∫ (2+3x)6 ________________________

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

Optimal. Leaf size=62 \[ \frac {243 x^3}{100}+\frac {13851 x^2}{1000}+\frac {473607 x}{10000}+\frac {117649}{3872 (1-2 x)}-\frac {1}{378125 (5 x+3)}+\frac {67228 \log (1-2 x)}{1331}+\frac {202 \log (5 x+3)}{4159375} \]

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Rubi [A] time = 0.03, antiderivative size = 62, 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 {243 x^3}{100}+\frac {13851 x^2}{1000}+\frac {473607 x}{10000}+\frac {117649}{3872 (1-2 x)}-\frac {1}{378125 (5 x+3)}+\frac {67228 \log (1-2 x)}{1331}+\frac {202 \log (5 x+3)}{4159375} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

117649/(3872*(1 - 2*x)) + (473607*x)/10000 + (13851*x^2)/1000 + (243*x^3)/100 - 1/(378125*(3 + 5*x)) + (67228*

Log[1 - 2*x])/1331 + (202*Log[3 + 5*x])/4159375

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 {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac {473607}{10000}+\frac {13851 x}{500}+\frac {729 x^2}{100}+\frac {117649}{1936 (-1+2 x)^2}+\frac {134456}{1331 (-1+2 x)}+\frac {1}{75625 (3+5 x)^2}+\frac {202}{831875 (3+5 x)}\right ) \, dx\\ &=\frac {117649}{3872 (1-2 x)}+\frac {473607 x}{10000}+\frac {13851 x^2}{1000}+\frac {243 x^3}{100}-\frac {1}{378125 (3+5 x)}+\frac {67228 \log (1-2 x)}{1331}+\frac {202 \log (3+5 x)}{4159375}\\ \end {align*}

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Mathematica [A] time = 0.04, size = 65, normalized size = 1.05 \begin {gather*} \frac {-\frac {11 (1838265689 x+1102959343)}{10 x^2+x-3}+11979000 (3 x+2)^3+132966900 (3 x+2)^2+1425620790 (3 x+2)+6722800000 \log (3-6 x)+6464 \log (-3 (5 x+3))}{133100000} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(1425620790*(2 + 3*x) + 132966900*(2 + 3*x)^2 + 11979000*(2 + 3*x)^3 - (11*(1102959343 + 1838265689*x))/(-3 +

x + 10*x^2) + 6722800000*Log[3 - 6*x] + 6464*Log[-3*(3 + 5*x)])/133100000

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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(2+3 x)^6}{(1-2 x)^2 (3+5 x)^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.54, size = 69, normalized size = 1.11 \begin {gather*} \frac {3234330000 \, x^{5} + 18759114000 \, x^{4} + 63910360800 \, x^{3} + 773004870 \, x^{2} + 6464 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) + 6722800000 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 39132050089 \, x - 12132552773}{133100000 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/133100000*(3234330000*x^5 + 18759114000*x^4 + 63910360800*x^3 + 773004870*x^2 + 6464*(10*x^2 + x - 3)*log(5*

x + 3) + 6722800000*(10*x^2 + x - 3)*log(2*x - 1) - 39132050089*x - 12132552773)/(10*x^2 + x - 3)

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giac [A] time = 1.17, size = 94, normalized size = 1.52 \begin {gather*} -\frac {{\left (5 \, x + 3\right )}^{3} {\left (\frac {4528062}{5 \, x + 3} + \frac {76330188}{{\left (5 \, x + 3\right )}^{2}} - \frac {840384278}{{\left (5 \, x + 3\right )}^{3}} + 323433\right )}}{8318750 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}} - \frac {1}{378125 \, {\left (5 \, x + 3\right )}} - \frac {157842}{3125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) + \frac {67228}{1331} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^2,x, algorithm="giac")

[Out]

-1/8318750*(5*x + 3)^3*(4528062/(5*x + 3) + 76330188/(5*x + 3)^2 - 840384278/(5*x + 3)^3 + 323433)/(11/(5*x +

3) - 2) - 1/378125/(5*x + 3) - 157842/3125*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 67228/1331*log(abs(-11/(5*x + 3

) + 2))

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maple [A] time = 0.01, size = 49, normalized size = 0.79 \begin {gather*} \frac {243 x^{3}}{100}+\frac {13851 x^{2}}{1000}+\frac {473607 x}{10000}+\frac {67228 \ln \left (2 x -1\right )}{1331}+\frac {202 \ln \left (5 x +3\right )}{4159375}-\frac {1}{378125 \left (5 x +3\right )}-\frac {117649}{3872 \left (2 x -1\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243/100*x^3+13851/1000*x^2+473607/10000*x-1/378125/(5*x+3)+202/4159375*ln(5*x+3)-117649/3872/(2*x-1)+67228/133

1*ln(2*x-1)

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maxima [A] time = 0.58, size = 47, normalized size = 0.76 \begin {gather*} \frac {243}{100} \, x^{3} + \frac {13851}{1000} \, x^{2} + \frac {473607}{10000} \, x - \frac {1838265689 \, x + 1102959343}{12100000 \, {\left (10 \, x^{2} + x - 3\right )}} + \frac {202}{4159375} \, \log \left (5 \, x + 3\right ) + \frac {67228}{1331} \, \log \left (2 \, x - 1\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6/(1-2*x)^2/(3+5*x)^2,x, algorithm="maxima")

[Out]

243/100*x^3 + 13851/1000*x^2 + 473607/10000*x - 1/12100000*(1838265689*x + 1102959343)/(10*x^2 + x - 3) + 202/

4159375*log(5*x + 3) + 67228/1331*log(2*x - 1)

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mupad [B] time = 0.04, size = 43, normalized size = 0.69 \begin {gather*} \frac {473607\,x}{10000}+\frac {67228\,\ln \left (x-\frac {1}{2}\right )}{1331}+\frac {202\,\ln \left (x+\frac {3}{5}\right )}{4159375}-\frac {\frac {1838265689\,x}{121000000}+\frac {1102959343}{121000000}}{x^2+\frac {x}{10}-\frac {3}{10}}+\frac {13851\,x^2}{1000}+\frac {243\,x^3}{100} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(473607*x)/10000 + (67228*log(x - 1/2))/1331 + (202*log(x + 3/5))/4159375 - ((1838265689*x)/121000000 + 110295

9343/121000000)/(x/10 + x^2 - 3/10) + (13851*x^2)/1000 + (243*x^3)/100

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sympy [A] time = 0.17, size = 54, normalized size = 0.87 \begin {gather*} \frac {243 x^{3}}{100} + \frac {13851 x^{2}}{1000} + \frac {473607 x}{10000} + \frac {- 1838265689 x - 1102959343}{121000000 x^{2} + 12100000 x - 36300000} + \frac {67228 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {202 \log {\left (x + \frac {3}{5} \right )}}{4159375} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

243*x**3/100 + 13851*x**2/1000 + 473607*x/10000 + (-1838265689*x - 1102959343)/(121000000*x**2 + 12100000*x -

36300000) + 67228*log(x - 1/2)/1331 + 202*log(x + 3/5)/4159375

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