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

Optimal. Leaf size=62 \[ \frac {1215 x^6}{8}+\frac {5103 x^5}{5}+\frac {210195 x^4}{64}+\frac {111501 x^3}{16}+\frac {1507977 x^2}{128}+\frac {661617 x}{32}+\frac {1294139}{256 (1-2 x)}+\frac {3916031}{256} \log (1-2 x) \]

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Rubi [A]  time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {1215 x^6}{8}+\frac {5103 x^5}{5}+\frac {210195 x^4}{64}+\frac {111501 x^3}{16}+\frac {1507977 x^2}{128}+\frac {661617 x}{32}+\frac {1294139}{256 (1-2 x)}+\frac {3916031}{256} \log (1-2 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

1294139/(256*(1 - 2*x)) + (661617*x)/32 + (1507977*x^2)/128 + (111501*x^3)/16 + (210195*x^4)/64 + (5103*x^5)/5
+ (1215*x^6)/8 + (3916031*Log[1 - 2*x])/256

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {(2+3 x)^6 (3+5 x)}{(1-2 x)^2} \, dx &=\int \left (\frac {661617}{32}+\frac {1507977 x}{64}+\frac {334503 x^2}{16}+\frac {210195 x^3}{16}+5103 x^4+\frac {3645 x^5}{4}+\frac {1294139}{128 (-1+2 x)^2}+\frac {3916031}{128 (-1+2 x)}\right ) \, dx\\ &=\frac {1294139}{256 (1-2 x)}+\frac {661617 x}{32}+\frac {1507977 x^2}{128}+\frac {111501 x^3}{16}+\frac {210195 x^4}{64}+\frac {5103 x^5}{5}+\frac {1215 x^6}{8}+\frac {3916031}{256} \log (1-2 x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 59, normalized size = 0.95 \begin {gather*} \frac {1555200 x^7+9673344 x^6+28405728 x^5+54545040 x^4+84957840 x^3+151398360 x^2-253249902 x+78320620 (2 x-1) \log (1-2 x)[phone]}{5120 (2 x-1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(47812811 - 253249902*x + 151398360*x^2 + 84957840*x^3 + 54545040*x^4 + 28405728*x^5 + 9673344*x^6 + 1555200*x
^7 + 78320620*(-1 + 2*x)*Log[1 - 2*x])/(5120*(-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)}{(1-2 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.93, size = 57, normalized size = 0.92 \begin {gather*} \frac {388800 \, x^{7} + 2418336 \, x^{6} + 7101432 \, x^{5} + 13636260 \, x^{4} + 21239460 \, x^{3} + 37849590 \, x^{2} + 19580155 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 26464680 \, x - 6470695}{1280 \, {\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)/(1-2*x)^2,x, algorithm="fricas")

[Out]

1/1280*(388800*x^7 + 2418336*x^6 + 7101432*x^5 + 13636260*x^4 + 21239460*x^3 + 37849590*x^2 + 19580155*(2*x -
1)*log(2*x - 1) - 26464680*x - 6470695)/(2*x - 1)

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giac [A]  time = 0.80, size = 84, normalized size = 1.35 \begin {gather*} \frac {9}{5120} \, {\left (2 \, x - 1\right )}^{6} {\left (\frac {26244}{2 \, x - 1} + \frac {227745}{{\left (2 \, x - 1\right )}^{2}} + \frac {1171100}{{\left (2 \, x - 1\right )}^{3}} + \frac {4064550}{{\left (2 \, x - 1\right )}^{4}} + \frac {11284700}{{\left (2 \, x - 1\right )}^{5}} + 1350\right )} - \frac {1294139}{256 \, {\left (2 \, x - 1\right )}} - \frac {3916031}{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)^6*(3+5*x)/(1-2*x)^2,x, algorithm="giac")

[Out]

9/5120*(2*x - 1)^6*(26244/(2*x - 1) + 227745/(2*x - 1)^2 + 1171100/(2*x - 1)^3 + 4064550/(2*x - 1)^4 + 1128470
0/(2*x - 1)^5 + 1350) - 1294139/256/(2*x - 1) - 3916031/256*log(1/2*abs(2*x - 1)/(2*x - 1)^2)

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maple [A]  time = 0.01, size = 47, normalized size = 0.76 \begin {gather*} \frac {1215 x^{6}}{8}+\frac {5103 x^{5}}{5}+\frac {210195 x^{4}}{64}+\frac {111501 x^{3}}{16}+\frac {1507977 x^{2}}{128}+\frac {661617 x}{32}+\frac {3916031 \ln \left (2 x -1\right )}{256}-\frac {1294139}{256 \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)/(1-2*x)^2,x)

[Out]

1215/8*x^6+5103/5*x^5+210195/64*x^4+111501/16*x^3+1507977/128*x^2+661617/32*x-1294139/256/(2*x-1)+3916031/256*
ln(2*x-1)

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maxima [A]  time = 0.49, size = 46, normalized size = 0.74 \begin {gather*} \frac {1215}{8} \, x^{6} + \frac {5103}{5} \, x^{5} + \frac {210195}{64} \, x^{4} + \frac {111501}{16} \, x^{3} + \frac {1507977}{128} \, x^{2} + \frac {661617}{32} \, x - \frac {1294139}{256 \, {\left (2 \, x - 1\right )}} + \frac {3916031}{256} \, \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)/(1-2*x)^2,x, algorithm="maxima")

[Out]

1215/8*x^6 + 5103/5*x^5 + 210195/64*x^4 + 111501/16*x^3 + 1507977/128*x^2 + 661617/32*x - 1294139/256/(2*x - 1
) + 3916031/256*log(2*x - 1)

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mupad [B]  time = 0.03, size = 44, normalized size = 0.71 \begin {gather*} \frac {661617\,x}{32}+\frac {3916031\,\ln \left (x-\frac {1}{2}\right )}{256}-\frac {1294139}{512\,\left (x-\frac {1}{2}\right )}+\frac {1507977\,x^2}{128}+\frac {111501\,x^3}{16}+\frac {210195\,x^4}{64}+\frac {5103\,x^5}{5}+\frac {1215\,x^6}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(661617*x)/32 + (3916031*log(x - 1/2))/256 - 1294139/(512*(x - 1/2)) + (1507977*x^2)/128 + (111501*x^3)/16 + (
210195*x^4)/64 + (5103*x^5)/5 + (1215*x^6)/8

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sympy [A]  time = 0.12, size = 54, normalized size = 0.87 \begin {gather*} \frac {1215 x^{6}}{8} + \frac {5103 x^{5}}{5} + \frac {210195 x^{4}}{64} + \frac {111501 x^{3}}{16} + \frac {1507977 x^{2}}{128} + \frac {661617 x}{32} + \frac {3916031 \log {\left (2 x - 1 \right )}}{256} - \frac {1294139}{512 x - 256} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1215*x**6/8 + 5103*x**5/5 + 210195*x**4/64 + 111501*x**3/16 + 1507977*x**2/128 + 661617*x/32 + 3916031*log(2*x
- 1)/256 - 1294139/(512*x - 256)

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