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

Optimal. Leaf size=73 \[ -\frac {6075 x^6}{16}-\frac {48843 x^5}{16}-\frac {770067 x^4}{64}-\frac {1024389 x^3}{32}-\frac {17700255 x^2}{256}-\frac {39980457 x}{256}-\frac {12386759}{128 (1-2 x)}+\frac {14235529}{1024 (1-2 x)^2}-\frac {18859855}{128} \log (1-2 x) \]

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Rubi [A]  time = 0.04, antiderivative size = 73, 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 {6075 x^6}{16}-\frac {48843 x^5}{16}-\frac {770067 x^4}{64}-\frac {1024389 x^3}{32}-\frac {17700255 x^2}{256}-\frac {39980457 x}{256}-\frac {12386759}{128 (1-2 x)}+\frac {14235529}{1024 (1-2 x)^2}-\frac {18859855}{128} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

14235529/(1024*(1 - 2*x)^2) - 12386759/(128*(1 - 2*x)) - (39980457*x)/256 - (17700255*x^2)/256 - (1024389*x^3)
/32 - (770067*x^4)/64 - (48843*x^5)/16 - (6075*x^6)/16 - (18859855*Log[1 - 2*x])/128

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 (3+5 x)^2}{(1-2 x)^3} \, dx &=\int \left (-\frac {39980457}{256}-\frac {17700255 x}{128}-\frac {3073167 x^2}{32}-\frac {770067 x^3}{16}-\frac {244215 x^4}{16}-\frac {18225 x^5}{8}-\frac {14235529}{256 (-1+2 x)^3}-\frac {12386759}{64 (-1+2 x)^2}-\frac {18859855}{64 (-1+2 x)}\right ) \, dx\\ &=\frac {14235529}{1024 (1-2 x)^2}-\frac {12386759}{128 (1-2 x)}-\frac {39980457 x}{256}-\frac {17700255 x^2}{256}-\frac {1024389 x^3}{32}-\frac {770067 x^4}{64}-\frac {48843 x^5}{16}-\frac {6075 x^6}{16}-\frac {18859855}{128} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 66, normalized size = 0.90 \begin {gather*} -\frac {777600 x^8+5474304 x^7+18584640 x^6+42481728 x^5+82201680 x^4+194631840 x^3-489708252 x^2+186131948 x+75439420 (1-2 x)^2 \log (1-2 x)-8887005}{512 (1-2 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/512*(-8887005 + 186131948*x - 489708252*x^2 + 194631840*x^3 + 82201680*x^4 + 42481728*x^5 + 18584640*x^6 +
5474304*x^7 + 777600*x^8 + 75439420*(1 - 2*x)^2*Log[1 - 2*x])/(1 - 2*x)^2

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.23, size = 72, normalized size = 0.99 \begin {gather*} -\frac {1555200 \, x^{8} + 10948608 \, x^{7} + 37169280 \, x^{6} + 84963456 \, x^{5} + 164403360 \, x^{4} + 389263680 \, x^{3} - 568886292 \, x^{2} + 150878840 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 38266316 \, x + 84858543}{1024 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024*(1555200*x^8 + 10948608*x^7 + 37169280*x^6 + 84963456*x^5 + 164403360*x^4 + 389263680*x^3 - 568886292*
x^2 + 150878840*(4*x^2 - 4*x + 1)*log(2*x - 1) - 38266316*x + 84858543)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.17, size = 52, normalized size = 0.71 \begin {gather*} -\frac {6075}{16} \, x^{6} - \frac {48843}{16} \, x^{5} - \frac {770067}{64} \, x^{4} - \frac {1024389}{32} \, x^{3} - \frac {17700255}{256} \, x^{2} - \frac {39980457}{256} \, x + \frac {184877 \, {\left (1072 \, x - 459\right )}}{1024 \, {\left (2 \, x - 1\right )}^{2}} - \frac {18859855}{128} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075/16*x^6 - 48843/16*x^5 - 770067/64*x^4 - 1024389/32*x^3 - 17700255/256*x^2 - 39980457/256*x + 184877/1024
*(1072*x - 459)/(2*x - 1)^2 - 18859855/128*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 56, normalized size = 0.77 \begin {gather*} -\frac {6075 x^{6}}{16}-\frac {48843 x^{5}}{16}-\frac {770067 x^{4}}{64}-\frac {1024389 x^{3}}{32}-\frac {17700255 x^{2}}{256}-\frac {39980457 x}{256}-\frac {18859855 \ln \left (2 x -1\right )}{128}+\frac {14235529}{1024 \left (2 x -1\right )^{2}}+\frac {12386759}{128 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075/16*x^6-48843/16*x^5-770067/64*x^4-1024389/32*x^3-17700255/256*x^2-39980457/256*x+14235529/1024/(2*x-1)^2
[phone]/128/(2*x-1)-18859855/128*ln(2*x-1)

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maxima [A]  time = 0.63, size = 56, normalized size = 0.77 \begin {gather*} -\frac {6075}{16} \, x^{6} - \frac {48843}{16} \, x^{5} - \frac {770067}{64} \, x^{4} - \frac {1024389}{32} \, x^{3} - \frac {17700255}{256} \, x^{2} - \frac {39980457}{256} \, x + \frac {184877 \, {\left (1072 \, x - 459\right )}}{1024 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac {18859855}{128} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075/16*x^6 - 48843/16*x^5 - 770067/64*x^4 - 1024389/32*x^3 - 17700255/256*x^2 - 39980457/256*x + 184877/1024
*(1072*x - 459)/(4*x^2 - 4*x + 1) - 18859855/128*log(2*x - 1)

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mupad [B]  time = 0.04, size = 51, normalized size = 0.70 \begin {gather*} \frac {\frac {12386759\,x}{256}-\frac {84858543}{4096}}{x^2-x+\frac {1}{4}}-\frac {18859855\,\ln \left (x-\frac {1}{2}\right )}{128}-\frac {39980457\,x}{256}-\frac {17700255\,x^2}{256}-\frac {1024389\,x^3}{32}-\frac {770067\,x^4}{64}-\frac {48843\,x^5}{16}-\frac {6075\,x^6}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((12386759*x)/256 - 84858543/4096)/(x^2 - x + 1/4) - (18859855*log(x - 1/2))/128 - (39980457*x)/256 - (1770025
5*x^2)/256 - (1024389*x^3)/32 - (770067*x^4)/64 - (48843*x^5)/16 - (6075*x^6)/16

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sympy [A]  time = 0.15, size = 65, normalized size = 0.89 \begin {gather*} - \frac {6075 x^{6}}{16} - \frac {48843 x^{5}}{16} - \frac {770067 x^{4}}{64} - \frac {1024389 x^{3}}{32} - \frac {17700255 x^{2}}{256} - \frac {39980457 x}{256} - \frac {84858543 - 198188144 x}{4096 x^{2} - 4096 x + 1024} - \frac {18859855 \log {\left (2 x - 1 \right )}}{128} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6075*x**6/16 - 48843*x**5/16 - 770067*x**4/64 - 1024389*x**3/32 - 17700255*x**2/256 - 39980457*x/256 - (84858
543 - 198188144*x)/(4096*x**2 - 4096*x + 1024) - 18859855*log(2*x - 1)/128

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