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

Optimal. Leaf size=34 \[ \frac {6 x^2}{25}-\frac {92 x}{125}-\frac {121}{625 (5 x+3)}+\frac {319}{625} \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, 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 {6 x^2}{25}-\frac {92 x}{125}-\frac {121}{625 (5 x+3)}+\frac {319}{625} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-92*x)/125 + (6*x^2)/25 - 121/(625*(3 + 5*x)) + (319*Log[3 + 5*x])/625

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 (2+3 x)}{(3+5 x)^2} \, dx &=\int \left (-\frac {92}{125}+\frac {12 x}{25}+\frac {121}{125 (3+5 x)^2}+\frac {319}{125 (3+5 x)}\right ) \, dx\\ &=-\frac {92 x}{125}+\frac {6 x^2}{25}-\frac {121}{625 (3+5 x)}+\frac {319}{625} \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 39, normalized size = 1.15 \begin {gather*} \frac {1500 x^3-3700 x^2-835 x+638 (5 x+3) \log (10 x+6)+913}{1250 (5 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(913 - 835*x - 3700*x^2 + 1500*x^3 + 638*(3 + 5*x)*Log[6 + 10*x])/(1250*(3 + 5*x))

________________________________________________________________________________________

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

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.19, size = 37, normalized size = 1.09 \begin {gather*} \frac {750 \, x^{3} - 1850 \, x^{2} + 319 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 1380 \, x - 121}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/625*(750*x^3 - 1850*x^2 + 319*(5*x + 3)*log(5*x + 3) - 1380*x - 121)/(5*x + 3)

________________________________________________________________________________________

giac [A]  time = 0.89, size = 48, normalized size = 1.41 \begin {gather*} -\frac {2}{625} \, {\left (5 \, x + 3\right )}^{2} {\left (\frac {64}{5 \, x + 3} - 3\right )} - \frac {121}{625 \, {\left (5 \, x + 3\right )}} - \frac {319}{625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/625*(5*x + 3)^2*(64/(5*x + 3) - 3) - 121/625/(5*x + 3) - 319/625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 27, normalized size = 0.79 \begin {gather*} \frac {6 x^{2}}{25}-\frac {92 x}{125}+\frac {319 \ln \left (5 x +3\right )}{625}-\frac {121}{625 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-92/125*x+6/25*x^2-121/625/(5*x+3)+319/625*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.59, size = 26, normalized size = 0.76 \begin {gather*} \frac {6}{25} \, x^{2} - \frac {92}{125} \, x - \frac {121}{625 \, {\left (5 \, x + 3\right )}} + \frac {319}{625} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/25*x^2 - 92/125*x - 121/625/(5*x + 3) + 319/625*log(5*x + 3)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 24, normalized size = 0.71 \begin {gather*} \frac {319\,\ln \left (x+\frac {3}{5}\right )}{625}-\frac {92\,x}{125}-\frac {121}{3125\,\left (x+\frac {3}{5}\right )}+\frac {6\,x^2}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(319*log(x + 3/5))/625 - (92*x)/125 - 121/(3125*(x + 3/5)) + (6*x^2)/25

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 27, normalized size = 0.79 \begin {gather*} \frac {6 x^{2}}{25} - \frac {92 x}{125} + \frac {319 \log {\left (5 x + 3 \right )}}{625} - \frac {121}{3125 x + 1875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6*x**2/25 - 92*x/125 + 319*log(5*x + 3)/625 - 121/(3125*x + 1875)

________________________________________________________________________________________