3.1251 \(\int \frac{(1-2 x)^2 (3+5 x)}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=45 \[ -\frac{20}{243 (3 x+2)^3}+\frac{4}{9 (3 x+2)^4}-\frac{91}{135 (3 x+2)^5}+\frac{49}{486 (3 x+2)^6} \]

[Out]

49/(486*(2 + 3*x)^6) - 91/(135*(2 + 3*x)^5) + 4/(9*(2 + 3*x)^4) - 20/(243*(2 + 3*x)^3)

________________________________________________________________________________________

Rubi [A]  time = 0.0166633, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {77} \[ -\frac{20}{243 (3 x+2)^3}+\frac{4}{9 (3 x+2)^4}-\frac{91}{135 (3 x+2)^5}+\frac{49}{486 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

49/(486*(2 + 3*x)^6) - 91/(135*(2 + 3*x)^5) + 4/(9*(2 + 3*x)^4) - 20/(243*(2 + 3*x)^3)

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 (3+5 x)}{(2+3 x)^7} \, dx &=\int \left (-\frac{49}{27 (2+3 x)^7}+\frac{91}{9 (2+3 x)^6}-\frac{16}{3 (2+3 x)^5}+\frac{20}{27 (2+3 x)^4}\right ) \, dx\\ &=\frac{49}{486 (2+3 x)^6}-\frac{91}{135 (2+3 x)^5}+\frac{4}{9 (2+3 x)^4}-\frac{20}{243 (2+3 x)^3}\\ \end{align*}

Mathematica [A]  time = 0.0123661, size = 26, normalized size = 0.58 \[ -\frac{5400 x^3+1080 x^2-846 x+311}{2430 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^2*(3 + 5*x))/(2 + 3*x)^7,x]

[Out]

-(311 - 846*x + 1080*x^2 + 5400*x^3)/(2430*(2 + 3*x)^6)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 38, normalized size = 0.8 \begin{align*}{\frac{49}{486\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{91}{135\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{4}{9\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{20}{243\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2*(3+5*x)/(2+3*x)^7,x)

[Out]

49/486/(2+3*x)^6-91/135/(2+3*x)^5+4/9/(2+3*x)^4-20/243/(2+3*x)^3

________________________________________________________________________________________

Maxima [A]  time = 1.03467, size = 66, normalized size = 1.47 \begin{align*} -\frac{5400 \, x^{3} + 1080 \, x^{2} - 846 \, x + 311}{2430 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

-1/2430*(5400*x^3 + 1080*x^2 - 846*x + 311)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

________________________________________________________________________________________

Fricas [A]  time = 1.54151, size = 151, normalized size = 3.36 \begin{align*} -\frac{5400 \, x^{3} + 1080 \, x^{2} - 846 \, x + 311}{2430 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/2430*(5400*x^3 + 1080*x^2 - 846*x + 311)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

________________________________________________________________________________________

Sympy [A]  time = 0.153905, size = 46, normalized size = 1.02 \begin{align*} - \frac{5400 x^{3} + 1080 x^{2} - 846 x + 311}{1771470 x^{6} + 7085880 x^{5} + 11809800 x^{4} + 10497600 x^{3} + 5248800 x^{2} + 1399680 x + 155520} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2*(3+5*x)/(2+3*x)**7,x)

[Out]

-(5400*x**3 + 1080*x**2 - 846*x + 311)/(1771470*x**6 + 7085880*x**5 + 11809800*x**4 + 10497600*x**3 + 5248800*
x**2 + 1399680*x + 155520)

________________________________________________________________________________________

Giac [A]  time = 2.28257, size = 32, normalized size = 0.71 \begin{align*} -\frac{5400 \, x^{3} + 1080 \, x^{2} - 846 \, x + 311}{2430 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2*(3+5*x)/(2+3*x)^7,x, algorithm="giac")

[Out]

-1/2430*(5400*x^3 + 1080*x^2 - 846*x + 311)/(3*x + 2)^6