3.1190 \(\int \frac{(1-2 x) (3+5 x)^3}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=55 \[ \frac{29 (5 x+3)^4}{36 (3 x+2)^4}+\frac{29 (5 x+3)^4}{45 (3 x+2)^5}+\frac{7 (5 x+3)^4}{18 (3 x+2)^6} \]

[Out]

(7*(3 + 5*x)^4)/(18*(2 + 3*x)^6) + (29*(3 + 5*x)^4)/(45*(2 + 3*x)^5) + (29*(3 + 5*x)^4)/(36*(2 + 3*x)^4)

________________________________________________________________________________________

Rubi [A]  time = 0.0103847, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {78, 45, 37} \[ \frac{29 (5 x+3)^4}{36 (3 x+2)^4}+\frac{29 (5 x+3)^4}{45 (3 x+2)^5}+\frac{7 (5 x+3)^4}{18 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(3 + 5*x)^4)/(18*(2 + 3*x)^6) + (29*(3 + 5*x)^4)/(45*(2 + 3*x)^5) + (29*(3 + 5*x)^4)/(36*(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 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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)^7} \, dx &=\frac{7 (3+5 x)^4}{18 (2+3 x)^6}+\frac{29}{9} \int \frac{(3+5 x)^3}{(2+3 x)^6} \, dx\\ &=\frac{7 (3+5 x)^4}{18 (2+3 x)^6}+\frac{29 (3+5 x)^4}{45 (2+3 x)^5}+\frac{29}{9} \int \frac{(3+5 x)^3}{(2+3 x)^5} \, dx\\ &=\frac{7 (3+5 x)^4}{18 (2+3 x)^6}+\frac{29 (3+5 x)^4}{45 (2+3 x)^5}+\frac{29 (3+5 x)^4}{36 (2+3 x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0089696, size = 31, normalized size = 0.56 \[ \frac{607500 x^4+1066500 x^3+587925 x^2+78048 x-13198}{14580 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13198 + 78048*x + 587925*x^2 + 1066500*x^3 + 607500*x^4)/(14580*(2 + 3*x)^6)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 47, normalized size = 0.9 \begin{align*}{\frac{7}{1458\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{107}{1215\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{125}{243\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{185}{324\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{1025}{729\, \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)^7,x)

[Out]

7/1458/(2+3*x)^6-107/1215/(2+3*x)^5+125/243/(2+3*x)^2+185/324/(2+3*x)^4-1025/729/(2+3*x)^3

________________________________________________________________________________________

Maxima [A]  time = 1.47541, size = 73, normalized size = 1.33 \begin{align*} \frac{607500 \, x^{4} + 1066500 \, x^{3} + 587925 \, x^{2} + 78048 \, x - 13198}{14580 \,{\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)*(3+5*x)^3/(2+3*x)^7,x, algorithm="maxima")

[Out]

1/14580*(607500*x^4 + 1066500*x^3 + 587925*x^2 + 78048*x - 13198)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

________________________________________________________________________________________

Fricas [A]  time = 1.52524, size = 181, normalized size = 3.29 \begin{align*} \frac{607500 \, x^{4} + 1066500 \, x^{3} + 587925 \, x^{2} + 78048 \, x - 13198}{14580 \,{\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)*(3+5*x)^3/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/14580*(607500*x^4 + 1066500*x^3 + 587925*x^2 + 78048*x - 13198)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 +
2160*x^2 + 576*x + 64)

________________________________________________________________________________________

Sympy [A]  time = 0.159341, size = 49, normalized size = 0.89 \begin{align*} \frac{607500 x^{4} + 1066500 x^{3} + 587925 x^{2} + 78048 x - 13198}{10628820 x^{6} + 42515280 x^{5} + 70858800 x^{4} + 62985600 x^{3} + 31492800 x^{2} + 8398080 x + 933120} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(607500*x**4 + 1066500*x**3 + 587925*x**2 + 78048*x - 13198)/(10628820*x**6 + 42515280*x**5 + 70858800*x**4 +
62985600*x**3 + 31492800*x**2 + 8398080*x + 933120)

________________________________________________________________________________________

Giac [A]  time = 1.86928, size = 39, normalized size = 0.71 \begin{align*} \frac{607500 \, x^{4} + 1066500 \, x^{3} + 587925 \, x^{2} + 78048 \, x - 13198}{14580 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14580*(607500*x^4 + 1066500*x^3 + 587925*x^2 + 78048*x - 13198)/(3*x + 2)^6