3.1522 \(\int \frac{(2+3 x)^2}{(1-2 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ -\frac{68}{3025 (5 x+3)}-\frac{1}{550 (5 x+3)^2}-\frac{49 \log (1-2 x)}{1331}+\frac{49 \log (5 x+3)}{1331} \]

[Out]

-1/(550*(3 + 5*x)^2) - 68/(3025*(3 + 5*x)) - (49*Log[1 - 2*x])/1331 + (49*Log[3 + 5*x])/1331

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Rubi [A]  time = 0.0193045, 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} \[ -\frac{68}{3025 (5 x+3)}-\frac{1}{550 (5 x+3)^2}-\frac{49 \log (1-2 x)}{1331}+\frac{49 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(550*(3 + 5*x)^2) - 68/(3025*(3 + 5*x)) - (49*Log[1 - 2*x])/1331 + (49*Log[3 + 5*x])/1331

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)^2}{(1-2 x) (3+5 x)^3} \, dx &=\int \left (-\frac{98}{1331 (-1+2 x)}+\frac{1}{55 (3+5 x)^3}+\frac{68}{605 (3+5 x)^2}+\frac{245}{1331 (3+5 x)}\right ) \, dx\\ &=-\frac{1}{550 (3+5 x)^2}-\frac{68}{3025 (3+5 x)}-\frac{49 \log (1-2 x)}{1331}+\frac{49 \log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0202736, size = 35, normalized size = 0.81 \[ \frac{-\frac{11 (680 x+419)}{(5 x+3)^2}-2450 \log (1-2 x)+2450 \log (10 x+6)}{66550} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(419 + 680*x))/(3 + 5*x)^2 - 2450*Log[1 - 2*x] + 2450*Log[6 + 10*x])/66550

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{49\,\ln \left ( 2\,x-1 \right ) }{1331}}-{\frac{1}{550\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{68}{9075+15125\,x}}+{\frac{49\,\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/1331*ln(2*x-1)-1/550/(3+5*x)^2-68/3025/(3+5*x)+49/1331*ln(3+5*x)

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Maxima [A]  time = 1.16188, size = 49, normalized size = 1.14 \begin{align*} -\frac{680 \, x + 419}{6050 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{49}{1331} \, \log \left (5 \, x + 3\right ) - \frac{49}{1331} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050*(680*x + 419)/(25*x^2 + 30*x + 9) + 49/1331*log(5*x + 3) - 49/1331*log(2*x - 1)

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Fricas [A]  time = 1.29664, size = 169, normalized size = 3.93 \begin{align*} \frac{2450 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 2450 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 7480 \, x - 4609}{66550 \,{\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)^2/(1-2*x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/66550*(2450*(25*x^2 + 30*x + 9)*log(5*x + 3) - 2450*(25*x^2 + 30*x + 9)*log(2*x - 1) - 7480*x - 4609)/(25*x^
2 + 30*x + 9)

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Sympy [A]  time = 0.146573, size = 34, normalized size = 0.79 \begin{align*} - \frac{680 x + 419}{151250 x^{2} + 181500 x + 54450} - \frac{49 \log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{49 \log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(680*x + 419)/(151250*x**2 + 181500*x + 54450) - 49*log(x - 1/2)/1331 + 49*log(x + 3/5)/1331

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Giac [A]  time = 1.26019, size = 45, normalized size = 1.05 \begin{align*} -\frac{680 \, x + 419}{6050 \,{\left (5 \, x + 3\right )}^{2}} + \frac{49}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{49}{1331} \, \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)^2/(1-2*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

-1/6050*(680*x + 419)/(5*x + 3)^2 + 49/1331*log(abs(5*x + 3)) - 49/1331*log(abs(2*x - 1))