3.1688 \(\int \frac{(2+3 x)^4}{(1-2 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=54 \[ -\frac{10633}{5324 (1-2 x)}-\frac{1}{33275 (5 x+3)}+\frac{2401}{1936 (1-2 x)^2}-\frac{47481 \log (1-2 x)}{117128}+\frac{138 \log (5 x+3)}{366025} \]

[Out]

2401/(1936*(1 - 2*x)^2) - 10633/(5324*(1 - 2*x)) - 1/(33275*(3 + 5*x)) - (47481*Log[1 - 2*x])/117128 + (138*Lo
g[3 + 5*x])/366025

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Rubi [A]  time = 0.0251708, antiderivative size = 54, 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} \[ -\frac{10633}{5324 (1-2 x)}-\frac{1}{33275 (5 x+3)}+\frac{2401}{1936 (1-2 x)^2}-\frac{47481 \log (1-2 x)}{117128}+\frac{138 \log (5 x+3)}{366025} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2401/(1936*(1 - 2*x)^2) - 10633/(5324*(1 - 2*x)) - 1/(33275*(3 + 5*x)) - (47481*Log[1 - 2*x])/117128 + (138*Lo
g[3 + 5*x])/366025

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)^3 (3+5 x)^2} \, dx &=\int \left (-\frac{2401}{484 (-1+2 x)^3}-\frac{10633}{2662 (-1+2 x)^2}-\frac{47481}{58564 (-1+2 x)}+\frac{1}{6655 (3+5 x)^2}+\frac{138}{73205 (3+5 x)}\right ) \, dx\\ &=\frac{2401}{1936 (1-2 x)^2}-\frac{10633}{5324 (1-2 x)}-\frac{1}{33275 (3+5 x)}-\frac{47481 \log (1-2 x)}{117128}+\frac{138 \log (3+5 x)}{366025}\\ \end{align*}

Mathematica [A]  time = 0.02889, size = 48, normalized size = 0.89 \[ \frac{\frac{11696300}{2 x-1}-\frac{176}{5 x+3}+\frac{7263025}{(1-2 x)^2}-2374050 \log (1-2 x)+2208 \log (10 x+6)}{5856400} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7263025/(1 - 2*x)^2 + 11696300/(-1 + 2*x) - 176/(3 + 5*x) - 2374050*Log[1 - 2*x] + 2208*Log[6 + 10*x])/585640
0

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Maple [A]  time = 0.008, size = 45, normalized size = 0.8 \begin{align*}{\frac{2401}{1936\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{10633}{10648\,x-5324}}-{\frac{47481\,\ln \left ( 2\,x-1 \right ) }{117128}}-{\frac{1}{99825+166375\,x}}+{\frac{138\,\ln \left ( 3+5\,x \right ) }{366025}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2401/1936/(2*x-1)^2+10633/5324/(2*x-1)-47481/117128*ln(2*x-1)-1/33275/(3+5*x)+138/366025*ln(3+5*x)

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Maxima [A]  time = 1.0606, size = 62, normalized size = 1.15 \begin{align*} \frac{10632936 \, x^{2} + 4364739 \, x - 1209091}{532400 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} + \frac{138}{366025} \, \log \left (5 \, x + 3\right ) - \frac{47481}{117128} \, \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)^3/(3+5*x)^2,x, algorithm="maxima")

[Out]

1/532400*(10632936*x^2 + 4364739*x - 1209091)/(20*x^3 - 8*x^2 - 7*x + 3) + 138/366025*log(5*x + 3) - 47481/117
128*log(2*x - 1)

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Fricas [A]  time = 1.51184, size = 236, normalized size = 4.37 \begin{align*} \frac{116962296 \, x^{2} + 2208 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 2374050 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 48012129 \, x - 13300001}{5856400 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5856400*(116962296*x^2 + 2208*(20*x^3 - 8*x^2 - 7*x + 3)*log(5*x + 3) - 2374050*(20*x^3 - 8*x^2 - 7*x + 3)*l
og(2*x - 1) + 48012129*x - 13300001)/(20*x^3 - 8*x^2 - 7*x + 3)

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Sympy [A]  time = 0.182854, size = 44, normalized size = 0.81 \begin{align*} \frac{10632936 x^{2} + 4364739 x - 1209091}{10648000 x^{3} - 4259200 x^{2} - 3726800 x + 1597200} - \frac{47481 \log{\left (x - \frac{1}{2} \right )}}{117128} + \frac{138 \log{\left (x + \frac{3}{5} \right )}}{366025} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(10632936*x**2 + 4364739*x - 1209091)/(10648000*x**3 - 4259200*x**2 - 3726800*x + 1597200) - 47481*log(x - 1/2
)/117128 + 138*log(x + 3/5)/366025

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Giac [A]  time = 3.0357, size = 93, normalized size = 1.72 \begin{align*} -\frac{1}{33275 \,{\left (5 \, x + 3\right )}} - \frac{1715 \,{\left (\frac{297}{5 \, x + 3} - 89\right )}}{58564 \,{\left (\frac{11}{5 \, x + 3} - 2\right )}^{2}} + \frac{81}{200} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{47481}{117128} \, \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)^3/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/33275/(5*x + 3) - 1715/58564*(297/(5*x + 3) - 89)/(11/(5*x + 3) - 2)^2 + 81/200*log(1/5*abs(5*x + 3)/(5*x +
3)^2) - 47481/117128*log(abs(-11/(5*x + 3) + 2))