3.1607 \(\int \frac{(2+3 x)^4}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{81 x}{100}+\frac{2401}{968 (1-2 x)}-\frac{1}{15125 (5 x+3)}+\frac{10633 \log (1-2 x)}{5324}+\frac{136 \log (5 x+3)}{166375} \]

[Out]

2401/(968*(1 - 2*x)) + (81*x)/100 - 1/(15125*(3 + 5*x)) + (10633*Log[1 - 2*x])/5324 + (136*Log[3 + 5*x])/16637
5

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Rubi [A]  time = 0.02227, antiderivative size = 48, 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} \[ \frac{81 x}{100}+\frac{2401}{968 (1-2 x)}-\frac{1}{15125 (5 x+3)}+\frac{10633 \log (1-2 x)}{5324}+\frac{136 \log (5 x+3)}{166375} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^2*(3 + 5*x)^2),x]

[Out]

2401/(968*(1 - 2*x)) + (81*x)/100 - 1/(15125*(3 + 5*x)) + (10633*Log[1 - 2*x])/5324 + (136*Log[3 + 5*x])/16637
5

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)^4}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{81}{100}+\frac{2401}{484 (-1+2 x)^2}+\frac{10633}{2662 (-1+2 x)}+\frac{1}{3025 (3+5 x)^2}+\frac{136}{33275 (3+5 x)}\right ) \, dx\\ &=\frac{2401}{968 (1-2 x)}+\frac{81 x}{100}-\frac{1}{15125 (3+5 x)}+\frac{10633 \log (1-2 x)}{5324}+\frac{136 \log (3+5 x)}{166375}\\ \end{align*}

Mathematica [A]  time = 0.029979, size = 47, normalized size = 0.98 \[ \frac{-\frac{11 (1500641 x+900367)}{10 x^2+x-3}+359370 (3 x+2)+2658250 \log (3-6 x)+1088 \log (-3 (5 x+3))}{1331000} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^2*(3 + 5*x)^2),x]

[Out]

(359370*(2 + 3*x) - (11*(900367 + 1500641*x))/(-3 + x + 10*x^2) + 2658250*Log[3 - 6*x] + 1088*Log[-3*(3 + 5*x)
])/1331000

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Maple [A]  time = 0.008, size = 39, normalized size = 0.8 \begin{align*}{\frac{81\,x}{100}}-{\frac{2401}{1936\,x-968}}+{\frac{10633\,\ln \left ( 2\,x-1 \right ) }{5324}}-{\frac{1}{45375+75625\,x}}+{\frac{136\,\ln \left ( 3+5\,x \right ) }{166375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^2/(3+5*x)^2,x)

[Out]

81/100*x-2401/968/(2*x-1)+10633/5324*ln(2*x-1)-1/15125/(3+5*x)+136/166375*ln(3+5*x)

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Maxima [A]  time = 1.04935, size = 50, normalized size = 1.04 \begin{align*} \frac{81}{100} \, x - \frac{1500641 \, x + 900367}{121000 \,{\left (10 \, x^{2} + x - 3\right )}} + \frac{136}{166375} \, \log \left (5 \, x + 3\right ) + \frac{10633}{5324} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

81/100*x - 1/121000*(1500641*x + 900367)/(10*x^2 + x - 3) + 136/166375*log(5*x + 3) + 10633/5324*log(2*x - 1)

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Fricas [A]  time = 1.30851, size = 212, normalized size = 4.42 \begin{align*} \frac{10781100 \, x^{3} + 1078110 \, x^{2} + 1088 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 2658250 \,{\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 19741381 \, x - 9904037}{1331000 \,{\left (10 \, x^{2} + x - 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^2/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/1331000*(10781100*x^3 + 1078110*x^2 + 1088*(10*x^2 + x - 3)*log(5*x + 3) + 2658250*(10*x^2 + x - 3)*log(2*x
- 1) - 19741381*x - 9904037)/(10*x^2 + x - 3)

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Sympy [A]  time = 0.151787, size = 39, normalized size = 0.81 \begin{align*} \frac{81 x}{100} - \frac{1500641 x + 900367}{1210000 x^{2} + 121000 x - 363000} + \frac{10633 \log{\left (x - \frac{1}{2} \right )}}{5324} + \frac{136 \log{\left (x + \frac{3}{5} \right )}}{166375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**2/(3+5*x)**2,x)

[Out]

81*x/100 - (1500641*x + 900367)/(1210000*x**2 + 121000*x - 363000) + 10633*log(x - 1/2)/5324 + 136*log(x + 3/5
)/166375

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Giac [A]  time = 1.43563, size = 100, normalized size = 2.08 \begin{align*} \frac{{\left (5 \, x + 3\right )}{\left (\frac{1343273}{5 \, x + 3} - 107811\right )}}{332750 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}} - \frac{1}{15125 \,{\left (5 \, x + 3\right )}} - \frac{999}{500} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{10633}{5324} \, \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)^4/(1-2*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

1/332750*(5*x + 3)*(1343273/(5*x + 3) - 107811)/(11/(5*x + 3) - 2) - 1/15125/(5*x + 3) - 999/500*log(1/5*abs(5
*x + 3)/(5*x + 3)^2) + 10633/5324*log(abs(-11/(5*x + 3) + 2))