∫ (1−2x)3(2+3x)5 ________________________

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

Optimal. Leaf size=69 \[ -\frac {1944 x^7}{175}-\frac {1026 x^6}{125}+\frac {44982 x^5}{3125}+\frac {108387 x^4}{12500}-\frac {26594 x^3}{3125}-\frac {507023 x^2}{156250}+\frac {1382328 x}{390625}-\frac {1331}{1953125 (5 x+3)}+\frac {19239 \log (5 x+3)}{1953125} \]

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Rubi [A] time = 0.03, antiderivative size = 69, 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 {1944 x^7}{175}-\frac {1026 x^6}{125}+\frac {44982 x^5}{3125}+\frac {108387 x^4}{12500}-\frac {26594 x^3}{3125}-\frac {507023 x^2}{156250}+\frac {1382328 x}{390625}-\frac {1331}{1953125 (5 x+3)}+\frac {19239 \log (5 x+3)}{1953125} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1382328*x)/390625 - (507023*x^2)/156250 - (26594*x^3)/3125 + (108387*x^4)/12500 + (44982*x^5)/3125 - (1026*x^

6)/125 - (1944*x^7)/175 - 1331/(1953125*(3 + 5*x)) + (19239*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)^2} \, dx &=\int \left (\frac {1382328}{390625}-\frac {507023 x}{78125}-\frac {79782 x^2}{3125}+\frac {108387 x^3}{3125}+\frac {44982 x^4}{625}-\frac {6156 x^5}{125}-\frac {1944 x^6}{25}+\frac {1331}{390625 (3+5 x)^2}+\frac {19239}{390625 (3+5 x)}\right ) \, dx\\ &=\frac {1382328 x}{390625}-\frac {507023 x^2}{156250}-\frac {26594 x^3}{3125}+\frac {108387 x^4}{12500}+\frac {44982 x^5}{3125}-\frac {1026 x^6}{125}-\frac {1944 x^7}{175}-\frac {1331}{1953125 (3+5 x)}+\frac {19239 \log (3+5 x)}{1953125}\\ \end {align*}

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Mathematica [A] time = 0.04, size = 66, normalized size = 0.96 \begin {gather*} \frac {-15187500000 x^8-20334375000 x^7+12946500000 x^6+23662603125 x^5-4521978125 x^4-11417376250 x^3+2176277250 x^2+4982083965 x+2693460 (5 x+3) \log (6 (5 x+3))+1247330759}{273437500 (5 x+3)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1247330759 + 4982083965*x + 2176277250*x^2 - 11417376250*x^3 - 4521978125*x^4 + 23662603125*x^5 + 12946500000

*x^6 - 20334375000*x^7 - 15187500000*x^8 + 2693460*(3 + 5*x)*Log[6*(3 + 5*x)])/(273437500*(3 + 5*x))

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

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.25, size = 62, normalized size = 0.90 \begin {gather*} -\frac {3037500000 \, x^{8} + 4066875000 \, x^{7} - 2589300000 \, x^{6} - 4732520625 \, x^{5} + 904395625 \, x^{4} + 2283475250 \, x^{3} - 435255450 \, x^{2} - 538692 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 580577760 \, x + 37268}{54687500 \, {\left (5 \, x + 3\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/54687500*(3037500000*x^8 + 4066875000*x^7 - 2589300000*x^6 - 4732520625*x^5 + 904395625*x^4 + 2283475250*x^

3 - 435255450*x^2 - 538692*(5*x + 3)*log(5*x + 3) - 580577760*x + 37268)/(5*x + 3)

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giac [A] time = 0.89, size = 93, normalized size = 1.35 \begin {gather*} \frac {1}{273437500} \, {\left (5 \, x + 3\right )}^{7} {\left (\frac {672840}{5 \, x + 3} - \frac {3503304}{{\left (5 \, x + 3\right )}^{2}} + \frac {2251305}{{\left (5 \, x + 3\right )}^{3}} + \frac {16557100}{{\left (5 \, x + 3\right )}^{4}} + \frac {20720140}{{\left (5 \, x + 3\right )}^{5}} + \frac {15264480}{{\left (5 \, x + 3\right )}^{6}} - 38880\right )} - \frac {1331}{1953125 \, {\left (5 \, x + 3\right )}} - \frac {19239}{1953125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/273437500*(5*x + 3)^7*(672840/(5*x + 3) - 3503304/(5*x + 3)^2 + 2251305/(5*x + 3)^3 + 16557100/(5*x + 3)^4 +

20720140/(5*x + 3)^5 + 15264480/(5*x + 3)^6 - 38880) - 1331/1953125/(5*x + 3) - 19239/1953125*log(1/5*abs(5*x

+ 3)/(5*x + 3)^2)

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maple [A] time = 0.01, size = 52, normalized size = 0.75 \begin {gather*} -\frac {1944 x^{7}}{175}-\frac {1026 x^{6}}{125}+\frac {44982 x^{5}}{3125}+\frac {108387 x^{4}}{12500}-\frac {26594 x^{3}}{3125}-\frac {507023 x^{2}}{156250}+\frac {1382328 x}{390625}+\frac {19239 \ln \left (5 x +3\right )}{1953125}-\frac {1331}{1953125 \left (5 x +3\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1382328/390625*x-507023/156250*x^2-26594/3125*x^3+108387/12500*x^4+44982/3125*x^5-1026/125*x^6-1944/175*x^7-13

31/1953125/(5*x+3)+19239/1953125*ln(5*x+3)

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maxima [A] time = 0.52, size = 51, normalized size = 0.74 \begin {gather*} -\frac {1944}{175} \, x^{7} - \frac {1026}{125} \, x^{6} + \frac {44982}{3125} \, x^{5} + \frac {108387}{12500} \, x^{4} - \frac {26594}{3125} \, x^{3} - \frac {507023}{156250} \, x^{2} + \frac {1382328}{390625} \, x - \frac {1331}{1953125 \, {\left (5 \, x + 3\right )}} + \frac {19239}{1953125} \, \log \left (5 \, x + 3\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1944/175*x^7 - 1026/125*x^6 + 44982/3125*x^5 + 108387/12500*x^4 - 26594/3125*x^3 - 507023/156250*x^2 + 138232

8/390625*x - 1331/1953125/(5*x + 3) + 19239/1953125*log(5*x + 3)

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mupad [B] time = 0.04, size = 49, normalized size = 0.71 \begin {gather*} \frac {1382328\,x}{390625}+\frac {19239\,\ln \left (x+\frac {3}{5}\right )}{1953125}-\frac {1331}{9765625\,\left (x+\frac {3}{5}\right )}-\frac {507023\,x^2}{156250}-\frac {26594\,x^3}{3125}+\frac {108387\,x^4}{12500}+\frac {44982\,x^5}{3125}-\frac {1026\,x^6}{125}-\frac {1944\,x^7}{175} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1382328*x)/390625 + (19239*log(x + 3/5))/1953125 - 1331/(9765625*(x + 3/5)) - (507023*x^2)/156250 - (26594*x^

3)/3125 + (108387*x^4)/12500 + (44982*x^5)/3125 - (1026*x^6)/125 - (1944*x^7)/175

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sympy [A] time = 0.12, size = 61, normalized size = 0.88 \begin {gather*} - \frac {1944 x^{7}}{175} - \frac {1026 x^{6}}{125} + \frac {44982 x^{5}}{3125} + \frac {108387 x^{4}}{12500} - \frac {26594 x^{3}}{3125} - \frac {507023 x^{2}}{156250} + \frac {1382328 x}{390625} + \frac {19239 \log {\left (5 x + 3 \right )}}{1953125} - \frac {1331}{9765625 x + 5859375} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1944*x**7/175 - 1026*x**6/125 + 44982*x**5/3125 + 108387*x**4/12500 - 26594*x**3/3125 - 507023*x**2/156250 +

1382328*x/390625 + 19239*log(5*x + 3)/1953125 - 1331/(9765625*x + 5859375)

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