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

Optimal. Leaf size=69 \[ \frac {30375 x^7}{28}+\frac {15525 x^6}{2}+\frac {423009 x^5}{16}+\frac {3724389 x^4}{64}+\frac {6179077 x^3}{64}+\frac {8881301 x^2}{64}+\frac {56291737 x}{256}+\frac {22370117}{512 (1-2 x)}+\frac {39220335}{256} \log (1-2 x) \]

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Rubi [A]  time = 0.04, 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 {30375 x^7}{28}+\frac {15525 x^6}{2}+\frac {423009 x^5}{16}+\frac {3724389 x^4}{64}+\frac {6179077 x^3}{64}+\frac {8881301 x^2}{64}+\frac {56291737 x}{256}+\frac {22370117}{512 (1-2 x)}+\frac {39220335}{256} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

22370117/(512*(1 - 2*x)) + (56291737*x)/256 + (8881301*x^2)/64 + (6179077*x^3)/64 + (3724389*x^4)/64 + (423009
*x^5)/16 + (15525*x^6)/2 + (30375*x^7)/28 + (39220335*Log[1 - 2*x])/256

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)^5 (3+5 x)^3}{(1-2 x)^2} \, dx &=\int \left (\frac {56291737}{256}+\frac {8881301 x}{32}+\frac {18537231 x^2}{64}+\frac {3724389 x^3}{16}+\frac {2115045 x^4}{16}+46575 x^5+\frac {30375 x^6}{4}+\frac {22370117}{256 (-1+2 x)^2}+\frac {39220335}{128 (-1+2 x)}\right ) \, dx\\ &=\frac {22370117}{512 (1-2 x)}+\frac {56291737 x}{256}+\frac {8881301 x^2}{64}+\frac {6179077 x^3}{64}+\frac {3724389 x^4}{64}+\frac {423009 x^5}{16}+\frac {15525 x^6}{2}+\frac {30375 x^7}{28}+\frac {39220335}{256} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 64, normalized size = 0.93 \begin {gather*} \frac {15552000 x^8+103507200 x^7+323374464 x^6+644755104 x^5+966981680 x^4+1297354800 x^3+2157631560 x^2-3888550282 x+1098169380 (2 x-1) \log (1-2 x)[phone]}{7168 (2 x-1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(843009185 - 3888550282*x + 2157631560*x^2 + 1297354800*x^3 + 966981680*x^4 + 644755104*x^5 + 323374464*x^6 +
103507200*x^7 + 15552000*x^8 + 1098169380*(-1 + 2*x)*Log[1 - 2*x])/(7168*(-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)^5 (3+5 x)^3}{(1-2 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.51, size = 62, normalized size = 0.90 \begin {gather*} \frac {7776000 \, x^{8} + 51753600 \, x^{7} + 161687232 \, x^{6} + 322377552 \, x^{5} + 483490840 \, x^{4} + 648677400 \, x^{3} + 1078815780 \, x^{2} + 549084690 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 788084318 \, x - 156590819}{3584 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3584*(7776000*x^8 + 51753600*x^7 + 161687232*x^6 + 322377552*x^5 + 483490840*x^4 + 648677400*x^3 + 107881578
0*x^2 + 549084690*(2*x - 1)*log(2*x - 1) - 788084318*x - 156590819)/(2*x - 1)

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giac [A]  time = 1.05, size = 93, normalized size = 1.35 \begin {gather*} \frac {1}{7168} \, {\left (2 \, x - 1\right )}^{7} {\left (\frac {1294650}{2 \, x - 1} + \frac {12414276}{{\left (2 \, x - 1\right )}^{2}} + \frac {70848603}{{\left (2 \, x - 1\right )}^{3}} + \frac {269525480}{{\left (2 \, x - 1\right )}^{4}} + \frac {738160010}{{\left (2 \, x - 1\right )}^{5}} + \frac {1684493580}{{\left (2 \, x - 1\right )}^{6}} + 60750\right )} - \frac {22370117}{512 \, {\left (2 \, x - 1\right )}} - \frac {39220335}{256} \, \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)^5*(3+5*x)^3/(1-2*x)^2,x, algorithm="giac")

[Out]

1/7168*(2*x - 1)^7*(1294650/(2*x - 1) + 12414276/(2*x - 1)^2 + 70848603/(2*x - 1)^3 + 269525480/(2*x - 1)^4 +
738160010/(2*x - 1)^5 + 1684493580/(2*x - 1)^6 + 60750) - 22370117/512/(2*x - 1) - 39220335/256*log(1/2*abs(2*
x - 1)/(2*x - 1)^2)

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maple [A]  time = 0.00, size = 52, normalized size = 0.75 \begin {gather*} \frac {30375 x^{7}}{28}+\frac {15525 x^{6}}{2}+\frac {423009 x^{5}}{16}+\frac {3724389 x^{4}}{64}+\frac {6179077 x^{3}}{64}+\frac {8881301 x^{2}}{64}+\frac {56291737 x}{256}+\frac {39220335 \ln \left (2 x -1\right )}{256}-\frac {22370117}{512 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30375/28*x^7+15525/2*x^6+423009/16*x^5+3724389/64*x^4+6179077/64*x^3+8881301/64*x^2+56291737/256*x-22370117/51
2/(2*x-1)[phone]/256*ln(2*x-1)

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maxima [A]  time = 0.50, size = 51, normalized size = 0.74 \begin {gather*} \frac {30375}{28} \, x^{7} + \frac {15525}{2} \, x^{6} + \frac {423009}{16} \, x^{5} + \frac {3724389}{64} \, x^{4} + \frac {6179077}{64} \, x^{3} + \frac {8881301}{64} \, x^{2} + \frac {56291737}{256} \, x - \frac {22370117}{512 \, {\left (2 \, x - 1\right )}} + \frac {39220335}{256} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30375/28*x^7 + 15525/2*x^6 + 423009/16*x^5 + 3724389/64*x^4 + 6179077/64*x^3 + 8881301/64*x^2 + 56291737/256*x
- 22370117/512/(2*x - 1) + 39220335/256*log(2*x - 1)

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mupad [B]  time = 0.04, size = 49, normalized size = 0.71 \begin {gather*} \frac {56291737\,x}{256}+\frac {39220335\,\ln \left (x-\frac {1}{2}\right )}{256}-\frac {22370117}{1024\,\left (x-\frac {1}{2}\right )}+\frac {8881301\,x^2}{64}+\frac {6179077\,x^3}{64}+\frac {3724389\,x^4}{64}+\frac {423009\,x^5}{16}+\frac {15525\,x^6}{2}+\frac {30375\,x^7}{28} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(56291737*x)/256 + (39220335*log(x - 1/2))/256 - 22370117/(1024*(x - 1/2)) + (8881301*x^2)/64 + (6179077*x^3)/
64 + (3724389*x^4)/64 + (423009*x^5)/16 + (15525*x^6)/2 + (30375*x^7)/28

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sympy [A]  time = 0.13, size = 61, normalized size = 0.88 \begin {gather*} \frac {30375 x^{7}}{28} + \frac {15525 x^{6}}{2} + \frac {423009 x^{5}}{16} + \frac {3724389 x^{4}}{64} + \frac {6179077 x^{3}}{64} + \frac {8881301 x^{2}}{64} + \frac {56291737 x}{256} + \frac {39220335 \log {\left (2 x - 1 \right )}}{256} - \frac {22370117}{1024 x - 512} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30375*x**7/28 + 15525*x**6/2 + 423009*x**5/16 + 3724389*x**4/64 + 6179077*x**3/64 + 8881301*x**2/64 + 56291737
*x/256 + 39220335*log(2*x - 1)/256 - 22370117/(1024*x - 512)

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