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)-147/14641*ln(1-2*x)+147/14641*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 54, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.04, 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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fricas [A]  time = 0.60, size = 75, normalized size = 1.39 \[ -\frac {2380785 \, x^{2} - 7350 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 7350 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 2832258 \, x + 842006}{732050 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \]

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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giac [A]  time = 1.20, size = 51, normalized size = 0.94 \[ -\frac {343}{2662 \, {\left (2 \, x - 1\right )}} + \frac {2 \, {\left (\frac {231}{2 \, x - 1} + 104\right )}}{14641 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} + \frac {147}{14641} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

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
))

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maple [A]  time = 0.01, size = 45, normalized size = 0.83 \[ -\frac {147 \ln \left (2 x -1\right )}{14641}+\frac {147 \ln \left (5 x +3\right )}{14641}-\frac {1}{6050 \left (5 x +3\right )^{2}}-\frac {103}{33275 \left (5 x +3\right )}-\frac {343}{2662 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.59, size = 46, normalized size = 0.85 \[ -\frac {216435 \, x^{2} + 257478 \, x + 76546}{66550 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {147}{14641} \, \log \left (5 \, x + 3\right ) - \frac {147}{14641} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.06, size = 37, normalized size = 0.69 \[ \frac {294\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{14641}+\frac {\frac {43287\,x^2}{665500}+\frac {128739\,x}{1663750}+\frac {38273}{1663750}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(294*atanh((20*x)/11 + 1/11))/14641 + ((128739*x)/1663750 + (43287*x^2)/665500 + 38273/1663750)/((6*x)/25 - (7
*x^2)/10 - x^3 + 9/50)

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sympy [A]  time = 0.16, size = 46, normalized size = 0.85 \[ \frac {- 216435 x^{2} - 257478 x - 76546}{3327500 x^{3} + 2329250 x^{2} - 798600 x - 598950} - \frac {147 \log {\left (x - \frac {1}{2} \right )}}{14641} + \frac {147 \log {\left (x + \frac {3}{5} \right )}}{14641} \]

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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