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

Optimal. Leaf size=48 \[ \frac {1125 x^4}{16}+\frac {775 x^3}{2}+\frac {35135 x^2}{32}+\frac {41537 x}{16}+\frac {65219}{64 (1-2 x)}+\frac {144837}{64} \log (1-2 x) \]

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Rubi [A]  time = 0.02, antiderivative size = 48, 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 {1125 x^4}{16}+\frac {775 x^3}{2}+\frac {35135 x^2}{32}+\frac {41537 x}{16}+\frac {65219}{64 (1-2 x)}+\frac {144837}{64} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

65219/(64*(1 - 2*x)) + (41537*x)/16 + (35135*x^2)/32 + (775*x^3)/2 + (1125*x^4)/16 + (144837*Log[1 - 2*x])/64

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)^2 (3+5 x)^3}{(1-2 x)^2} \, dx &=\int \left (\frac {41537}{16}+\frac {35135 x}{16}+\frac {2325 x^2}{2}+\frac {1125 x^3}{4}+\frac {65219}{32 (-1+2 x)^2}+\frac {144837}{32 (-1+2 x)}\right ) \, dx\\ &=\frac {65219}{64 (1-2 x)}+\frac {41537 x}{16}+\frac {35135 x^2}{32}+\frac {775 x^3}{2}+\frac {1125 x^4}{16}+\frac {144837}{64} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 49, normalized size = 1.02 \begin {gather*} \frac {36000 x^5+180400 x^4+462960 x^3+1048104 x^2-1496774 x+579348 (2 x-1) \log (1-2 x)+155215}{256 (2 x-1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(155215 - 1496774*x + 1048104*x^2 + 462960*x^3 + 180400*x^4 + 36000*x^5 + 579348*(-1 + 2*x)*Log[1 - 2*x])/(256
*(-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)^2 (3+5 x)^3}{(1-2 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.92, size = 47, normalized size = 0.98 \begin {gather*} \frac {9000 \, x^{5} + 45100 \, x^{4} + 115740 \, x^{3} + 262026 \, x^{2} + 144837 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 166148 \, x - 65219}{64 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64*(9000*x^5 + 45100*x^4 + 115740*x^3 + 262026*x^2 + 144837*(2*x - 1)*log(2*x - 1) - 166148*x - 65219)/(2*x
- 1)

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giac [A]  time = 1.15, size = 66, normalized size = 1.38 \begin {gather*} \frac {1}{256} \, {\left (2 \, x - 1\right )}^{4} {\left (\frac {16900}{2 \, x - 1} + \frac {114220}{{\left (2 \, x - 1\right )}^{2}} + \frac {514536}{{\left (2 \, x - 1\right )}^{3}} + 1125\right )} - \frac {65219}{64 \, {\left (2 \, x - 1\right )}} - \frac {144837}{64} \, \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)^2*(3+5*x)^3/(1-2*x)^2,x, algorithm="giac")

[Out]

1/256*(2*x - 1)^4*(16900/(2*x - 1) + 114220/(2*x - 1)^2 + 514536/(2*x - 1)^3 + 1125) - 65219/64/(2*x - 1) - 14
4837/64*log(1/2*abs(2*x - 1)/(2*x - 1)^2)

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maple [A]  time = 0.01, size = 37, normalized size = 0.77 \begin {gather*} \frac {1125 x^{4}}{16}+\frac {775 x^{3}}{2}+\frac {35135 x^{2}}{32}+\frac {41537 x}{16}+\frac {144837 \ln \left (2 x -1\right )}{64}-\frac {65219}{64 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/16*x^4+775/2*x^3+35135/32*x^2+41537/16*x-65219/64/(2*x-1)+144837/64*ln(2*x-1)

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maxima [A]  time = 0.59, size = 36, normalized size = 0.75 \begin {gather*} \frac {1125}{16} \, x^{4} + \frac {775}{2} \, x^{3} + \frac {35135}{32} \, x^{2} + \frac {41537}{16} \, x - \frac {65219}{64 \, {\left (2 \, x - 1\right )}} + \frac {144837}{64} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125/16*x^4 + 775/2*x^3 + 35135/32*x^2 + 41537/16*x - 65219/64/(2*x - 1) + 144837/64*log(2*x - 1)

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mupad [B]  time = 0.03, size = 34, normalized size = 0.71 \begin {gather*} \frac {41537\,x}{16}+\frac {144837\,\ln \left (x-\frac {1}{2}\right )}{64}-\frac {65219}{128\,\left (x-\frac {1}{2}\right )}+\frac {35135\,x^2}{32}+\frac {775\,x^3}{2}+\frac {1125\,x^4}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(41537*x)/16 + (144837*log(x - 1/2))/64 - 65219/(128*(x - 1/2)) + (35135*x^2)/32 + (775*x^3)/2 + (1125*x^4)/16

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sympy [A]  time = 0.11, size = 41, normalized size = 0.85 \begin {gather*} \frac {1125 x^{4}}{16} + \frac {775 x^{3}}{2} + \frac {35135 x^{2}}{32} + \frac {41537 x}{16} + \frac {144837 \log {\left (2 x - 1 \right )}}{64} - \frac {65219}{128 x - 64} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1125*x**4/16 + 775*x**3/2 + 35135*x**2/32 + 41537*x/16 + 144837*log(2*x - 1)/64 - 65219/(128*x - 64)

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