3.1675 \(\int \frac{(2+3 x)^4}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=48 \[ -\frac{81 x}{40}-\frac{33271}{1936 (1-2 x)}+\frac{2401}{352 (1-2 x)^2}-\frac{153811 \log (1-2 x)}{21296}+\frac{\log (5 x+3)}{33275} \]

[Out]

2401/(352*(1 - 2*x)^2) - 33271/(1936*(1 - 2*x)) - (81*x)/40 - (153811*Log[1 - 2*x])/21296 + Log[3 + 5*x]/33275

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Rubi [A]  time = 0.0199077, 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} \[ -\frac{81 x}{40}-\frac{33271}{1936 (1-2 x)}+\frac{2401}{352 (1-2 x)^2}-\frac{153811 \log (1-2 x)}{21296}+\frac{\log (5 x+3)}{33275} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2401/(352*(1 - 2*x)^2) - 33271/(1936*(1 - 2*x)) - (81*x)/40 - (153811*Log[1 - 2*x])/21296 + Log[3 + 5*x]/33275

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)} \, dx &=\int \left (-\frac{81}{40}-\frac{2401}{88 (-1+2 x)^3}-\frac{33271}{968 (-1+2 x)^2}-\frac{153811}{10648 (-1+2 x)}+\frac{1}{6655 (3+5 x)}\right ) \, dx\\ &=\frac{2401}{352 (1-2 x)^2}-\frac{33271}{1936 (1-2 x)}-\frac{81 x}{40}-\frac{153811 \log (1-2 x)}{21296}+\frac{\log (3+5 x)}{33275}\\ \end{align*}

Mathematica [A]  time = 0.0292742, size = 46, normalized size = 0.96 \[ \frac{-431244 (5 x+3)+\frac{18299050}{2 x-1}+\frac{7263025}{(1-2 x)^2}-7690550 \log (5-10 x)+32 \log (5 x+3)}{1064800} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7263025/(1 - 2*x)^2 + 18299050/(-1 + 2*x) - 431244*(3 + 5*x) - 7690550*Log[5 - 10*x] + 32*Log[3 + 5*x])/10648
00

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Maple [A]  time = 0.007, size = 39, normalized size = 0.8 \begin{align*} -{\frac{81\,x}{40}}+{\frac{2401}{352\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{33271}{3872\,x-1936}}-{\frac{153811\,\ln \left ( 2\,x-1 \right ) }{21296}}+{\frac{\ln \left ( 3+5\,x \right ) }{33275}} \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),x)

[Out]

-81/40*x+2401/352/(2*x-1)^2+33271/1936/(2*x-1)-153811/21296*ln(2*x-1)+1/33275*ln(3+5*x)

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Maxima [A]  time = 1.0504, size = 53, normalized size = 1.1 \begin{align*} -\frac{81}{40} \, x + \frac{343 \,{\left (388 \, x - 117\right )}}{3872 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{33275} \, \log \left (5 \, x + 3\right ) - \frac{153811}{21296} \, \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),x, algorithm="maxima")

[Out]

-81/40*x + 343/3872*(388*x - 117)/(4*x^2 - 4*x + 1) + 1/33275*log(5*x + 3) - 153811/21296*log(2*x - 1)

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Fricas [A]  time = 1.49601, size = 215, normalized size = 4.48 \begin{align*} -\frac{8624880 \, x^{3} - 8624880 \, x^{2} - 32 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) + 7690550 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 34441880 \, x + 11036025}{1064800 \,{\left (4 \, x^{2} - 4 \, 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),x, algorithm="fricas")

[Out]

-1/1064800*(8624880*x^3 - 8624880*x^2 - 32*(4*x^2 - 4*x + 1)*log(5*x + 3) + 7690550*(4*x^2 - 4*x + 1)*log(2*x
- 1) - 34441880*x + 11036025)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.159726, size = 37, normalized size = 0.77 \begin{align*} - \frac{81 x}{40} + \frac{133084 x - 40131}{15488 x^{2} - 15488 x + 3872} - \frac{153811 \log{\left (x - \frac{1}{2} \right )}}{21296} + \frac{\log{\left (x + \frac{3}{5} \right )}}{33275} \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),x)

[Out]

-81*x/40 + (133084*x - 40131)/(15488*x**2 - 15488*x + 3872) - 153811*log(x - 1/2)/21296 + log(x + 3/5)/33275

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Giac [A]  time = 1.79984, size = 49, normalized size = 1.02 \begin{align*} -\frac{81}{40} \, x + \frac{343 \,{\left (388 \, x - 117\right )}}{3872 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{33275} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{153811}{21296} \, \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)^4/(1-2*x)^3/(3+5*x),x, algorithm="giac")

[Out]

-81/40*x + 343/3872*(388*x - 117)/(2*x - 1)^2 + 1/33275*log(abs(5*x + 3)) - 153811/21296*log(abs(2*x - 1))