3.1521 \(\int \frac{(2+3 x)^3}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{101}{15125 (5 x+3)}-\frac{1}{2750 (5 x+3)^2}-\frac{343 \log (1-2 x)}{2662}+\frac{3469 \log (5 x+3)}{166375} \]

[Out]

-1/(2750*(3 + 5*x)^2) - 101/(15125*(3 + 5*x)) - (343*Log[1 - 2*x])/2662 + (3469*Log[3 + 5*x])/166375

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Rubi [A]  time = 0.0181204, antiderivative size = 43, 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} \[ -\frac{101}{15125 (5 x+3)}-\frac{1}{2750 (5 x+3)^2}-\frac{343 \log (1-2 x)}{2662}+\frac{3469 \log (5 x+3)}{166375} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(2750*(3 + 5*x)^2) - 101/(15125*(3 + 5*x)) - (343*Log[1 - 2*x])/2662 + (3469*Log[3 + 5*x])/166375

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)^3}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac{343}{1331 (-1+2 x)}+\frac{1}{275 (3+5 x)^3}+\frac{101}{3025 (3+5 x)^2}+\frac{3469}{33275 (3+5 x)}\right ) \, dx\\ &=-\frac{1}{2750 (3+5 x)^2}-\frac{101}{15125 (3+5 x)}-\frac{343 \log (1-2 x)}{2662}+\frac{3469 \log (3+5 x)}{166375}\\ \end{align*}

Mathematica [A]  time = 0.0199605, size = 35, normalized size = 0.81 \[ \frac{-\frac{11 (1010 x+617)}{(5 x+3)^2}-42875 \log (1-2 x)+6938 \log (10 x+6)}{332750} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(617 + 1010*x))/(3 + 5*x)^2 - 42875*Log[1 - 2*x] + 6938*Log[6 + 10*x])/332750

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*} -{\frac{343\,\ln \left ( 2\,x-1 \right ) }{2662}}-{\frac{1}{2750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{101}{45375+75625\,x}}+{\frac{3469\,\ln \left ( 3+5\,x \right ) }{166375}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/2662*ln(2*x-1)-1/2750/(3+5*x)^2-101/15125/(3+5*x)+3469/166375*ln(3+5*x)

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Maxima [A]  time = 1.1355, size = 49, normalized size = 1.14 \begin{align*} -\frac{1010 \, x + 617}{30250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{3469}{166375} \, \log \left (5 \, x + 3\right ) - \frac{343}{2662} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30250*(1010*x + 617)/(25*x^2 + 30*x + 9) + 3469/166375*log(5*x + 3) - 343/2662*log(2*x - 1)

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Fricas [A]  time = 1.4412, size = 173, normalized size = 4.02 \begin{align*} \frac{6938 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 42875 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 11110 \, x - 6787}{332750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/332750*(6938*(25*x^2 + 30*x + 9)*log(5*x + 3) - 42875*(25*x^2 + 30*x + 9)*log(2*x - 1) - 11110*x - 6787)/(25
*x^2 + 30*x + 9)

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Sympy [A]  time = 0.158581, size = 34, normalized size = 0.79 \begin{align*} - \frac{1010 x + 617}{756250 x^{2} + 907500 x + 272250} - \frac{343 \log{\left (x - \frac{1}{2} \right )}}{2662} + \frac{3469 \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)**3/(1-2*x)/(3+5*x)**3,x)

[Out]

-(1010*x + 617)/(756250*x**2 + 907500*x + 272250) - 343*log(x - 1/2)/2662 + 3469*log(x + 3/5)/166375

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Giac [A]  time = 1.234, size = 45, normalized size = 1.05 \begin{align*} -\frac{1010 \, x + 617}{30250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{3469}{166375} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{343}{2662} \, \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)^3/(1-2*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

-1/30250*(1010*x + 617)/(5*x + 3)^2 + 3469/166375*log(abs(5*x + 3)) - 343/2662*log(abs(2*x - 1))