3.1622 \(\int \frac{(2+3 x)^3}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{343}{2662 (1-2 x)}-\frac{103}{33275 (5 x+3)}-\frac{1}{6050 (5 x+3)^2}-\frac{147 \log (1-2 x)}{14641}+\frac{147 \log (5 x+3)}{14641} \]

[Out]

343/(2662*(1 - 2*x)) - 1/(6050*(3 + 5*x)^2) - 103/(33275*(3 + 5*x)) - (147*Log[1 - 2*x])/14641 + (147*Log[3 +
5*x])/14641

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Rubi [A]  time = 0.0241942, 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} \[ \frac{343}{2662 (1-2 x)}-\frac{103}{33275 (5 x+3)}-\frac{1}{6050 (5 x+3)^2}-\frac{147 \log (1-2 x)}{14641}+\frac{147 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(2662*(1 - 2*x)) - 1/(6050*(3 + 5*x)^2) - 103/(33275*(3 + 5*x)) - (147*Log[1 - 2*x])/14641 + (147*Log[3 +
5*x])/14641

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)^2 (3+5 x)^3} \, dx &=\int \left (\frac{343}{1331 (-1+2 x)^2}-\frac{294}{14641 (-1+2 x)}+\frac{1}{605 (3+5 x)^3}+\frac{103}{6655 (3+5 x)^2}+\frac{735}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{343}{2662 (1-2 x)}-\frac{1}{6050 (3+5 x)^2}-\frac{103}{33275 (3+5 x)}-\frac{147 \log (1-2 x)}{14641}+\frac{147 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0263734, size = 47, normalized size = 0.87 \[ \frac{-\frac{11 \left (216435 x^2+257478 x+76546\right )}{(2 x-1) (5 x+3)^2}-7350 \log (1-2 x)+7350 \log (10 x+6)}{732050} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(76546 + 257478*x + 216435*x^2))/((-1 + 2*x)*(3 + 5*x)^2) - 7350*Log[1 - 2*x] + 7350*Log[6 + 10*x])/7320
50

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Maple [A]  time = 0.009, size = 45, normalized size = 0.8 \begin{align*} -{\frac{343}{5324\,x-2662}}-{\frac{147\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{1}{6050\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{103}{99825+166375\,x}}+{\frac{147\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/2662/(2*x-1)-147/14641*ln(2*x-1)-1/6050/(3+5*x)^2-103/33275/(3+5*x)+147/14641*ln(3+5*x)

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Maxima [A]  time = 1.07944, size = 62, normalized size = 1.15 \begin{align*} -\frac{216435 \, x^{2} + 257478 \, x + 76546}{66550 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{147}{14641} \, \log \left (5 \, x + 3\right ) - \frac{147}{14641} \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

-1/66550*(216435*x^2 + 257478*x + 76546)/(50*x^3 + 35*x^2 - 12*x - 9) + 147/14641*log(5*x + 3) - 147/14641*log
(2*x - 1)

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Fricas [A]  time = 1.31739, size = 234, normalized size = 4.33 \begin{align*} -\frac{2380785 \, x^{2} - 7350 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 7350 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 2832258 \, x + 842006}{732050 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/732050*(2380785*x^2 - 7350*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 7350*(50*x^3 + 35*x^2 - 12*x - 9)*lo
g(2*x - 1) + 2832258*x + 842006)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 0.153363, size = 44, normalized size = 0.81 \begin{align*} - \frac{216435 x^{2} + 257478 x + 76546}{3327500 x^{3} + 2329250 x^{2} - 798600 x - 598950} - \frac{147 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{147 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(216435*x**2 + 257478*x + 76546)/(3327500*x**3 + 2329250*x**2 - 798600*x - 598950) - 147*log(x - 1/2)/14641 +
147*log(x + 3/5)/14641

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Giac [A]  time = 2.69635, size = 69, normalized size = 1.28 \begin{align*} -\frac{343}{2662 \,{\left (2 \, x - 1\right )}} + \frac{2 \,{\left (\frac{231}{2 \, x - 1} + 104\right )}}{14641 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} + \frac{147}{14641} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343/2662/(2*x - 1) + 2/14641*(231/(2*x - 1) + 104)/(11/(2*x - 1) + 5)^2 + 147/14641*log(abs(-11/(2*x - 1) - 5
))