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

Optimal. Leaf size=59 \[ -\frac {243 x^4}{250}-\frac {837 x^3}{625}+\frac {1971 x^2}{6250}+\frac {3636 x}{3125}-\frac {163}{78125 (5 x+3)}-\frac {11}{156250 (5 x+3)^2}+\frac {192 \log (5 x+3)}{15625} \]

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Rubi [A]  time = 0.03, antiderivative size = 59, 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 {243 x^4}{250}-\frac {837 x^3}{625}+\frac {1971 x^2}{6250}+\frac {3636 x}{3125}-\frac {163}{78125 (5 x+3)}-\frac {11}{156250 (5 x+3)^2}+\frac {192 \log (5 x+3)}{15625} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3636*x)/3125 + (1971*x^2)/6250 - (837*x^3)/625 - (243*x^4)/250 - 11/(156250*(3 + 5*x)^2) - 163/(78125*(3 + 5*
x)) + (192*Log[3 + 5*x])/15625

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 {(1-2 x) (2+3 x)^5}{(3+5 x)^3} \, dx &=\int \left (\frac {3636}{3125}+\frac {1971 x}{3125}-\frac {2511 x^2}{625}-\frac {486 x^3}{125}+\frac {11}{15625 (3+5 x)^3}+\frac {163}{15625 (3+5 x)^2}+\frac {192}{3125 (3+5 x)}\right ) \, dx\\ &=\frac {3636 x}{3125}+\frac {1971 x^2}{6250}-\frac {837 x^3}{625}-\frac {243 x^4}{250}-\frac {11}{156250 (3+5 x)^2}-\frac {163}{78125 (3+5 x)}+\frac {192 \log (3+5 x)}{15625}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 58, normalized size = 0.98 \begin {gather*} \frac {-3796875 x^6-9787500 x^5-6412500 x^4+4140000 x^3+7579975 x^2+3653570 x+1920 (5 x+3)^2 \log (-3 (5 x+3))+604711}{156250 (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(604711 + 3653570*x + 7579975*x^2 + 4140000*x^3 - 6412500*x^4 - 9787500*x^5 - 3796875*x^6 + 1920*(3 + 5*x)^2*L
og[-3*(3 + 5*x)])/(156250*(3 + 5*x)^2)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.28, size = 62, normalized size = 1.05 \begin {gather*} -\frac {3796875 \, x^{6} + 9787500 \, x^{5} + 6412500 \, x^{4} - 4140000 \, x^{3} - 5897475 \, x^{2} - 1920 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 1634570 \, x + 989}{156250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/156250*(3796875*x^6 + 9787500*x^5 + 6412500*x^4 - 4140000*x^3 - 5897475*x^2 - 1920*(25*x^2 + 30*x + 9)*log(
5*x + 3) - 1634570*x + 989)/(25*x^2 + 30*x + 9)

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giac [A]  time = 1.14, size = 42, normalized size = 0.71 \begin {gather*} -\frac {243}{250} \, x^{4} - \frac {837}{625} \, x^{3} + \frac {1971}{6250} \, x^{2} + \frac {3636}{3125} \, x - \frac {1630 \, x + 989}{156250 \, {\left (5 \, x + 3\right )}^{2}} + \frac {192}{15625} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/250*x^4 - 837/625*x^3 + 1971/6250*x^2 + 3636/3125*x - 1/156250*(1630*x + 989)/(5*x + 3)^2 + 192/15625*log
(abs(5*x + 3))

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maple [A]  time = 0.01, size = 46, normalized size = 0.78 \begin {gather*} -\frac {243 x^{4}}{250}-\frac {837 x^{3}}{625}+\frac {1971 x^{2}}{6250}+\frac {3636 x}{3125}+\frac {192 \ln \left (5 x +3\right )}{15625}-\frac {11}{156250 \left (5 x +3\right )^{2}}-\frac {163}{78125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3636/3125*x+1971/6250*x^2-837/625*x^3-243/250*x^4-11/156250/(5*x+3)^2-163/78125/(5*x+3)+192/15625*ln(5*x+3)

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maxima [A]  time = 0.45, size = 46, normalized size = 0.78 \begin {gather*} -\frac {243}{250} \, x^{4} - \frac {837}{625} \, x^{3} + \frac {1971}{6250} \, x^{2} + \frac {3636}{3125} \, x - \frac {1630 \, x + 989}{156250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {192}{15625} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/250*x^4 - 837/625*x^3 + 1971/6250*x^2 + 3636/3125*x - 1/156250*(1630*x + 989)/(25*x^2 + 30*x + 9) + 192/1
5625*log(5*x + 3)

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mupad [B]  time = 0.03, size = 42, normalized size = 0.71 \begin {gather*} \frac {3636\,x}{3125}+\frac {192\,\ln \left (x+\frac {3}{5}\right )}{15625}-\frac {\frac {163\,x}{390625}+\frac {989}{3906250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}+\frac {1971\,x^2}{6250}-\frac {837\,x^3}{625}-\frac {243\,x^4}{250} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3636*x)/3125 + (192*log(x + 3/5))/15625 - ((163*x)/390625 + 989/3906250)/((6*x)/5 + x^2 + 9/25) + (1971*x^2)/
6250 - (837*x^3)/625 - (243*x^4)/250

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sympy [A]  time = 0.13, size = 49, normalized size = 0.83 \begin {gather*} - \frac {243 x^{4}}{250} - \frac {837 x^{3}}{625} + \frac {1971 x^{2}}{6250} + \frac {3636 x}{3125} - \frac {1630 x + 989}{3906250 x^{2} + 4687500 x + 1406250} + \frac {192 \log {\left (5 x + 3 \right )}}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243*x**4/250 - 837*x**3/625 + 1971*x**2/6250 + 3636*x/3125 - (1630*x + 989)/(3906250*x**2 + 4687500*x + 14062
50) + 192*log(5*x + 3)/15625

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