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

Optimal. Leaf size=64 \[ \frac {8}{9317 (1-2 x)}+\frac {725}{1331 (5 x+3)}-\frac {25}{242 (5 x+3)^2}-\frac {1104 \log (1-2 x)}{717409}-\frac {81}{49} \log (3 x+2)+\frac {24225 \log (5 x+3)}{14641} \]

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Rubi [A]  time = 0.03, antiderivative size = 64, 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} \begin {gather*} \frac {8}{9317 (1-2 x)}+\frac {725}{1331 (5 x+3)}-\frac {25}{242 (5 x+3)^2}-\frac {1104 \log (1-2 x)}{717409}-\frac {81}{49} \log (3 x+2)+\frac {24225 \log (5 x+3)}{14641} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

8/(9317*(1 - 2*x)) - 25/(242*(3 + 5*x)^2) + 725/(1331*(3 + 5*x)) - (1104*Log[1 - 2*x])/717409 - (81*Log[2 + 3*
x])/49 + (24225*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 {1}{(1-2 x)^2 (2+3 x) (3+5 x)^3} \, dx &=\int \left (\frac {16}{9317 (-1+2 x)^2}-\frac {2208}{717409 (-1+2 x)}-\frac {243}{49 (2+3 x)}+\frac {125}{121 (3+5 x)^3}-\frac {3625}{1331 (3+5 x)^2}+\frac {121125}{14641 (3+5 x)}\right ) \, dx\\ &=\frac {8}{9317 (1-2 x)}-\frac {25}{242 (3+5 x)^2}+\frac {725}{1331 (3+5 x)}-\frac {1104 \log (1-2 x)}{717409}-\frac {81}{49} \log (2+3 x)+\frac {24225 \log (3+5 x)}{14641}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 60, normalized size = 0.94 \begin {gather*} \frac {3 \left (\frac {1232}{3-6 x}+\frac {781550}{15 x+9}-\frac {148225}{3 (5 x+3)^2}-736 \log (3-6 x)-790614 \log (3 x+2)+791350 \log (-3 (5 x+3))\right )}{1434818} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(1232/(3 - 6*x) - 148225/(3*(3 + 5*x)^2) + 781550/(9 + 15*x) - 736*Log[3 - 6*x] - 790614*Log[2 + 3*x] + 791
350*Log[-3*(3 + 5*x)]))/1434818

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(1-2 x)^2 (2+3 x) (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.27, size = 98, normalized size = 1.53 \begin {gather*} \frac {7784700 \, x^{2} + 2374050 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) - 2371842 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (3 \, x + 2\right ) - 2208 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 448140 \, x - 2207513}{1434818 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1434818*(7784700*x^2 + 2374050*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) - 2371842*(50*x^3 + 35*x^2 - 12*x -
9)*log(3*x + 2) - 2208*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) + 448140*x - 2207513)/(50*x^3 + 35*x^2 - 12*
x - 9)

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giac [A]  time = 1.28, size = 66, normalized size = 1.03 \begin {gather*} -\frac {8}{9317 \, {\left (2 \, x - 1\right )}} - \frac {250 \, {\left (\frac {297}{2 \, x - 1} + 140\right )}}{14641 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {81}{49} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) + \frac {24225}{14641} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/9317/(2*x - 1) - 250/14641*(297/(2*x - 1) + 140)/(11/(2*x - 1) + 5)^2 - 81/49*log(abs(-7/(2*x - 1) - 3)) +
24225/14641*log(abs(-11/(2*x - 1) - 5))

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maple [A]  time = 0.01, size = 53, normalized size = 0.83 \begin {gather*} -\frac {1104 \ln \left (2 x -1\right )}{717409}-\frac {81 \ln \left (3 x +2\right )}{49}+\frac {24225 \ln \left (5 x +3\right )}{14641}-\frac {25}{242 \left (5 x +3\right )^{2}}+\frac {725}{1331 \left (5 x +3\right )}-\frac {8}{9317 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-25/242/(5*x+3)^2+725/1331/(5*x+3)+24225/14641*ln(5*x+3)-81/49*ln(3*x+2)-8/9317/(2*x-1)-1104/717409*ln(2*x-1)

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maxima [A]  time = 0.48, size = 54, normalized size = 0.84 \begin {gather*} \frac {101100 \, x^{2} + 5820 \, x - 28669}{18634 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {24225}{14641} \, \log \left (5 \, x + 3\right ) - \frac {81}{49} \, \log \left (3 \, x + 2\right ) - \frac {1104}{717409} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/18634*(101100*x^2 + 5820*x - 28669)/(50*x^3 + 35*x^2 - 12*x - 9) + 24225/14641*log(5*x + 3) - 81/49*log(3*x
+ 2) - 1104/717409*log(2*x - 1)

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mupad [B]  time = 1.05, size = 48, normalized size = 0.75 \begin {gather*} \frac {24225\,\ln \left (x+\frac {3}{5}\right )}{14641}-\frac {81\,\ln \left (x+\frac {2}{3}\right )}{49}-\frac {1104\,\ln \left (x-\frac {1}{2}\right )}{717409}-\frac {\frac {1011\,x^2}{9317}+\frac {291\,x}{46585}-\frac {28669}{931700}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(24225*log(x + 3/5))/14641 - (81*log(x + 2/3))/49 - (1104*log(x - 1/2))/717409 - ((291*x)/46585 + (1011*x^2)/9
317 - 28669/931700)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50)

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sympy [A]  time = 0.22, size = 54, normalized size = 0.84 \begin {gather*} \frac {101100 x^{2} + 5820 x - 28669}{931700 x^{3} + 652190 x^{2} - 223608 x - 167706} - \frac {1104 \log {\left (x - \frac {1}{2} \right )}}{717409} + \frac {24225 \log {\left (x + \frac {3}{5} \right )}}{14641} - \frac {81 \log {\left (x + \frac {2}{3} \right )}}{49} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(101100*x**2 + 5820*x - 28669)/(931700*x**3 + 652190*x**2 - 223608*x - 167706) - 1104*log(x - 1/2)/717409 + 24
225*log(x + 3/5)/14641 - 81*log(x + 2/3)/49

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