1974861}{20000} \, x^{2} + \frac{6156243}{25000} \, x - \frac{64339297003 \, x + 38603578061}{121000000 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{47}{4159375} \, \log \left (5 \, x + 3\right ) + \frac{18941489}{85184} \, \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)^2,x, algorithm="maxima")

[Out]

2187/400*x^4 + 16281/500*x^3 + 1974861/20000*x^2 + 6156243/25000*x - 1/121000000*(64339297003*x + 38603578061)
/(10*x^2 + x - 3) + 47/4159375*log(5*x + 3) + 18941489/85184*log(2*x - 1)

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Fricas [A]  time = 1.34537, size = 327, normalized size = 4.74 \begin{align*} \frac{72772425000 \, x^{6} + 440677462500 \, x^{5} + 1335778290000 \, x^{4} + 3278990706750 \, x^{3} - 66522621330 \, x^{2} + 15040 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 295960765625 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 1691007398993 \, x - 424639358671}{1331000000 \,{\left (10 \, x^{2} + x - 3\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)^2,x, algorithm="fricas")

[Out]

1/1331000000*(72772425000*x^6 + 440677462500*x^5 + 1335778290000*x^4 + 3278990706750*x^3 - 66522621330*x^2 + 1
5040*(10*x^2 + x - 3)*log(5*x + 3) + 295960765625*(10*x^2 + x - 3)*log(2*x - 1) - 1691007398993*x - 4246393586
71)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.162484, size = 60, normalized size = 0.87 \begin{align*} \frac{2187 x^{4}}{400} + \frac{16281 x^{3}}{500} + \frac{1974861 x^{2}}{20000} + \frac{6156243 x}{25000} - \frac{64339297003 x + 38603578061}{1210000000 x^{2} + 121000000 x - 363000000} + \frac{18941489 \log{\left (x - \frac{1}{2} \right )}}{85184} + \frac{47 \log{\left (x + \frac{3}{5} \right )}}{4159375} \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)**2,x)

[Out]

2187*x**4/400 + 16281*x**3/500 + 1974861*x**2/20000 + 6156243*x/25000 - (64339297003*x + 38603578061)/(1210000
000*x**2 + 121000000*x - 363000000) + 18941489*log(x - 1/2)/85184 + 47*log(x + 3/5)/4159375

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Giac [A]  time = 2.16637, size = 139, normalized size = 2.01 \begin{align*} -\frac{{\left (5 \, x + 3\right )}^{4}{\left (\frac{142957386}{5 \, x + 3} + \frac{1626867990}{{\left (5 \, x + 3\right )}^{2}} + \frac{26903695995}{{\left (5 \, x + 3\right )}^{3}} - \frac{295961527385}{{\left (5 \, x + 3\right )}^{4}} + 11643588\right )}}{665500000 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{1}{1890625 \,{\left (5 \, x + 3\right )}} - \frac{44471943}{200000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{18941489}{85184} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \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)^2,x, algorithm="giac")

[Out]

-1/665500000*(5*x + 3)^4*(142957386/(5*x + 3) + 1626867990/(5*x + 3)^2 + 26903695995/(5*x + 3)^3 - 29596152738
5/(5*x + 3)^4 + 11643588)/(11/(5*x + 3) - 2) - 1/1890625/(5*x + 3) - 44471943/200000*log(1/5*abs(5*x + 3)/(5*x
+ 3)^2) + 18941489/85184*log(abs(-11/(5*x + 3) + 2))