3.1185 \(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=41 \[ -\frac {250 x^3}{27}+\frac {25 x^2}{54}+\frac {55 x}{9}+\frac {7}{243 (3 x+2)}+\frac {107}{243} \log (3 x+2) \]

[Out]

55/9*x+25/54*x^2-250/27*x^3+7/243/(2+3*x)+107/243*ln(2+3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 41, 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} \[ -\frac {250 x^3}{27}+\frac {25 x^2}{54}+\frac {55 x}{9}+\frac {7}{243 (3 x+2)}+\frac {107}{243} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(55*x)/9 + (25*x^2)/54 - (250*x^3)/27 + 7/(243*(2 + 3*x)) + (107*Log[2 + 3*x])/243

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+5 x)^3}{(2+3 x)^2} \, dx &=\int \left (\frac {55}{9}+\frac {25 x}{27}-\frac {250 x^2}{9}-\frac {7}{81 (2+3 x)^2}+\frac {107}{81 (2+3 x)}\right ) \, dx\\ &=\frac {55 x}{9}+\frac {25 x^2}{54}-\frac {250 x^3}{27}+\frac {7}{243 (2+3 x)}+\frac {107}{243} \log (2+3 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 44, normalized size = 1.07 \[ \frac {-40500 x^4-24975 x^3+28080 x^2+22740 x+642 (3 x+2) \log (3 x+2)+3322}{1458 (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3322 + 22740*x + 28080*x^2 - 24975*x^3 - 40500*x^4 + 642*(2 + 3*x)*Log[2 + 3*x])/(1458*(2 + 3*x))

________________________________________________________________________________________

fricas [A]  time = 0.69, size = 42, normalized size = 1.02 \[ -\frac {13500 \, x^{4} + 8325 \, x^{3} - 9360 \, x^{2} - 214 \, {\left (3 \, x + 2\right )} \log \left (3 \, x + 2\right ) - 5940 \, x - 14}{486 \, {\left (3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/486*(13500*x^4 + 8325*x^3 - 9360*x^2 - 214*(3*x + 2)*log(3*x + 2) - 5940*x - 14)/(3*x + 2)

________________________________________________________________________________________

giac [A]  time = 1.18, size = 57, normalized size = 1.39 \[ \frac {5}{1458} \, {\left (3 \, x + 2\right )}^{3} {\left (\frac {615}{3 \, x + 2} - \frac {666}{{\left (3 \, x + 2\right )}^{2}} - 100\right )} + \frac {7}{243 \, {\left (3 \, x + 2\right )}} - \frac {107}{243} \, \log \left (\frac {{\left | 3 \, x + 2 \right |}}{3 \, {\left (3 \, x + 2\right )}^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/1458*(3*x + 2)^3*(615/(3*x + 2) - 666/(3*x + 2)^2 - 100) + 7/243/(3*x + 2) - 107/243*log(1/3*abs(3*x + 2)/(3
*x + 2)^2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 32, normalized size = 0.78 \[ -\frac {250 x^{3}}{27}+\frac {25 x^{2}}{54}+\frac {55 x}{9}+\frac {107 \ln \left (3 x +2\right )}{243}+\frac {7}{243 \left (3 x +2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

55/9*x+25/54*x^2-250/27*x^3+7/243/(3*x+2)+107/243*ln(3*x+2)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 31, normalized size = 0.76 \[ -\frac {250}{27} \, x^{3} + \frac {25}{54} \, x^{2} + \frac {55}{9} \, x + \frac {7}{243 \, {\left (3 \, x + 2\right )}} + \frac {107}{243} \, \log \left (3 \, x + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250/27*x^3 + 25/54*x^2 + 55/9*x + 7/243/(3*x + 2) + 107/243*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 29, normalized size = 0.71 \[ \frac {55\,x}{9}+\frac {107\,\ln \left (x+\frac {2}{3}\right )}{243}+\frac {7}{729\,\left (x+\frac {2}{3}\right )}+\frac {25\,x^2}{54}-\frac {250\,x^3}{27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(55*x)/9 + (107*log(x + 2/3))/243 + 7/(729*(x + 2/3)) + (25*x^2)/54 - (250*x^3)/27

________________________________________________________________________________________

sympy [A]  time = 0.12, size = 34, normalized size = 0.83 \[ - \frac {250 x^{3}}{27} + \frac {25 x^{2}}{54} + \frac {55 x}{9} + \frac {107 \log {\left (3 x + 2 \right )}}{243} + \frac {7}{729 x + 486} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-250*x**3/27 + 25*x**2/54 + 55*x/9 + 107*log(3*x + 2)/243 + 7/(729*x + 486)

________________________________________________________________________________________