3.1389 \(\int \frac {(1-2 x)^3 (2+3 x)^5}{3+5 x} \, dx\)

Optimal. Leaf size=65 \[ -\frac {243 x^8}{5}-\frac {11988 x^7}{175}+\frac {4419 x^6}{125}+\frac {243333 x^5}{3125}-\frac {73749 x^4}{12500}-\frac {1703753 x^3}{46875}-\frac {138741 x^2}{156250}+\frac {4166223 x}{390625}+\frac {1331 \log (5 x+3)}{1953125} \]

[Out]

4166223/390625*x-138741/156250*x^2-1703753/46875*x^3-73749/12500*x^4+243333/3125*x^5+4419/125*x^6-11988/175*x^
7-243/5*x^8+1331/1953125*ln(3+5*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} \[ -\frac {243 x^8}{5}-\frac {11988 x^7}{175}+\frac {4419 x^6}{125}+\frac {243333 x^5}{3125}-\frac {73749 x^4}{12500}-\frac {1703753 x^3}{46875}-\frac {138741 x^2}{156250}+\frac {4166223 x}{390625}+\frac {1331 \log (5 x+3)}{1953125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4166223*x)/390625 - (138741*x^2)/156250 - (1703753*x^3)/46875 - (73749*x^4)/12500 + (243333*x^5)/3125 + (4419
*x^6)/125 - (11988*x^7)/175 - (243*x^8)/5 + (1331*Log[3 + 5*x])/1953125

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 {(1-2 x)^3 (2+3 x)^5}{3+5 x} \, dx &=\int \left (\frac {4166223}{390625}-\frac {138741 x}{78125}-\frac {1703753 x^2}{15625}-\frac {73749 x^3}{3125}+\frac {243333 x^4}{625}+\frac {26514 x^5}{125}-\frac {11988 x^6}{25}-\frac {1944 x^7}{5}+\frac {1331}{390625 (3+5 x)}\right ) \, dx\\ &=\frac {4166223 x}{390625}-\frac {138741 x^2}{156250}-\frac {1703753 x^3}{46875}-\frac {73749 x^4}{12500}+\frac {243333 x^5}{3125}+\frac {4419 x^6}{125}-\frac {11988 x^7}{175}-\frac {243 x^8}{5}+\frac {1331 \log (3+5 x)}{1953125}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 52, normalized size = 0.80 \[ \frac {-39867187500 x^8-56193750000 x^7+28999687500 x^6+63874912500 x^5-4839778125 x^4-29815677500 x^3-728390250 x^2+8749068300 x+559020 \log (5 x+3)[phone]}{820312500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2409164451 + 8749068300*x - 728390250*x^2 - 29815677500*x^3 - 4839778125*x^4 + 63874912500*x^5 + 28999687500*
x^6 - 56193750000*x^7 - 39867187500*x^8 + 559020*Log[3 + 5*x])/820312500

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fricas [A]  time = 0.56, size = 47, normalized size = 0.72 \[ -\frac {243}{5} \, x^{8} - \frac {11988}{175} \, x^{7} + \frac {4419}{125} \, x^{6} + \frac {243333}{3125} \, x^{5} - \frac {73749}{12500} \, x^{4} - \frac {1703753}{46875} \, x^{3} - \frac {138741}{156250} \, x^{2} + \frac {4166223}{390625} \, x + \frac {1331}{1953125} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/5*x^8 - 11988/175*x^7 + 4419/125*x^6 + 243333/3125*x^5 - 73749/12500*x^4 - 1703753/46875*x^3 - 138741/156
250*x^2 + 4166223/390625*x + 1331/1953125*log(5*x + 3)

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giac [A]  time = 0.98, size = 48, normalized size = 0.74 \[ -\frac {243}{5} \, x^{8} - \frac {11988}{175} \, x^{7} + \frac {4419}{125} \, x^{6} + \frac {243333}{3125} \, x^{5} - \frac {73749}{12500} \, x^{4} - \frac {1703753}{46875} \, x^{3} - \frac {138741}{156250} \, x^{2} + \frac {4166223}{390625} \, x + \frac {1331}{1953125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/5*x^8 - 11988/175*x^7 + 4419/125*x^6 + 243333/3125*x^5 - 73749/12500*x^4 - 1703753/46875*x^3 - 138741/156
250*x^2 + 4166223/390625*x + 1331/1953125*log(abs(5*x + 3))

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maple [A]  time = 0.00, size = 48, normalized size = 0.74 \[ -\frac {243 x^{8}}{5}-\frac {11988 x^{7}}{175}+\frac {4419 x^{6}}{125}+\frac {243333 x^{5}}{3125}-\frac {73749 x^{4}}{12500}-\frac {1703753 x^{3}}{46875}-\frac {138741 x^{2}}{156250}+\frac {4166223 x}{390625}+\frac {1331 \ln \left (5 x +3\right )}{1953125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4166223/390625*x-138741/156250*x^2-1703753/46875*x^3-73749/12500*x^4+243333/3125*x^5+4419/125*x^6-11988/175*x^
7-243/5*x^8+1331/1953125*ln(5*x+3)

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maxima [A]  time = 0.43, size = 47, normalized size = 0.72 \[ -\frac {243}{5} \, x^{8} - \frac {11988}{175} \, x^{7} + \frac {4419}{125} \, x^{6} + \frac {243333}{3125} \, x^{5} - \frac {73749}{12500} \, x^{4} - \frac {1703753}{46875} \, x^{3} - \frac {138741}{156250} \, x^{2} + \frac {4166223}{390625} \, x + \frac {1331}{1953125} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/5*x^8 - 11988/175*x^7 + 4419/125*x^6 + 243333/3125*x^5 - 73749/12500*x^4 - 1703753/46875*x^3 - 138741/156
250*x^2 + 4166223/390625*x + 1331/1953125*log(5*x + 3)

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mupad [B]  time = 0.04, size = 45, normalized size = 0.69 \[ \frac {4166223\,x}{390625}+\frac {1331\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {138741\,x^2}{156250}-\frac {1703753\,x^3}{46875}-\frac {73749\,x^4}{12500}+\frac {243333\,x^5}{3125}+\frac {4419\,x^6}{125}-\frac {11988\,x^7}{175}-\frac {243\,x^8}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4166223*x)/390625 + (1331*log(x + 3/5))/1953125 - (138741*x^2)/156250 - (1703753*x^3)/46875 - (73749*x^4)/125
00 + (243333*x^5)/3125 + (4419*x^6)/125 - (11988*x^7)/175 - (243*x^8)/5

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sympy [A]  time = 0.11, size = 61, normalized size = 0.94 \[ - \frac {243 x^{8}}{5} - \frac {11988 x^{7}}{175} + \frac {4419 x^{6}}{125} + \frac {243333 x^{5}}{3125} - \frac {73749 x^{4}}{12500} - \frac {1703753 x^{3}}{46875} - \frac {138741 x^{2}}{156250} + \frac {4166223 x}{390625} + \frac {1331 \log {\left (5 x + 3 \right )}}{1953125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**8/5 - 11988*x**7/175 + 4419*x**6/125 + 243333*x**5/3125 - 73749*x**4/12500 - 1703753*x**3/46875 - 1387
41*x**2/156250 + 4166223*x/390625 + 1331*log(5*x + 3)/1953125

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