3.11.48 \(\int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=49 \[ -\frac {250 x}{81}+\frac {185}{81 (3 x+2)}-\frac {107}{486 (3 x+2)^2}+\frac {7}{729 (3 x+2)^3}+\frac {1025}{243} \log (3 x+2) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 49, 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 {250 x}{81}+\frac {185}{81 (3 x+2)}-\frac {107}{486 (3 x+2)^2}+\frac {7}{729 (3 x+2)^3}+\frac {1025}{243} \log (3 x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(-250*x)/81 + 7/(729*(2 + 3*x)^3) - 107/(486*(2 + 3*x)^2) + 185/(81*(2 + 3*x)) + (1025*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)^4} \, dx &=\int \left (-\frac {250}{81}-\frac {7}{81 (2+3 x)^4}+\frac {107}{81 (2+3 x)^3}-\frac {185}{27 (2+3 x)^2}+\frac {1025}{81 (2+3 x)}\right ) \, dx\\ &=-\frac {250 x}{81}+\frac {7}{729 (2+3 x)^3}-\frac {107}{486 (2+3 x)^2}+\frac {185}{81 (2+3 x)}+\frac {1025}{243} \log (2+3 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 47, normalized size = 0.96 \begin {gather*} \frac {-1500 (3 x+2)+\frac {3330}{3 x+2}-\frac {321}{(3 x+2)^2}+\frac {14}{(3 x+2)^3}+6150 \log (3 x+2)}{1458} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

(14/(2 + 3*x)^3 - 321/(2 + 3*x)^2 + 3330/(2 + 3*x) - 1500*(2 + 3*x) + 6150*Log[2 + 3*x])/1458

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x) (3+5 x)^3}{(2+3 x)^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^4,x]

[Out]

IntegrateAlgebraic[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^4, x]

________________________________________________________________________________________

fricas [A]  time = 1.35, size = 62, normalized size = 1.27 \begin {gather*} -\frac {121500 \, x^{4} + 243000 \, x^{3} + 132030 \, x^{2} - 6150 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) - 2997 \, x - 12692}{1458 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/1458*(121500*x^4 + 243000*x^3 + 132030*x^2 - 6150*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x + 2) - 2997*x - 1269
2)/(27*x^3 + 54*x^2 + 36*x + 8)

________________________________________________________________________________________

giac [A]  time = 1.11, size = 32, normalized size = 0.65 \begin {gather*} -\frac {250}{81} \, x + \frac {29970 \, x^{2} + 38997 \, x + 12692}{1458 \, {\left (3 \, x + 2\right )}^{3}} + \frac {1025}{243} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^4,x, algorithm="giac")

[Out]

-250/81*x + 1/1458*(29970*x^2 + 38997*x + 12692)/(3*x + 2)^3 + 1025/243*log(abs(3*x + 2))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 40, normalized size = 0.82 \begin {gather*} -\frac {250 x}{81}+\frac {1025 \ln \left (3 x +2\right )}{243}+\frac {7}{729 \left (3 x +2\right )^{3}}-\frac {107}{486 \left (3 x +2\right )^{2}}+\frac {185}{81 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(5*x+3)^3/(3*x+2)^4,x)

[Out]

-250/81*x+7/729/(3*x+2)^3-107/486/(3*x+2)^2+185/81/(3*x+2)+1025/243*ln(3*x+2)

________________________________________________________________________________________

maxima [A]  time = 0.47, size = 41, normalized size = 0.84 \begin {gather*} -\frac {250}{81} \, x + \frac {29970 \, x^{2} + 38997 \, x + 12692}{1458 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1025}{243} \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^4,x, algorithm="maxima")

[Out]

-250/81*x + 1/1458*(29970*x^2 + 38997*x + 12692)/(27*x^3 + 54*x^2 + 36*x + 8) + 1025/243*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 0.04, size = 36, normalized size = 0.73 \begin {gather*} \frac {1025\,\ln \left (x+\frac {2}{3}\right )}{243}-\frac {250\,x}{81}+\frac {\frac {185\,x^2}{243}+\frac {4333\,x}{4374}+\frac {6346}{19683}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((2*x - 1)*(5*x + 3)^3)/(3*x + 2)^4,x)

[Out]

(1025*log(x + 2/3))/243 - (250*x)/81 + ((4333*x)/4374 + (185*x^2)/243 + 6346/19683)/((4*x)/3 + 2*x^2 + x^3 + 8
/27)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 41, normalized size = 0.84 \begin {gather*} - \frac {250 x}{81} - \frac {- 29970 x^{2} - 38997 x - 12692}{39366 x^{3} + 78732 x^{2} + 52488 x + 11664} + \frac {1025 \log {\left (3 x + 2 \right )}}{243} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**3/(2+3*x)**4,x)

[Out]

-250*x/81 - (-29970*x**2 - 38997*x - 12692)/(39366*x**3 + 78732*x**2 + 52488*x + 11664) + 1025*log(3*x + 2)/24
3

________________________________________________________________________________________