∫ (3+5x)3 ______________________

3.14.44 \(\int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx\)

Optimal. Leaf size=98 \[ -\frac {10648}{823543 (3 x+2)}-\frac {2662}{117649 (3 x+2)^2}-\frac {2662}{50421 (3 x+2)^3}-\frac {1331}{9604 (3 x+2)^4}+\frac {3469}{46305 (3 x+2)^5}-\frac {103}{7938 (3 x+2)^6}+\frac {1}{1323 (3 x+2)^7}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (3 x+2)}{5764801} \]

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Rubi [A] time = 0.04, antiderivative size = 98, 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 {10648}{823543 (3 x+2)}-\frac {2662}{117649 (3 x+2)^2}-\frac {2662}{50421 (3 x+2)^3}-\frac {1331}{9604 (3 x+2)^4}+\frac {3469}{46305 (3 x+2)^5}-\frac {103}{7938 (3 x+2)^6}+\frac {1}{1323 (3 x+2)^7}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (3 x+2)}{5764801} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

1/(1323*(2 + 3*x)^7) - 103/(7938*(2 + 3*x)^6) + 3469/(46305*(2 + 3*x)^5) - 1331/(9604*(2 + 3*x)^4) - 2662/(504

21*(2 + 3*x)^3) - 2662/(117649*(2 + 3*x)^2) - 10648/(823543*(2 + 3*x)) - (21296*Log[1 - 2*x])/5764801 + (21296

*Log[2 + 3*x])/5764801

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+3 x)^8} \, dx &=\int \left (-\frac {42592}{5764801 (-1+2 x)}-\frac {1}{63 (2+3 x)^8}+\frac {103}{441 (2+3 x)^7}-\frac {3469}{3087 (2+3 x)^6}+\frac {3993}{2401 (2+3 x)^5}+\frac {7986}{16807 (2+3 x)^4}+\frac {15972}{117649 (2+3 x)^3}+\frac {31944}{823543 (2+3 x)^2}+\frac {63888}{5764801 (2+3 x)}\right ) \, dx\\ &=\frac {1}{1323 (2+3 x)^7}-\frac {103}{7938 (2+3 x)^6}+\frac {3469}{46305 (2+3 x)^5}-\frac {1331}{9604 (2+3 x)^4}-\frac {2662}{50421 (2+3 x)^3}-\frac {2662}{117649 (2+3 x)^2}-\frac {10648}{823543 (2+3 x)}-\frac {21296 \log (1-2 x)}{5764801}+\frac {21296 \log (2+3 x)}{5764801}\\ \end {align*}

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Mathematica [A] time = 0.11, size = 62, normalized size = 0.63 \begin {gather*} \frac {4 \left (-\frac {7 \left (12575075040 x^6+57635760600 x^5+113990726520 x^4+127327486275 x^3+83293304778 x^2+29451465714 x+4309941128\right )}{16 (3 x+2)^7}-2156220 \log (1-2 x)+2156220 \log (6 x+4)\right )}{2334744405} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

(4*((-7*(4309941128 + 29451465714*x + 83293304778*x^2 + 127327486275*x^3 + 113990726520*x^4 + 57635760600*x^5

+ 12575075040*x^6))/(16*(2 + 3*x)^7) - 2156220*Log[1 - 2*x] + 2156220*Log[4 + 6*x]))/2334744405

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^8} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^8),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^8), x]

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fricas [A] time = 1.52, size = 155, normalized size = 1.58 \begin {gather*} -\frac {88025525280 \, x^{6} + 403450324200 \, x^{5} + 797935085640 \, x^{4} + 891292403925 \, x^{3} + 583053133446 \, x^{2} - 34499520 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (3 \, x + 2\right ) + 34499520 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} \log \left (2 \, x - 1\right ) + 206160259998 \, x + 30169587896}{9338977620 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^8,x, algorithm="fricas")

[Out]

-1/9338977620*(88025525280*x^6 + 403450324200*x^5 + 797935085640*x^4 + 891292403925*x^3 + 583053133446*x^2 - 3

4499520*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(3*x + 2) + 34

499520*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*log(2*x - 1) + 206

160259998*x + 30169587896)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128

)

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giac [A] time = 0.95, size = 58, normalized size = 0.59 \begin {gather*} -\frac {12575075040 \, x^{6} + 57635760600 \, x^{5} + 113990726520 \, x^{4} + 127327486275 \, x^{3} + 83293304778 \, x^{2} + 29451465714 \, x + 4309941128}{1334139660 \, {\left (3 \, x + 2\right )}^{7}} + \frac {21296}{5764801} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {21296}{5764801} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^8,x, algorithm="giac")

[Out]

-1/1334139660*(12575075040*x^6 + 57635760600*x^5 + 113990726520*x^4 + 127327486275*x^3 + 83293304778*x^2 + 294

51465714*x + 4309941128)/(3*x + 2)^7 + 21296/5764801*log(abs(3*x + 2)) - 21296/5764801*log(abs(2*x - 1))

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maple [A] time = 0.01, size = 81, normalized size = 0.83 \begin {gather*} -\frac {21296 \ln \left (2 x -1\right )}{5764801}+\frac {21296 \ln \left (3 x +2\right )}{5764801}+\frac {1}{1323 \left (3 x +2\right )^{7}}-\frac {103}{7938 \left (3 x +2\right )^{6}}+\frac {3469}{46305 \left (3 x +2\right )^{5}}-\frac {1331}{9604 \left (3 x +2\right )^{4}}-\frac {2662}{50421 \left (3 x +2\right )^{3}}-\frac {2662}{117649 \left (3 x +2\right )^{2}}-\frac {10648}{823543 \left (3 x +2\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^3/(1-2*x)/(3*x+2)^8,x)

[Out]

1/1323/(3*x+2)^7-103/7938/(3*x+2)^6+3469/46305/(3*x+2)^5-1331/9604/(3*x+2)^4-2662/50421/(3*x+2)^3-2662/117649/

(3*x+2)^2-10648/823543/(3*x+2)+21296/5764801*ln(3*x+2)-21296/5764801*ln(2*x-1)

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maxima [A] time = 0.54, size = 86, normalized size = 0.88 \begin {gather*} -\frac {12575075040 \, x^{6} + 57635760600 \, x^{5} + 113990726520 \, x^{4} + 127327486275 \, x^{3} + 83293304778 \, x^{2} + 29451465714 \, x + 4309941128}{1334139660 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} + \frac {21296}{5764801} \, \log \left (3 \, x + 2\right ) - \frac {21296}{5764801} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^3/(1-2*x)/(2+3*x)^8,x, algorithm="maxima")

[Out]

-1/1334139660*(12575075040*x^6 + 57635760600*x^5 + 113990726520*x^4 + 127327486275*x^3 + 83293304778*x^2 + 294

51465714*x + 4309941128)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)

+ 21296/5764801*log(3*x + 2) - 21296/5764801*log(2*x - 1)

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mupad [B] time = 0.04, size = 76, normalized size = 0.78 \begin {gather*} \frac {42592\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{5764801}-\frac {\frac {10648\,x^6}{2470629}+\frac {146410\,x^5}{7411887}+\frac {2606098\,x^4}{66706983}+\frac {34932095\,x^3}{800483796}+\frac {1542468607\,x^2}{54032656230}+\frac {4908577619\,x}{486293906070}+\frac {1077485282}{729440859105}}{x^7+\frac {14\,x^6}{3}+\frac {28\,x^5}{3}+\frac {280\,x^4}{27}+\frac {560\,x^3}{81}+\frac {224\,x^2}{81}+\frac {448\,x}{729}+\frac {128}{2187}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)^3/((2*x - 1)*(3*x + 2)^8),x)

[Out]

(42592*atanh((12*x)/7 + 1/7))/5764801 - ((4908577619*x)/486293906070 + (1542468607*x^2)/54032656230 + (3493209

5*x^3)/800483796 + (2606098*x^4)/66706983 + (146410*x^5)/7411887 + (10648*x^6)/2470629 + 1077485282/7294408591

05)/((448*x)/729 + (224*x^2)/81 + (560*x^3)/81 + (280*x^4)/27 + (28*x^5)/3 + (14*x^6)/3 + x^7 + 128/2187)

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sympy [A] time = 0.24, size = 85, normalized size = 0.87 \begin {gather*} - \frac {12575075040 x^{6} + 57635760600 x^{5} + 113990726520 x^{4} + 127327486275 x^{3} + 83293304778 x^{2} + 29451465714 x + 4309941128}{2917763436420 x^{7} + 13616229369960 x^{6} + 27232458739920 x^{5} + 30258287488800 x^{4} + 20172191659200 x^{3} + 8068876663680 x^{2} + 1793083703040 x + 170769876480} - \frac {21296 \log {\left (x - \frac {1}{2} \right )}}{5764801} + \frac {21296 \log {\left (x + \frac {2}{3} \right )}}{5764801} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)/(2+3*x)**8,x)

[Out]

-(12575075040*x**6 + 57635760600*x**5 + 113990726520*x**4 + 127327486275*x**3 + 83293304778*x**2 + 29451465714

*x + 4309941128)/(2917763436420*x**7 + 13616229369960*x**6 + 27232458739920*x**5 + 30258287488800*x**4 + 20172

191659200*x**3 + 8068876663680*x**2 + 1793083703040*x + 170769876480) - 21296*log(x - 1/2)/5764801 + 21296*log

(x + 2/3)/5764801

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