∫ (2+3x)6(3+5x)3 ________________________

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

Optimal. Leaf size=76 \[ \frac {91125 x^8}{32}+\frac {309825 x^7}{14}+\frac {2611845 x^6}{32}+\frac {15403257 x^5}{80}+\frac {85406805 x^4}{256}+\frac {7530189 x^3}{16}+\frac {310976027 x^2}{512}+\frac {230244479 x}{256}+\frac {156590819}{1024 (1-2 x)}+\frac {616195041 \log (1-2 x)}{1024} \]

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Rubi [A] time = 0.04, antiderivative size = 76, 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 {91125 x^8}{32}+\frac {309825 x^7}{14}+\frac {2611845 x^6}{32}+\frac {15403257 x^5}{80}+\frac {85406805 x^4}{256}+\frac {7530189 x^3}{16}+\frac {310976027 x^2}{512}+\frac {230244479 x}{256}+\frac {156590819}{1024 (1-2 x)}+\frac {616195041 \log (1-2 x)}{1024} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

156590819/(1024*(1 - 2*x)) + (230244479*x)/256 + (310976027*x^2)/512 + (7530189*x^3)/16 + (85406805*x^4)/256 +

(15403257*x^5)/80 + (2611845*x^6)/32 + (309825*x^7)/14 + (91125*x^8)/32 + (616195041*Log[1 - 2*x])/1024

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)^3}{(1-2 x)^2} \, dx &=\int \left (\frac {230244479}{256}+\frac {310976027 x}{256}+\frac {22590567 x^2}{16}+\frac {85406805 x^3}{64}+\frac {15403257 x^4}{16}+\frac {7835535 x^5}{16}+\frac {309825 x^6}{2}+\frac {91125 x^7}{4}+\frac {156590819}{512 (-1+2 x)^2}+\frac {616195041}{512 (-1+2 x)}\right ) \, dx\\ &=\frac {156590819}{1024 (1-2 x)}+\frac {230244479 x}{256}+\frac {310976027 x^2}{512}+\frac {7530189 x^3}{16}+\frac {85406805 x^4}{256}+\frac {15403257 x^5}{80}+\frac {2611845 x^6}{32}+\frac {309825 x^7}{14}+\frac {91125 x^8}{32}+\frac {616195041 \log (1-2 x)}{1024}\\ \end {align*}

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Mathematica [A] time = 0.02, size = 69, normalized size = 0.91 \begin {gather*} \frac {1632960000 x^9+11873952000 x^8+40459046400 x^7+87008414976 x^6+136105970112 x^5+174226352160 x^4+213352163360 x^3+341601057840 x^2-652800288858 x+172534611480 (2 x-1) \log (1-2 x)+153617806869}{286720 (2 x-1)} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(153617806869 - 652800288858*x + 341601057840*x^2 + 213352163360*x^3 + 174226352160*x^4 + 136105970112*x^5 + 8

7008414976*x^6 + 40459046400*x^7 + 11873952000*x^8 + 1632960000*x^9 + 172534611480*(-1 + 2*x)*Log[1 - 2*x])/(2

86720*(-1 + 2*x))

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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)^3}{(1-2 x)^2} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.32, size = 67, normalized size = 0.88 \begin {gather*} \frac {204120000 \, x^{9} + 1484244000 \, x^{8} + 5057380800 \, x^{7} + 10876051872 \, x^{6} + 17013246264 \, x^{5} + 21778294020 \, x^{4} + 26669020420 \, x^{3} + 42700132230 \, x^{2} + 21566826435 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 32234227060 \, x - 5480678665}{35840 \, {\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)^3/(1-2*x)^2,x, algorithm="fricas")

[Out]

1/35840*(204120000*x^9 + 1484244000*x^8 + 5057380800*x^7 + 10876051872*x^6 + 17013246264*x^5 + 21778294020*x^4

+ 26669020420*x^3 + 42700132230*x^2 + 21566826435*(2*x - 1)*log(2*x - 1) - 32234227060*x - 5480678665)/(2*x -

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giac [A] time = 0.92, size = 102, normalized size = 1.34 \begin {gather*} \frac {1}{286720} \, {\left (2 \, x - 1\right )}^{8} {\left (\frac {75087000}{2 \, x - 1} + \frac {801964800}{{\left (2 \, x - 1\right )}^{2}} + \frac {5138731584}{{\left (2 \, x - 1\right )}^{3}} + \frac {22047451020}{{\left (2 \, x - 1\right )}^{4}} + \frac {67259967600}{{\left (2 \, x - 1\right )}^{5}} + \frac {153877208800}{{\left (2 \, x - 1\right )}^{6}} + \frac {301719264000}{{\left (2 \, x - 1\right )}^{7}} + 3189375\right )} - \frac {156590819}{1024 \, {\left (2 \, x - 1\right )}} - \frac {616195041}{1024} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/286720*(2*x - 1)^8*(75087000/(2*x - 1) + 801964800/(2*x - 1)^2 + 5138731584/(2*x - 1)^3 + 22047451020/(2*x -

1)^4 + 67259967600/(2*x - 1)^5 + 153877208800/(2*x - 1)^6 + 301719264000/(2*x - 1)^7 + 3189375) - 156590819/1

024/(2*x - 1) - 616195041/1024*log(1/2*abs(2*x - 1)/(2*x - 1)^2)

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maple [A] time = 0.01, size = 57, normalized size = 0.75 \begin {gather*} \frac {91125 x^{8}}{32}+\frac {309825 x^{7}}{14}+\frac {2611845 x^{6}}{32}+\frac {15403257 x^{5}}{80}+\frac {85406805 x^{4}}{256}+\frac {7530189 x^{3}}{16}+\frac {310976027 x^{2}}{512}+\frac {230244479 x}{256}+\frac {616195041 \ln \left (2 x -1\right )}{1024}-\frac {156590819}{1024 \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)^3/(1-2*x)^2,x)

[Out]

91125/32*x^8+309825/14*x^7+2611845/32*x^6+15403257/80*x^5+85406805/256*x^4+7530189/16*x^3+310976027/512*x^2+23

0244479/256*x-156590819/1024/(2*x-1)+616195041/1024*ln(2*x-1)

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maxima [A] time = 0.72, size = 56, normalized size = 0.74 \begin {gather*} \frac {91125}{32} \, x^{8} + \frac {309825}{14} \, x^{7} + \frac {2611845}{32} \, x^{6} + \frac {15403257}{80} \, x^{5} + \frac {85406805}{256} \, x^{4} + \frac {7530189}{16} \, x^{3} + \frac {310976027}{512} \, x^{2} + \frac {230244479}{256} \, x - \frac {156590819}{1024 \, {\left (2 \, x - 1\right )}} + \frac {616195041}{1024} \, \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)^3/(1-2*x)^2,x, algorithm="maxima")

[Out]

91125/32*x^8 + 309825/14*x^7 + 2611845/32*x^6 + 15403257/80*x^5 + 85406805/256*x^4 + 7530189/16*x^3 + 31097602

7/512*x^2 + 230244479/256*x - 156590819/1024/(2*x - 1) + 616195041/1024*log(2*x - 1)

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mupad [B] time = 0.05, size = 54, normalized size = 0.71 \begin {gather*} \frac {230244479\,x}{256}+\frac {616195041\,\ln \left (x-\frac {1}{2}\right )}{1024}-\frac {156590819}{2048\,\left (x-\frac {1}{2}\right )}+\frac {310976027\,x^2}{512}+\frac {7530189\,x^3}{16}+\frac {85406805\,x^4}{256}+\frac {15403257\,x^5}{80}+\frac {2611845\,x^6}{32}+\frac {309825\,x^7}{14}+\frac {91125\,x^8}{32} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(230244479*x)/256 + (616195041*log(x - 1/2))/1024 - 156590819/(2048*(x - 1/2)) + (310976027*x^2)/512 + (753018

9*x^3)/16 + (85406805*x^4)/256 + (15403257*x^5)/80 + (2611845*x^6)/32 + (309825*x^7)/14 + (91125*x^8)/32

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sympy [A] time = 0.13, size = 68, normalized size = 0.89 \begin {gather*} \frac {91125 x^{8}}{32} + \frac {309825 x^{7}}{14} + \frac {2611845 x^{6}}{32} + \frac {15403257 x^{5}}{80} + \frac {85406805 x^{4}}{256} + \frac {7530189 x^{3}}{16} + \frac {310976027 x^{2}}{512} + \frac {230244479 x}{256} + \frac {616195041 \log {\left (2 x - 1 \right )}}{1024} - \frac {156590819}{2048 x - 1024} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

91125*x**8/32 + 309825*x**7/14 + 2611845*x**6/32 + 15403257*x**5/80 + 85406805*x**4/256 + 7530189*x**3/16 + 31

0976027*x**2/512 + 230244479*x/256 + 616195041*log(2*x - 1)/1024 - 156590819/(2048*x - 1024)

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