3.1407 \(\int \frac {(1-2 x)^3 (2+3 x)^2}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=48 \[ -\frac {18 x^4}{25}+\frac {164 x^3}{125}-\frac {427 x^2}{625}-\frac {1179 x}{3125}-\frac {1331}{15625 (5 x+3)}+\frac {1452 \log (5 x+3)}{3125} \]

[Out]

-1179/3125*x-427/625*x^2+164/125*x^3-18/25*x^4-1331/15625/(3+5*x)+1452/3125*ln(3+5*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} \[ -\frac {18 x^4}{25}+\frac {164 x^3}{125}-\frac {427 x^2}{625}-\frac {1179 x}{3125}-\frac {1331}{15625 (5 x+3)}+\frac {1452 \log (5 x+3)}{3125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1179*x)/3125 - (427*x^2)/625 + (164*x^3)/125 - (18*x^4)/25 - 1331/(15625*(3 + 5*x)) + (1452*Log[3 + 5*x])/31
25

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 {(1-2 x)^3 (2+3 x)^2}{(3+5 x)^2} \, dx &=\int \left (-\frac {1179}{3125}-\frac {854 x}{625}+\frac {492 x^2}{125}-\frac {72 x^3}{25}+\frac {1331}{3125 (3+5 x)^2}+\frac {1452}{625 (3+5 x)}\right ) \, dx\\ &=-\frac {1179 x}{3125}-\frac {427 x^2}{625}+\frac {164 x^3}{125}-\frac {18 x^4}{25}-\frac {1331}{15625 (3+5 x)}+\frac {1452 \log (3+5 x)}{3125}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 51, normalized size = 1.06 \[ \frac {-11250 x^5+13750 x^4+1625 x^3-12300 x^2+2655 x+1452 (5 x+3) \log (6 (5 x+3))+3449}{3125 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3449 + 2655*x - 12300*x^2 + 1625*x^3 + 13750*x^4 - 11250*x^5 + 1452*(3 + 5*x)*Log[6*(3 + 5*x)])/(3125*(3 + 5*
x))

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fricas [A]  time = 0.51, size = 47, normalized size = 0.98 \[ -\frac {56250 \, x^{5} - 68750 \, x^{4} - 8125 \, x^{3} + 61500 \, x^{2} - 7260 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 17685 \, x + 1331}{15625 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15625*(56250*x^5 - 68750*x^4 - 8125*x^3 + 61500*x^2 - 7260*(5*x + 3)*log(5*x + 3) + 17685*x + 1331)/(5*x +
3)

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giac [A]  time = 1.02, size = 66, normalized size = 1.38 \[ \frac {1}{15625} \, {\left (5 \, x + 3\right )}^{4} {\left (\frac {380}{5 \, x + 3} - \frac {2875}{{\left (5 \, x + 3\right )}^{2}} + \frac {7755}{{\left (5 \, x + 3\right )}^{3}} - 18\right )} - \frac {1331}{15625 \, {\left (5 \, x + 3\right )}} - \frac {1452}{3125} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15625*(5*x + 3)^4*(380/(5*x + 3) - 2875/(5*x + 3)^2 + 7755/(5*x + 3)^3 - 18) - 1331/15625/(5*x + 3) - 1452/3
125*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

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maple [A]  time = 0.01, size = 37, normalized size = 0.77 \[ -\frac {18 x^{4}}{25}+\frac {164 x^{3}}{125}-\frac {427 x^{2}}{625}-\frac {1179 x}{3125}+\frac {1452 \ln \left (5 x +3\right )}{3125}-\frac {1331}{15625 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1179/3125*x-427/625*x^2+164/125*x^3-18/25*x^4-1331/15625/(5*x+3)+1452/3125*ln(5*x+3)

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maxima [A]  time = 0.54, size = 36, normalized size = 0.75 \[ -\frac {18}{25} \, x^{4} + \frac {164}{125} \, x^{3} - \frac {427}{625} \, x^{2} - \frac {1179}{3125} \, x - \frac {1331}{15625 \, {\left (5 \, x + 3\right )}} + \frac {1452}{3125} \, \log \left (5 \, x + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-18/25*x^4 + 164/125*x^3 - 427/625*x^2 - 1179/3125*x - 1331/15625/(5*x + 3) + 1452/3125*log(5*x + 3)

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mupad [B]  time = 0.03, size = 34, normalized size = 0.71 \[ \frac {1452\,\ln \left (x+\frac {3}{5}\right )}{3125}-\frac {1179\,x}{3125}-\frac {1331}{78125\,\left (x+\frac {3}{5}\right )}-\frac {427\,x^2}{625}+\frac {164\,x^3}{125}-\frac {18\,x^4}{25} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1452*log(x + 3/5))/3125 - (1179*x)/3125 - 1331/(78125*(x + 3/5)) - (427*x^2)/625 + (164*x^3)/125 - (18*x^4)/2
5

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sympy [A]  time = 0.12, size = 41, normalized size = 0.85 \[ - \frac {18 x^{4}}{25} + \frac {164 x^{3}}{125} - \frac {427 x^{2}}{625} - \frac {1179 x}{3125} + \frac {1452 \log {\left (5 x + 3 \right )}}{3125} - \frac {1331}{78125 x + 46875} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-18*x**4/25 + 164*x**3/125 - 427*x**2/625 - 1179*x/3125 + 1452*log(5*x + 3)/3125 - 1331/(78125*x + 46875)

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