∫ (2+3x)9 ________________________

3.1697 \(\int \frac{(2+3 x)^9}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=91 \[ -\frac{19683 x^4}{4000}-\frac{373977 x^3}{10000}-\frac{7459857 x^2}{50000}-\frac{50150097 x}{100000}-\frac{17294403}{29282 (1-2 x)}-\frac{303}{1143828125 (5 x+3)}+\frac{40353607}{340736 (1-2 x)^2}-\frac{1}{207968750 (5 x+3)^2}-\frac{12657032367 \log (1-2 x)}{20614528}+\frac{8202 \log (5 x+3)}{2516421875} \]

[Out]

40353607/(340736*(1 - 2*x)^2) - 17294403/(29282*(1 - 2*x)) - (50150097*x)/100000 - (7459857*x^2)/50000 - (3739

77*x^3)/10000 - (19683*x^4)/4000 - 1/(207968750*(3 + 5*x)^2) - 303/(1143828125*(3 + 5*x)) - (12657032367*Log[1

- 2*x])/20614528 + (8202*Log[3 + 5*x])/2516421875

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Rubi [A] time = 0.0531208, antiderivative size = 91, 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} \[ -\frac{19683 x^4}{4000}-\frac{373977 x^3}{10000}-\frac{7459857 x^2}{50000}-\frac{50150097 x}{100000}-\frac{17294403}{29282 (1-2 x)}-\frac{303}{1143828125 (5 x+3)}+\frac{40353607}{340736 (1-2 x)^2}-\frac{1}{207968750 (5 x+3)^2}-\frac{12657032367 \log (1-2 x)}{20614528}+\frac{8202 \log (5 x+3)}{2516421875} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(2 + 3*x)^9/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

40353607/(340736*(1 - 2*x)^2) - 17294403/(29282*(1 - 2*x)) - (50150097*x)/100000 - (7459857*x^2)/50000 - (3739

77*x^3)/10000 - (19683*x^4)/4000 - 1/(207968750*(3 + 5*x)^2) - 303/(1143828125*(3 + 5*x)) - (12657032367*Log[1

- 2*x])/20614528 + (8202*Log[3 + 5*x])/2516421875

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)^9}{(1-2 x)^3 (3+5 x)^3} \, dx &=\int \left (-\frac{50150097}{100000}-\frac{7459857 x}{25000}-\frac{1121931 x^2}{10000}-\frac{19683 x^3}{1000}-\frac{40353607}{85184 (-1+2 x)^3}-\frac{17294403}{14641 (-1+2 x)^2}-\frac{12657032367}{10307264 (-1+2 x)}+\frac{1}{20796875 (3+5 x)^3}+\frac{303}{228765625 (3+5 x)^2}+\frac{8202}{503284375 (3+5 x)}\right ) \, dx\\ &=\frac{40353607}{340736 (1-2 x)^2}-\frac{17294403}{29282 (1-2 x)}-\frac{50150097 x}{100000}-\frac{7459857 x^2}{50000}-\frac{373977 x^3}{10000}-\frac{19683 x^4}{4000}-\frac{1}{207968750 (3+5 x)^2}-\frac{303}{1143828125 (3+5 x)}-\frac{12657032367 \log (1-2 x)}{20614528}+\frac{8202 \log (3+5 x)}{2516421875}\\ \end{align*}

Mathematica [A] time = 0.0441248, size = 75, normalized size = 0.82 \[ \frac{-\frac{11 \left (144089401500000 x^8+1123897331700000 x^7+4502793796875000 x^6+14903967293820000 x^5+8450955823285800 x^4-15846035365304040 x^3-12213049363361937 x^2+1867950230356442 x+1977372328510687\right )}{\left (10 x^2+x-3\right )^2}-1977661307343750 \log (3-6 x)+10498560 \log (-3 (5 x+3))}{3221020000000} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(2 + 3*x)^9/((1 - 2*x)^3*(3 + 5*x)^3),x]

[Out]

((-11*(1977372328510687 + 1867950230356442*x - 12213049363361937*x^2 - 15846035365304040*x^3 + 845095582328580

0*x^4 + 14903967293820000*x^5 + 4502793796875000*x^6 + 1123897331700000*x^7 + 144089401500000*x^8))/(-3 + x +

10*x^2)^2 - 1977661307343750*Log[3 - 6*x] + 10498560*Log[-3*(3 + 5*x)])/3221020000000

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Maple [A] time = 0.01, size = 72, normalized size = 0.8 \begin{align*} -{\frac{19683\,{x}^{4}}{4000}}-{\frac{373977\,{x}^{3}}{10000}}-{\frac{7459857\,{x}^{2}}{50000}}-{\frac{50150097\,x}{100000}}+{\frac{40353607}{340736\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{17294403}{58564\,x-29282}}-{\frac{12657032367\,\ln \left ( 2\,x-1 \right ) }{20614528}}-{\frac{1}{207968750\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{303}{3431484375+5719140625\,x}}+{\frac{8202\,\ln \left ( 3+5\,x \right ) }{2516421875}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x)

[Out]

-19683/4000*x^4-373977/10000*x^3-7459857/50000*x^2-50150097/100000*x+40353607/340736/(2*x-1)^2+17294403/29282/

(2*x-1)-12657032367/20614528*ln(2*x-1)-1/207968750/(3+5*x)^2-303/1143828125/(3+5*x)+8202/2516421875*ln(3+5*x)

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Maxima [A] time = 1.08051, size = 100, normalized size = 1.1 \begin{align*} -\frac{19683}{4000} \, x^{4} - \frac{373977}{10000} \, x^{3} - \frac{7459857}{50000} \, x^{2} - \frac{50150097}{100000} \, x + \frac{8647201498448640 \, x^{3} + 6920013076005537 \, x^{2} - 1034961928982642 \, x - 1244386341093487}{292820000000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac{8202}{2516421875} \, \log \left (5 \, x + 3\right ) - \frac{12657032367}{20614528} \, \log \left (2 \, x - 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x, algorithm="maxima")

[Out]

-19683/4000*x^4 - 373977/10000*x^3 - 7459857/50000*x^2 - 50150097/100000*x + 1/292820000000*(8647201498448640*

x^3 + 6920013076005537*x^2 - 1034961928982642*x - 1244386341093487)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 82

02/2516421875*log(5*x + 3) - 12657032367/20614528*log(2*x - 1)

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Fricas [A] time = 1.52635, size = 537, normalized size = 5.9 \begin{align*} -\frac{1584983416500000 \, x^{8} + 12362870648700000 \, x^{7} + 49530731765625000 \, x^{6} + 163943640232020000 \, x^{5} + 3373337816263800 \, x^{4} - 192223824266320440 \, x^{3} - 81487109015452107 \, x^{2} - 10498560 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 1977661307343750 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) + 25922683108313662 \, x + 13688249752028357}{3221020000000 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^9/(1-2*x)^3/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/3221020000000*(1584983416500000*x^8 + 12362870648700000*x^7 + 49530731765625000*x^6 + 163943640232020000*x^

5 + 3373337816263800*x^4 - 192223824266320440*x^3 - 81487109015452107*x^2 - 10498560*(100*x^4 + 20*x^3 - 59*x^

2 - 6*x + 9)*log(5*x + 3) + 1977661307343750*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) + 259226831083

13662*x + 13688249752028357)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [A] time = 0.207334, size = 80, normalized size = 0.88 \begin{align*} - \frac{19683 x^{4}}{4000} - \frac{373977 x^{3}}{10000} - \frac{7459857 x^{2}}{50000} - \frac{50150097 x}{100000} + \frac{8647201498448640 x^{3} + 6920013076005537 x^{2} - 1034961928982642 x - 1244386341093487}{29282000000000 x^{4} + 5856400000000 x^{3} - 17276380000000 x^{2} - 1756920000000 x + 2635380000000} - \frac{12657032367 \log{\left (x - \frac{1}{2} \right )}}{20614528} + \frac{8202 \log{\left (x + \frac{3}{5} \right )}}{2516421875} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**9/(1-2*x)**3/(3+5*x)**3,x)

[Out]

-19683*x**4/4000 - 373977*x**3/10000 - 7459857*x**2/50000 - 50150097*x/100000 + (8647201498448640*x**3 + 69200

13076005537*x**2 - 1034961928982642*x - 1244386341093487)/(29282000000000*x**4 + 5856400000000*x**3 - 17276380

000000*x**2 - 1756920000000*x + 2635380000000) - 12657032367*log(x - 1/2)/20614528 + 8202*log(x + 3/5)/2516421

875

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Giac [A] time = 2.0431, size = 92, normalized size = 1.01 \begin{align*} -\frac{19683}{4000} \, x^{4} - \frac{373977}{10000} \, x^{3} - \frac{7459857}{50000} \, x^{2} - \frac{50150097}{100000} \, x + \frac{8647201498448640 \, x^{3} + 6920013076005537 \, x^{2} - 1034961928982642 \, x - 1244386341093487}{292820000000 \,{\left (5 \, x + 3\right )}^{2}{\left (2 \, x - 1\right )}^{2}} + \frac{8202}{2516421875} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{12657032367}{20614528} \, \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)^9/(1-2*x)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-19683/4000*x^4 - 373977/10000*x^3 - 7459857/50000*x^2 - 50150097/100000*x + 1/292820000000*(8647201498448640*

x^3 + 6920013076005537*x^2 - 1034961928982642*x - 1244386341093487)/((5*x + 3)^2*(2*x - 1)^2) + 8202/251642187

5*log(abs(5*x + 3)) - 12657032367/20614528*log(abs(2*x - 1))

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