3.1189 \(\int \frac{(1-2 x) (3+5 x)^3}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=37 \[ \frac{5 (5 x+3)^4}{12 (3 x+2)^4}+\frac{7 (5 x+3)^4}{15 (3 x+2)^5} \]

[Out]

(7*(3 + 5*x)^4)/(15*(2 + 3*x)^5) + (5*(3 + 5*x)^4)/(12*(2 + 3*x)^4)

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Rubi [A]  time = 0.0056935, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {78, 37} \[ \frac{5 (5 x+3)^4}{12 (3 x+2)^4}+\frac{7 (5 x+3)^4}{15 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

(7*(3 + 5*x)^4)/(15*(2 + 3*x)^5) + (5*(3 + 5*x)^4)/(12*(2 + 3*x)^4)

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(1-2 x) (3+5 x)^3}{(2+3 x)^6} \, dx &=\frac{7 (3+5 x)^4}{15 (2+3 x)^5}+\frac{5}{3} \int \frac{(3+5 x)^3}{(2+3 x)^5} \, dx\\ &=\frac{7 (3+5 x)^4}{15 (2+3 x)^5}+\frac{5 (3+5 x)^4}{12 (2+3 x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0095094, size = 31, normalized size = 0.84 \[ \frac{405000 x^4+803250 x^3+559800 x^2+153795 x+11758}{4860 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)*(3 + 5*x)^3)/(2 + 3*x)^6,x]

[Out]

(11758 + 153795*x + 559800*x^2 + 803250*x^3 + 405000*x^4)/(4860*(2 + 3*x)^5)

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Maple [A]  time = 0.005, size = 47, normalized size = 1.3 \begin{align*}{\frac{7}{1215\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{1025}{486\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{107}{972\, \left ( 2+3\,x \right ) ^{4}}}+{\frac{250}{486+729\,x}}+{\frac{185}{243\, \left ( 2+3\,x \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)*(3+5*x)^3/(2+3*x)^6,x)

[Out]

7/1215/(2+3*x)^5-1025/486/(2+3*x)^2-107/972/(2+3*x)^4+250/243/(2+3*x)+185/243/(2+3*x)^3

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Maxima [A]  time = 1.09334, size = 66, normalized size = 1.78 \begin{align*} \frac{405000 \, x^{4} + 803250 \, x^{3} + 559800 \, x^{2} + 153795 \, x + 11758}{4860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^6,x, algorithm="maxima")

[Out]

1/4860*(405000*x^4 + 803250*x^3 + 559800*x^2 + 153795*x + 11758)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240
*x + 32)

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Fricas [A]  time = 1.48484, size = 162, normalized size = 4.38 \begin{align*} \frac{405000 \, x^{4} + 803250 \, x^{3} + 559800 \, x^{2} + 153795 \, x + 11758}{4860 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^6,x, algorithm="fricas")

[Out]

1/4860*(405000*x^4 + 803250*x^3 + 559800*x^2 + 153795*x + 11758)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240
*x + 32)

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Sympy [A]  time = 0.150514, size = 44, normalized size = 1.19 \begin{align*} \frac{405000 x^{4} + 803250 x^{3} + 559800 x^{2} + 153795 x + 11758}{1180980 x^{5} + 3936600 x^{4} + 5248800 x^{3} + 3499200 x^{2} + 1166400 x + 155520} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)**3/(2+3*x)**6,x)

[Out]

(405000*x**4 + 803250*x**3 + 559800*x**2 + 153795*x + 11758)/(1180980*x**5 + 3936600*x**4 + 5248800*x**3 + 349
9200*x**2 + 1166400*x + 155520)

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Giac [A]  time = 2.80235, size = 39, normalized size = 1.05 \begin{align*} \frac{405000 \, x^{4} + 803250 \, x^{3} + 559800 \, x^{2} + 153795 \, x + 11758}{4860 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)*(3+5*x)^3/(2+3*x)^6,x, algorithm="giac")

[Out]

1/4860*(405000*x^4 + 803250*x^3 + 559800*x^2 + 153795*x + 11758)/(3*x + 2)^5