3.1353 \(\int \frac{(1-2 x)^3 (3+5 x)}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=55 \[ -\frac{103 (1-2 x)^4}{30870 (3 x+2)^4}-\frac{103 (1-2 x)^4}{2205 (3 x+2)^5}+\frac{(1-2 x)^4}{126 (3 x+2)^6} \]

[Out]

(1 - 2*x)^4/(126*(2 + 3*x)^6) - (103*(1 - 2*x)^4)/(2205*(2 + 3*x)^5) - (103*(1 - 2*x)^4)/(30870*(2 + 3*x)^4)

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Rubi [A]  time = 0.0100002, 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} \[ -\frac{103 (1-2 x)^4}{30870 (3 x+2)^4}-\frac{103 (1-2 x)^4}{2205 (3 x+2)^5}+\frac{(1-2 x)^4}{126 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^4/(126*(2 + 3*x)^6) - (103*(1 - 2*x)^4)/(2205*(2 + 3*x)^5) - (103*(1 - 2*x)^4)/(30870*(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 (3+5 x)}{(2+3 x)^7} \, dx &=\frac{(1-2 x)^4}{126 (2+3 x)^6}+\frac{103}{63} \int \frac{(1-2 x)^3}{(2+3 x)^6} \, dx\\ &=\frac{(1-2 x)^4}{126 (2+3 x)^6}-\frac{103 (1-2 x)^4}{2205 (2+3 x)^5}+\frac{206 \int \frac{(1-2 x)^3}{(2+3 x)^5} \, dx}{2205}\\ &=\frac{(1-2 x)^4}{126 (2+3 x)^6}-\frac{103 (1-2 x)^4}{2205 (2+3 x)^5}-\frac{103 (1-2 x)^4}{30870 (2+3 x)^4}\\ \end{align*}

Mathematica [A]  time = 0.0121659, size = 31, normalized size = 0.56 \[ \frac{48600 x^4+14040 x^3+3375 x^2+7218 x-413}{7290 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-413 + 7218*x + 3375*x^2 + 14040*x^3 + 48600*x^4)/(7290*(2 + 3*x)^6)

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Maple [A]  time = 0.006, size = 47, normalized size = 0.9 \begin{align*}{\frac{343}{1458\, \left ( 2+3\,x \right ) ^{6}}}-{\frac{2009}{1215\, \left ( 2+3\,x \right ) ^{5}}}+{\frac{20}{243\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{259}{162\, \left ( 2+3\,x \right ) ^{4}}}-{\frac{428}{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*(3+5*x)/(2+3*x)^7,x)

[Out]

343/1458/(2+3*x)^6-2009/1215/(2+3*x)^5+20/243/(2+3*x)^2+259/162/(2+3*x)^4-428/729/(2+3*x)^3

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Maxima [A]  time = 1.00445, size = 73, normalized size = 1.33 \begin{align*} \frac{48600 \, x^{4} + 14040 \, x^{3} + 3375 \, x^{2} + 7218 \, x - 413}{7290 \,{\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*(3+5*x)/(2+3*x)^7,x, algorithm="maxima")

[Out]

1/7290*(48600*x^4 + 14040*x^3 + 3375*x^2 + 7218*x - 413)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64)

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Fricas [A]  time = 1.20815, size = 169, normalized size = 3.07 \begin{align*} \frac{48600 \, x^{4} + 14040 \, x^{3} + 3375 \, x^{2} + 7218 \, x - 413}{7290 \,{\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*(3+5*x)/(2+3*x)^7,x, algorithm="fricas")

[Out]

1/7290*(48600*x^4 + 14040*x^3 + 3375*x^2 + 7218*x - 413)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2
+ 576*x + 64)

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Sympy [A]  time = 0.16052, size = 49, normalized size = 0.89 \begin{align*} \frac{48600 x^{4} + 14040 x^{3} + 3375 x^{2} + 7218 x - 413}{5314410 x^{6} + 21257640 x^{5} + 35429400 x^{4} + 31492800 x^{3} + 15746400 x^{2} + 4199040 x + 466560} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(48600*x**4 + 14040*x**3 + 3375*x**2 + 7218*x - 413)/(5314410*x**6 + 21257640*x**5 + 35429400*x**4 + 31492800*
x**3 + 15746400*x**2 + 4199040*x + 466560)

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Giac [A]  time = 2.42264, size = 39, normalized size = 0.71 \begin{align*} \frac{48600 \, x^{4} + 14040 \, x^{3} + 3375 \, x^{2} + 7218 \, x - 413}{7290 \,{\left (3 \, x + 2\right )}^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7290*(48600*x^4 + 14040*x^3 + 3375*x^2 + 7218*x - 413)/(3*x + 2)^6