3.14.42 \(\int \frac {(3+5 x)^3}{(1-2 x) (2+3 x)^6} \, dx\)

Optimal. Leaf size=76 \[ -\frac {2662}{16807 (3 x+2)}-\frac {1331}{4802 (3 x+2)^2}+\frac {3469}{27783 (3 x+2)^3}-\frac {103}{5292 (3 x+2)^4}+\frac {1}{945 (3 x+2)^5}-\frac {5324 \log (1-2 x)}{117649}+\frac {5324 \log (3 x+2)}{117649} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 76, 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 {2662}{16807 (3 x+2)}-\frac {1331}{4802 (3 x+2)^2}+\frac {3469}{27783 (3 x+2)^3}-\frac {103}{5292 (3 x+2)^4}+\frac {1}{945 (3 x+2)^5}-\frac {5324 \log (1-2 x)}{117649}+\frac {5324 \log (3 x+2)}{117649} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

1/(945*(2 + 3*x)^5) - 103/(5292*(2 + 3*x)^4) + 3469/(27783*(2 + 3*x)^3) - 1331/(4802*(2 + 3*x)^2) - 2662/(1680
7*(2 + 3*x)) - (5324*Log[1 - 2*x])/117649 + (5324*Log[2 + 3*x])/117649

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)^6} \, dx &=\int \left (-\frac {10648}{117649 (-1+2 x)}-\frac {1}{63 (2+3 x)^6}+\frac {103}{441 (2+3 x)^5}-\frac {3469}{3087 (2+3 x)^4}+\frac {3993}{2401 (2+3 x)^3}+\frac {7986}{16807 (2+3 x)^2}+\frac {15972}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {1}{945 (2+3 x)^5}-\frac {103}{5292 (2+3 x)^4}+\frac {3469}{27783 (2+3 x)^3}-\frac {1331}{4802 (2+3 x)^2}-\frac {2662}{16807 (2+3 x)}-\frac {5324 \log (1-2 x)}{117649}+\frac {5324 \log (2+3 x)}{117649}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.08, size = 52, normalized size = 0.68 \begin {gather*} \frac {2 \left (-\frac {7 \left (349307640 x^4+1135249830 x^3+1308416040 x^2+646472325 x+116805778\right )}{8 (3 x+2)^5}-1078110 \log (1-2 x)+1078110 \log (6 x+4)\right )}{47647845} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

(2*((-7*(116805778 + 646472325*x + 1308416040*x^2 + 1135249830*x^3 + 349307640*x^4))/(8*(2 + 3*x)^5) - 1078110
*Log[1 - 2*x] + 1078110*Log[4 + 6*x]))/47647845

________________________________________________________________________________________

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)^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^6),x]

[Out]

IntegrateAlgebraic[(3 + 5*x)^3/((1 - 2*x)*(2 + 3*x)^6), x]

________________________________________________________________________________________

fricas [A]  time = 1.26, size = 115, normalized size = 1.51 \begin {gather*} -\frac {2445153480 \, x^{4} + 7946748810 \, x^{3} + 9158912280 \, x^{2} - 8624880 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (3 \, x + 2\right ) + 8624880 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (2 \, x - 1\right ) + 4525306275 \, x + 817640446}{190591380 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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)^6,x, algorithm="fricas")

[Out]

-1/190591380*(2445153480*x^4 + 7946748810*x^3 + 9158912280*x^2 - 8624880*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x
^2 + 240*x + 32)*log(3*x + 2) + 8624880*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log(2*x - 1) + 4
525306275*x + 817640446)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

________________________________________________________________________________________

giac [A]  time = 1.09, size = 48, normalized size = 0.63 \begin {gather*} -\frac {349307640 \, x^{4} + 1135249830 \, x^{3} + 1308416040 \, x^{2} + 646472325 \, x + 116805778}{27227340 \, {\left (3 \, x + 2\right )}^{5}} + \frac {5324}{117649} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {5324}{117649} \, \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)^6,x, algorithm="giac")

[Out]

-1/27227340*(349307640*x^4 + 1135249830*x^3 + 1308416040*x^2 + 646472325*x + 116805778)/(3*x + 2)^5 + 5324/117
649*log(abs(3*x + 2)) - 5324/117649*log(abs(2*x - 1))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 63, normalized size = 0.83 \begin {gather*} -\frac {5324 \ln \left (2 x -1\right )}{117649}+\frac {5324 \ln \left (3 x +2\right )}{117649}+\frac {1}{945 \left (3 x +2\right )^{5}}-\frac {103}{5292 \left (3 x +2\right )^{4}}+\frac {3469}{27783 \left (3 x +2\right )^{3}}-\frac {1331}{4802 \left (3 x +2\right )^{2}}-\frac {2662}{16807 \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)^6,x)

[Out]

1/945/(3*x+2)^5-103/5292/(3*x+2)^4+3469/27783/(3*x+2)^3-1331/4802/(3*x+2)^2-2662/16807/(3*x+2)+5324/117649*ln(
3*x+2)-5324/117649*ln(2*x-1)

________________________________________________________________________________________

maxima [A]  time = 0.53, size = 66, normalized size = 0.87 \begin {gather*} -\frac {349307640 \, x^{4} + 1135249830 \, x^{3} + 1308416040 \, x^{2} + 646472325 \, x + 116805778}{27227340 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {5324}{117649} \, \log \left (3 \, x + 2\right ) - \frac {5324}{117649} \, \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)^6,x, algorithm="maxima")

[Out]

-1/27227340*(349307640*x^4 + 1135249830*x^3 + 1308416040*x^2 + 646472325*x + 116805778)/(243*x^5 + 810*x^4 + 1
080*x^3 + 720*x^2 + 240*x + 32) + 5324/117649*log(3*x + 2) - 5324/117649*log(2*x - 1)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 56, normalized size = 0.74 \begin {gather*} \frac {10648\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}-\frac {\frac {2662\,x^4}{50421}+\frac {17303\,x^3}{100842}+\frac {7268978\,x^2}{36756909}+\frac {43098155\,x}{441082908}+\frac {58402889}{3308121810}}{x^5+\frac {10\,x^4}{3}+\frac {40\,x^3}{9}+\frac {80\,x^2}{27}+\frac {80\,x}{81}+\frac {32}{243}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(5*x + 3)^3/((2*x - 1)*(3*x + 2)^6),x)

[Out]

(10648*atanh((12*x)/7 + 1/7))/117649 - ((43098155*x)/441082908 + (7268978*x^2)/36756909 + (17303*x^3)/100842 +
(2662*x^4)/50421 + 58402889/3308121810)/((80*x)/81 + (80*x^2)/27 + (40*x^3)/9 + (10*x^4)/3 + x^5 + 32/243)

________________________________________________________________________________________

sympy [A]  time = 0.20, size = 65, normalized size = 0.86 \begin {gather*} - \frac {349307640 x^{4} + 1135249830 x^{3} + 1308416040 x^{2} + 646472325 x + 116805778}{6616243620 x^{5} + 22054145400 x^{4} + 29405527200 x^{3} + 19603684800 x^{2} + 6534561600 x + 871274880} - \frac {5324 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {5324 \log {\left (x + \frac {2}{3} \right )}}{117649} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**3/(1-2*x)/(2+3*x)**6,x)

[Out]

-(349307640*x**4 + 1135249830*x**3 + 1308416040*x**2 + 646472325*x + 116805778)/(6616243620*x**5 + 22054145400
*x**4 + 29405527200*x**3 + 19603684800*x**2 + 6534561600*x + 871274880) - 5324*log(x - 1/2)/117649 + 5324*log(
x + 2/3)/117649

________________________________________________________________________________________