3.1653 \(\int \frac{(3+5 x)^2}{(1-2 x)^3 (2+3 x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{407}{196 (1-2 x)}+\frac{121}{56 (1-2 x)^2}-\frac{1}{343} \log (1-2 x)+\frac{1}{343} \log (3 x+2) \]

[Out]

121/(56*(1 - 2*x)^2) - 407/(196*(1 - 2*x)) - Log[1 - 2*x]/343 + Log[2 + 3*x]/343

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Rubi [A]  time = 0.017587, 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} \[ -\frac{407}{196 (1-2 x)}+\frac{121}{56 (1-2 x)^2}-\frac{1}{343} \log (1-2 x)+\frac{1}{343} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(56*(1 - 2*x)^2) - 407/(196*(1 - 2*x)) - Log[1 - 2*x]/343 + Log[2 + 3*x]/343

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{(3+5 x)^2}{(1-2 x)^3 (2+3 x)} \, dx &=\int \left (-\frac{121}{14 (-1+2 x)^3}-\frac{407}{98 (-1+2 x)^2}-\frac{2}{343 (-1+2 x)}+\frac{3}{343 (2+3 x)}\right ) \, dx\\ &=\frac{121}{56 (1-2 x)^2}-\frac{407}{196 (1-2 x)}-\frac{1}{343} \log (1-2 x)+\frac{1}{343} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0222259, size = 35, normalized size = 0.81 \[ \frac{\frac{77 (148 x+3)}{(1-2 x)^2}-8 \log (3-6 x)+8 \log (3 x+2)}{2744} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(3 + 148*x))/(1 - 2*x)^2 - 8*Log[3 - 6*x] + 8*Log[2 + 3*x])/2744

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{121}{56\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{407}{392\,x-196}}-{\frac{\ln \left ( 2\,x-1 \right ) }{343}}+{\frac{\ln \left ( 2+3\,x \right ) }{343}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/56/(2*x-1)^2+407/196/(2*x-1)-1/343*ln(2*x-1)+1/343*ln(2+3*x)

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Maxima [A]  time = 1.06975, size = 49, normalized size = 1.14 \begin{align*} \frac{11 \,{\left (148 \, x + 3\right )}}{392 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{343} \, \log \left (3 \, x + 2\right ) - \frac{1}{343} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/392*(148*x + 3)/(4*x^2 - 4*x + 1) + 1/343*log(3*x + 2) - 1/343*log(2*x - 1)

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Fricas [A]  time = 1.65345, size = 151, normalized size = 3.51 \begin{align*} \frac{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (3 \, x + 2\right ) - 8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) + 11396 \, x + 231}{2744 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2744*(8*(4*x^2 - 4*x + 1)*log(3*x + 2) - 8*(4*x^2 - 4*x + 1)*log(2*x - 1) + 11396*x + 231)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.138477, size = 31, normalized size = 0.72 \begin{align*} \frac{1628 x + 33}{1568 x^{2} - 1568 x + 392} - \frac{\log{\left (x - \frac{1}{2} \right )}}{343} + \frac{\log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1628*x + 33)/(1568*x**2 - 1568*x + 392) - log(x - 1/2)/343 + log(x + 2/3)/343

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Giac [A]  time = 1.98953, size = 45, normalized size = 1.05 \begin{align*} \frac{11 \,{\left (148 \, x + 3\right )}}{392 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{1}{343} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/392*(148*x + 3)/(2*x - 1)^2 + 1/343*log(abs(3*x + 2)) - 1/343*log(abs(2*x - 1))