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

Optimal. Leaf size=76 \[ \frac {2662}{16807 (1-2 x)}-\frac {3267}{16807 (3 x+2)}+\frac {363}{4802 (3 x+2)^2}-\frac {101}{9261 (3 x+2)^3}+\frac {1}{1764 (3 x+2)^4}-\frac {14520 \log (1-2 x)}{117649}+\frac {14520 \log (3 x+2)}{117649} \]

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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 (1-2 x)}-\frac {3267}{16807 (3 x+2)}+\frac {363}{4802 (3 x+2)^2}-\frac {101}{9261 (3 x+2)^3}+\frac {1}{1764 (3 x+2)^4}-\frac {14520 \log (1-2 x)}{117649}+\frac {14520 \log (3 x+2)}{117649} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2662/(16807*(1 - 2*x)) + 1/(1764*(2 + 3*x)^4) - 101/(9261*(2 + 3*x)^3) + 363/(4802*(2 + 3*x)^2) - 3267/(16807*
(2 + 3*x)) - (14520*Log[1 - 2*x])/117649 + (14520*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 (2+3 x)^5} \, dx &=\int \left (\frac {5324}{16807 (-1+2 x)^2}-\frac {29040}{117649 (-1+2 x)}-\frac {1}{147 (2+3 x)^5}+\frac {101}{1029 (2+3 x)^4}-\frac {1089}{2401 (2+3 x)^3}+\frac {9801}{16807 (2+3 x)^2}+\frac {43560}{117649 (2+3 x)}\right ) \, dx\\ &=\frac {2662}{16807 (1-2 x)}+\frac {1}{1764 (2+3 x)^4}-\frac {101}{9261 (2+3 x)^3}+\frac {363}{4802 (2+3 x)^2}-\frac {3267}{16807 (2+3 x)}-\frac {14520 \log (1-2 x)}{117649}+\frac {14520 \log (2+3 x)}{117649}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 59, normalized size = 0.78 \begin {gather*} \frac {2 \left (-\frac {7 \left (42340320 x^4+88209000 x^3+66510750 x^2+21109490 x+2287541\right )}{8 (2 x-1) (3 x+2)^4}-196020 \log (1-2 x)+196020 \log (6 x+4)\right )}{3176523} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-7*(2287541 + 21109490*x + 66510750*x^2 + 88209000*x^3 + 42340320*x^4))/(8*(-1 + 2*x)*(2 + 3*x)^4) - 1960
20*Log[1 - 2*x] + 196020*Log[4 + 6*x]))/3176523

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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 (2+3 x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.70, size = 115, normalized size = 1.51 \begin {gather*} -\frac {296382240 \, x^{4} + 617463000 \, x^{3} + 465575250 \, x^{2} - 1568160 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (3 \, x + 2\right ) + 1568160 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \log \left (2 \, x - 1\right ) + 147766430 \, x + 16012787}{12706092 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12706092*(296382240*x^4 + 617463000*x^3 + 465575250*x^2 - 1568160*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 6
4*x - 16)*log(3*x + 2) + 1568160*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*log(2*x - 1) + 147766430*x
+ 16012787)/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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giac [A]  time = 0.90, size = 67, normalized size = 0.88 \begin {gather*} -\frac {3267}{16807 \, {\left (3 \, x + 2\right )}} + \frac {15972}{117649 \, {\left (\frac {7}{3 \, x + 2} - 2\right )}} + \frac {363}{4802 \, {\left (3 \, x + 2\right )}^{2}} - \frac {101}{9261 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1}{1764 \, {\left (3 \, x + 2\right )}^{4}} - \frac {14520}{117649} \, \log \left ({\left | -\frac {7}{3 \, x + 2} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3267/16807/(3*x + 2) + 15972/117649/(7/(3*x + 2) - 2) + 363/4802/(3*x + 2)^2 - 101/9261/(3*x + 2)^3 + 1/1764/
(3*x + 2)^4 - 14520/117649*log(abs(-7/(3*x + 2) + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.83 \begin {gather*} -\frac {14520 \ln \left (2 x -1\right )}{117649}+\frac {14520 \ln \left (3 x +2\right )}{117649}+\frac {1}{1764 \left (3 x +2\right )^{4}}-\frac {101}{9261 \left (3 x +2\right )^{3}}+\frac {363}{4802 \left (3 x +2\right )^{2}}-\frac {3267}{16807 \left (3 x +2\right )}-\frac {2662}{16807 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1764/(3*x+2)^4-101/9261/(3*x+2)^3+363/4802/(3*x+2)^2-3267/16807/(3*x+2)+14520/117649*ln(3*x+2)-2662/16807/(2
*x-1)-14520/117649*ln(2*x-1)

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maxima [A]  time = 0.57, size = 66, normalized size = 0.87 \begin {gather*} -\frac {42340320 \, x^{4} + 88209000 \, x^{3} + 66510750 \, x^{2} + 21109490 \, x + 2287541}{1815156 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} + \frac {14520}{117649} \, \log \left (3 \, x + 2\right ) - \frac {14520}{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/(2+3*x)^5,x, algorithm="maxima")

[Out]

-1/1815156*(42340320*x^4 + 88209000*x^3 + 66510750*x^2 + 21109490*x + 2287541)/(162*x^5 + 351*x^4 + 216*x^3 -
24*x^2 - 64*x - 16) + 14520/117649*log(3*x + 2) - 14520/117649*log(2*x - 1)

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mupad [B]  time = 1.10, size = 57, normalized size = 0.75 \begin {gather*} \frac {29040\,\mathrm {atanh}\left (\frac {12\,x}{7}+\frac {1}{7}\right )}{117649}+\frac {\frac {2420\,x^4}{16807}+\frac {15125\,x^3}{50421}+\frac {11085125\,x^2}{49009212}+\frac {10554745\,x}{147027636}+\frac {2287541}{294055272}}{-x^5-\frac {13\,x^4}{6}-\frac {4\,x^3}{3}+\frac {4\,x^2}{27}+\frac {32\,x}{81}+\frac {8}{81}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(29040*atanh((12*x)/7 + 1/7))/117649 + ((10554745*x)/147027636 + (11085125*x^2)/49009212 + (15125*x^3)/50421 +
(2420*x^4)/16807 + 2287541/294055272)/((32*x)/81 + (4*x^2)/27 - (4*x^3)/3 - (13*x^4)/6 - x^5 + 8/81)

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sympy [A]  time = 0.19, size = 66, normalized size = 0.87 \begin {gather*} \frac {- 42340320 x^{4} - 88209000 x^{3} - 66510750 x^{2} - 21109490 x - 2287541}{294055272 x^{5} + 637119756 x^{4} + 392073696 x^{3} - 43563744 x^{2} - 116169984 x - 29042496} - \frac {14520 \log {\left (x - \frac {1}{2} \right )}}{117649} + \frac {14520 \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/(2+3*x)**5,x)

[Out]

(-42340320*x**4 - 88209000*x**3 - 66510750*x**2 - 21109490*x - 2287541)/(294055272*x**5 + 637119756*x**4 + 392
073696*x**3 - 43563744*x**2 - 116169984*x - 29042496) - 14520*log(x - 1/2)/117649 + 14520*log(x + 2/3)/117649

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