3.1583 \(\int \frac {(3+5 x)^3}{(1-2 x)^2 (2+3 x)^4} \, dx\)

Optimal. Leaf size=65 \[ \frac {1331}{2401 (1-2 x)}+\frac {363}{2401 (3 x+2)}-\frac {101}{6174 (3 x+2)^2}+\frac {1}{1323 (3 x+2)^3}-\frac {3267 \log (1-2 x)}{16807}+\frac {3267 \log (3 x+2)}{16807} \]

[Out]

1331/2401/(1-2*x)+1/1323/(2+3*x)^3-101/6174/(2+3*x)^2+363/2401/(2+3*x)-3267/16807*ln(1-2*x)+3267/16807*ln(2+3*
x)

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Rubi [A]  time = 0.03, antiderivative size = 65, 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} \[ \frac {1331}{2401 (1-2 x)}+\frac {363}{2401 (3 x+2)}-\frac {101}{6174 (3 x+2)^2}+\frac {1}{1323 (3 x+2)^3}-\frac {3267 \log (1-2 x)}{16807}+\frac {3267 \log (3 x+2)}{16807} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

1331/(2401*(1 - 2*x)) + 1/(1323*(2 + 3*x)^3) - 101/(6174*(2 + 3*x)^2) + 363/(2401*(2 + 3*x)) - (3267*Log[1 - 2
*x])/16807 + (3267*Log[2 + 3*x])/16807

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)^3}{(1-2 x)^2 (2+3 x)^4} \, dx &=\int \left (\frac {2662}{2401 (-1+2 x)^2}-\frac {6534}{16807 (-1+2 x)}-\frac {1}{147 (2+3 x)^4}+\frac {101}{1029 (2+3 x)^3}-\frac {1089}{2401 (2+3 x)^2}+\frac {9801}{16807 (2+3 x)}\right ) \, dx\\ &=\frac {1331}{2401 (1-2 x)}+\frac {1}{1323 (2+3 x)^3}-\frac {101}{6174 (2+3 x)^2}+\frac {363}{2401 (2+3 x)}-\frac {3267 \log (1-2 x)}{16807}+\frac {3267 \log (2+3 x)}{16807}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 62, normalized size = 0.95 \[ \frac {\frac {7}{2} \left (\frac {19602}{3 x+2}-\frac {2121}{(3 x+2)^2}+\frac {98}{(3 x+2)^3}+\frac {71874}{1-2 x}\right )-88209 \log (1-2 x)+88209 \log (6 x+4)}{453789} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)^2*(2 + 3*x)^4),x]

[Out]

((7*(71874/(1 - 2*x) + 98/(2 + 3*x)^3 - 2121/(2 + 3*x)^2 + 19602/(2 + 3*x)))/2 - 88209*Log[1 - 2*x] + 88209*Lo
g[4 + 6*x])/453789

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fricas [A]  time = 0.75, size = 95, normalized size = 1.46 \[ -\frac {11114334 \, x^{3} + 25199244 \, x^{2} - 176418 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (3 \, x + 2\right ) + 176418 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (2 \, x - 1\right ) + 18674579 \, x + 4544792}{907578 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/907578*(11114334*x^3 + 25199244*x^2 - 176418*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(3*x + 2) + 176418*(5
4*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(2*x - 1) + 18674579*x + 4544792)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)

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giac [A]  time = 0.94, size = 60, normalized size = 0.92 \[ -\frac {1331}{2401 \, {\left (2 \, x - 1\right )}} - \frac {2 \, {\left (\frac {43645}{2 \, x - 1} + \frac {50127}{{\left (2 \, x - 1\right )}^{2}} + 9502\right )}}{16807 \, {\left (\frac {7}{2 \, x - 1} + 3\right )}^{3}} + \frac {3267}{16807} \, \log \left ({\left | -\frac {7}{2 \, x - 1} - 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^4,x, algorithm="giac")

[Out]

-1331/2401/(2*x - 1) - 2/16807*(43645/(2*x - 1) + 50127/(2*x - 1)^2 + 9502)/(7/(2*x - 1) + 3)^3 + 3267/16807*l
og(abs(-7/(2*x - 1) - 3))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \[ -\frac {3267 \ln \left (2 x -1\right )}{16807}+\frac {3267 \ln \left (3 x +2\right )}{16807}+\frac {1}{1323 \left (3 x +2\right )^{3}}-\frac {101}{6174 \left (3 x +2\right )^{2}}+\frac {363}{2401 \left (3 x +2\right )}-\frac {1331}{2401 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(1-2*x)^2/(3*x+2)^4,x)

[Out]

1/1323/(3*x+2)^3-101/6174/(3*x+2)^2+363/2401/(3*x+2)+3267/16807*ln(3*x+2)-1331/2401/(2*x-1)-3267/16807*ln(2*x-
1)

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maxima [A]  time = 0.65, size = 56, normalized size = 0.86 \[ -\frac {1587762 \, x^{3} + 3599892 \, x^{2} + 2667797 \, x + 649256}{129654 \, {\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} + \frac {3267}{16807} \, \log \left (3 \, x + 2\right ) - \frac {3267}{16807} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)^2/(2+3*x)^4,x, algorithm="maxima")

[Out]

-1/129654*(1587762*x^3 + 3599892*x^2 + 2667797*x + 649256)/(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8) + 3267/16807*
log(3*x + 2) - 3267/16807*log(2*x - 1)

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mupad [B]  time = 1.18, size = 46, normalized size = 0.71 \[ \frac {6534\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{16807}-\frac {\frac {1089\,x^3}{4802}+\frac {99997\,x^2}{194481}+\frac {2667797\,x}{7001316}+\frac {162314}{1750329}}{x^4+\frac {3\,x^3}{2}+\frac {x^2}{3}-\frac {10\,x}{27}-\frac {4}{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^3/((2*x - 1)^2*(3*x + 2)^4),x)

[Out]

(6534*atanh((12*x)/7 + 1/7))/16807 - ((2667797*x)/7001316 + (99997*x^2)/194481 + (1089*x^3)/4802 + 162314/1750
329)/(x^2/3 - (10*x)/27 + (3*x^3)/2 + x^4 - 4/27)

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sympy [A]  time = 0.18, size = 56, normalized size = 0.86 \[ \frac {- 1587762 x^{3} - 3599892 x^{2} - 2667797 x - 649256}{7001316 x^{4} + 10501974 x^{3} + 2333772 x^{2} - 2593080 x - 1037232} - \frac {3267 \log {\left (x - \frac {1}{2} \right )}}{16807} + \frac {3267 \log {\left (x + \frac {2}{3} \right )}}{16807} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)**2/(2+3*x)**4,x)

[Out]

(-1587762*x**3 - 3599892*x**2 - 2667797*x - 649256)/(7001316*x**4 + 10501974*x**3 + 2333772*x**2 - 2593080*x -
1037232) - 3267*log(x - 1/2)/16807 + 3267*log(x + 2/3)/16807

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