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

Optimal. Leaf size=45 \[ -\frac {12 x^2}{125}+\frac {316 x}{625}-\frac {3267}{3125 (5 x+3)}-\frac {1331}{6250 (5 x+3)^2}-\frac {2046 \log (5 x+3)}{3125} \]

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Rubi [A]  time = 0.02, antiderivative size = 45, 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 {12 x^2}{125}+\frac {316 x}{625}-\frac {3267}{3125 (5 x+3)}-\frac {1331}{6250 (5 x+3)^2}-\frac {2046 \log (5 x+3)}{3125} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(316*x)/625 - (12*x^2)/125 - 1331/(6250*(3 + 5*x)^2) - 3267/(3125*(3 + 5*x)) - (2046*Log[3 + 5*x])/3125

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)^3 (2+3 x)}{(3+5 x)^3} \, dx &=\int \left (\frac {316}{625}-\frac {24 x}{125}+\frac {1331}{625 (3+5 x)^3}+\frac {3267}{625 (3+5 x)^2}-\frac {2046}{625 (3+5 x)}\right ) \, dx\\ &=\frac {316 x}{625}-\frac {12 x^2}{125}-\frac {1331}{6250 (3+5 x)^2}-\frac {3267}{3125 (3+5 x)}-\frac {2046 \log (3+5 x)}{3125}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 1.02 \begin {gather*} -\frac {15000 x^4-61000 x^3-53650 x^2+47130 x+4092 (5 x+3)^2 \log (10 x+6)+33803}{6250 (5 x+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6250*(33803 + 47130*x - 53650*x^2 - 61000*x^3 + 15000*x^4 + 4092*(3 + 5*x)^2*Log[6 + 10*x])/(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)^3 (2+3 x)}{(3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.28, size = 52, normalized size = 1.16 \begin {gather*} -\frac {15000 \, x^{4} - 61000 \, x^{3} - 89400 \, x^{2} + 4092 \, {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 4230 \, x + 20933}{6250 \, {\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)^3*(2+3*x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

-1/6250*(15000*x^4 - 61000*x^3 - 89400*x^2 + 4092*(25*x^2 + 30*x + 9)*log(5*x + 3) + 4230*x + 20933)/(25*x^2 +
30*x + 9)

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giac [A]  time = 0.91, size = 32, normalized size = 0.71 \begin {gather*} -\frac {12}{125} \, x^{2} + \frac {316}{625} \, x - \frac {121 \, {\left (270 \, x + 173\right )}}{6250 \, {\left (5 \, x + 3\right )}^{2}} - \frac {2046}{3125} \, \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)^3*(2+3*x)/(3+5*x)^3,x, algorithm="giac")

[Out]

-12/125*x^2 + 316/625*x - 121/6250*(270*x + 173)/(5*x + 3)^2 - 2046/3125*log(abs(5*x + 3))

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maple [A]  time = 0.01, size = 36, normalized size = 0.80 \begin {gather*} -\frac {12 x^{2}}{125}+\frac {316 x}{625}-\frac {2046 \ln \left (5 x +3\right )}{3125}-\frac {1331}{6250 \left (5 x +3\right )^{2}}-\frac {3267}{3125 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

316/625*x-12/125*x^2-1331/6250/(5*x+3)^2-3267/3125/(5*x+3)-2046/3125*ln(5*x+3)

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maxima [A]  time = 0.49, size = 36, normalized size = 0.80 \begin {gather*} -\frac {12}{125} \, x^{2} + \frac {316}{625} \, x - \frac {121 \, {\left (270 \, x + 173\right )}}{6250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {2046}{3125} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12/125*x^2 + 316/625*x - 121/6250*(270*x + 173)/(25*x^2 + 30*x + 9) - 2046/3125*log(5*x + 3)

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mupad [B]  time = 0.03, size = 32, normalized size = 0.71 \begin {gather*} \frac {316\,x}{625}-\frac {2046\,\ln \left (x+\frac {3}{5}\right )}{3125}-\frac {\frac {3267\,x}{15625}+\frac {20933}{156250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}}-\frac {12\,x^2}{125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(316*x)/625 - (2046*log(x + 3/5))/3125 - ((3267*x)/15625 + 20933/156250)/((6*x)/5 + x^2 + 9/25) - (12*x^2)/125

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sympy [A]  time = 0.13, size = 36, normalized size = 0.80 \begin {gather*} - \frac {12 x^{2}}{125} + \frac {316 x}{625} - \frac {32670 x + 20933}{156250 x^{2} + 187500 x + 56250} - \frac {2046 \log {\left (5 x + 3 \right )}}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-12*x**2/125 + 316*x/625 - (32670*x + 20933)/(156250*x**2 + 187500*x + 56250) - 2046*log(5*x + 3)/3125

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