∫ (2+3x)6 ________________________

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

Optimal. Leaf size=66 \[ \frac {729 x^2}{1000}+\frac {2916 x}{625}+\frac {117649}{21296 (1-2 x)}-\frac {202}{4159375 (5 x+3)}-\frac {1}{756250 (5 x+3)^2}+\frac {1563051 \log (1-2 x)}{234256}+\frac {17139 \log (5 x+3)}{45753125} \]

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Rubi [A] time = 0.03, antiderivative size = 66, 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 {729 x^2}{1000}+\frac {2916 x}{625}+\frac {117649}{21296 (1-2 x)}-\frac {202}{4159375 (5 x+3)}-\frac {1}{756250 (5 x+3)^2}+\frac {1563051 \log (1-2 x)}{234256}+\frac {17139 \log (5 x+3)}{45753125} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

117649/(21296*(1 - 2*x)) + (2916*x)/625 + (729*x^2)/1000 - 1/(756250*(3 + 5*x)^2) - 202/(4159375*(3 + 5*x)) +

(1563051*Log[1 - 2*x])/234256 + (17139*Log[3 + 5*x])/45753125

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)^3} \, dx &=\int \left (\frac {2916}{625}+\frac {729 x}{500}+\frac {117649}{10648 (-1+2 x)^2}+\frac {1563051}{117128 (-1+2 x)}+\frac {1}{75625 (3+5 x)^3}+\frac {202}{831875 (3+5 x)^2}+\frac {17139}{9150625 (3+5 x)}\right ) \, dx\\ &=\frac {117649}{21296 (1-2 x)}+\frac {2916 x}{625}+\frac {729 x^2}{1000}-\frac {1}{756250 (3+5 x)^2}-\frac {202}{4159375 (3+5 x)}+\frac {1563051 \log (1-2 x)}{234256}+\frac {17139 \log (3+5 x)}{45753125}\\ \end {align*}

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Mathematica [A] time = 0.06, size = 60, normalized size = 0.91 \begin {gather*} \frac {11 \left (97029900 x^2+620991360 x+\frac {735306250}{1-2 x}-\frac {6464}{5 x+3}-\frac {176}{(5 x+3)^2}-334753155\right )+9769068750 \log (1-2 x)+548448 \log (10 x+6)}{1464100000} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(11*(-334753155 + 735306250/(1 - 2*x) + 620991360*x + 97029900*x^2 - 176/(3 + 5*x)^2 - 6464/(3 + 5*x)) + 97690

68750*Log[1 - 2*x] + 548448*Log[6 + 10*x])/1464100000

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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)^3} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.67, size = 90, normalized size = 1.36 \begin {gather*} \frac {26683222500 \, x^{5} + 189450879750 \, x^{4} + 113136863400 \, x^{3} - 146893374705 \, x^{2} + 274224 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 4884534375 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) - 152064641058 \, x - 36397551751}{732050000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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)^3,x, algorithm="fricas")

[Out]

1/732050000*(26683222500*x^5 + 189450879750*x^4 + 113136863400*x^3 - 146893374705*x^2 + 274224*(50*x^3 + 35*x^

2 - 12*x - 9)*log(5*x + 3) + 4884534375*(50*x^3 + 35*x^2 - 12*x - 9)*log(2*x - 1) - 152064641058*x - 363975517

51)/(50*x^3 + 35*x^2 - 12*x - 9)

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giac [A] time = 1.21, size = 94, normalized size = 1.42 \begin {gather*} \frac {{\left (2 \, x - 1\right )}^{2} {\left (\frac {25615893600}{2 \, x - 1} + \frac {93337977265}{{\left (2 \, x - 1\right )}^{2}} + \frac {95568773322}{{\left (2 \, x - 1\right )}^{3}} + 1334161125\right )}}{292820000 \, {\left (\frac {11}{2 \, x - 1} + 5\right )}^{2}} - \frac {117649}{21296 \, {\left (2 \, x - 1\right )}} - \frac {333639}{50000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {17139}{45753125} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \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)^3,x, algorithm="giac")

[Out]

1/292820000*(2*x - 1)^2*(25615893600/(2*x - 1) + 93337977265/(2*x - 1)^2 + 95568773322/(2*x - 1)^3 + 133416112

5)/(11/(2*x - 1) + 5)^2 - 117649/21296/(2*x - 1) - 333639/50000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) + 17139/4575

3125*log(abs(-11/(2*x - 1) - 5))

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maple [A] time = 0.01, size = 53, normalized size = 0.80 \begin {gather*} \frac {729 x^{2}}{1000}+\frac {2916 x}{625}+\frac {1563051 \ln \left (2 x -1\right )}{234256}+\frac {17139 \ln \left (5 x +3\right )}{45753125}-\frac {1}{756250 \left (5 x +3\right )^{2}}-\frac {202}{4159375 \left (5 x +3\right )}-\frac {117649}{21296 \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)^3,x)

[Out]

729/1000*x^2+2916/625*x-1/756250/(5*x+3)^2-202/4159375/(5*x+3)+17139/45753125*ln(5*x+3)-117649/21296/(2*x-1)+1

563051/234256*ln(2*x-1)

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maxima [A] time = 0.71, size = 54, normalized size = 0.82 \begin {gather*} \frac {729}{1000} \, x^{2} + \frac {2916}{625} \, x - \frac {9191360445 \, x^{2} + 11029597158 \, x + 3308868341}{66550000 \, {\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac {17139}{45753125} \, \log \left (5 \, x + 3\right ) + \frac {1563051}{234256} \, \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)^3,x, algorithm="maxima")

[Out]

729/1000*x^2 + 2916/625*x - 1/66550000*(9191360445*x^2 + 11029597158*x + 3308868341)/(50*x^3 + 35*x^2 - 12*x -

9) + 17139/45753125*log(5*x + 3) + 1563051/234256*log(2*x - 1)

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mupad [B] time = 0.04, size = 49, normalized size = 0.74 \begin {gather*} \frac {2916\,x}{625}+\frac {1563051\,\ln \left (x-\frac {1}{2}\right )}{234256}+\frac {17139\,\ln \left (x+\frac {3}{5}\right )}{45753125}+\frac {\frac {1838272089\,x^2}{665500000}+\frac {5514798579\,x}{1663750000}+\frac {3308868341}{3327500000}}{-x^3-\frac {7\,x^2}{10}+\frac {6\,x}{25}+\frac {9}{50}}+\frac {729\,x^2}{1000} \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)^3),x)

[Out]

(2916*x)/625 + (1563051*log(x - 1/2))/234256 + (17139*log(x + 3/5))/45753125 + ((5514798579*x)/1663750000 + (1

838272089*x^2)/665500000 + 3308868341/3327500000)/((6*x)/25 - (7*x^2)/10 - x^3 + 9/50) + (729*x^2)/1000

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sympy [A] time = 0.19, size = 58, normalized size = 0.88 \begin {gather*} \frac {729 x^{2}}{1000} + \frac {2916 x}{625} + \frac {- 9191360445 x^{2} - 11029597158 x - 3308868341}{3327500000 x^{3} + 2329250000 x^{2} - 798600000 x - 598950000} + \frac {1563051 \log {\left (x - \frac {1}{2} \right )}}{234256} + \frac {17139 \log {\left (x + \frac {3}{5} \right )}}{45753125} \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)**3,x)

[Out]

729*x**2/1000 + 2916*x/625 + (-9191360445*x**2 - 11029597158*x - 3308868341)/(3327500000*x**3 + 2329250000*x**

2 - 798600000*x - 598950000) + 1563051*log(x - 1/2)/234256 + 17139*log(x + 3/5)/45753125

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