3.1877 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=94 \[ \frac{47 (1-2 x)^{5/2}}{294 (3 x+2)}-\frac{(1-2 x)^{5/2}}{126 (3 x+2)^2}+\frac{2873 (1-2 x)^{3/2}}{3969}+\frac{2873}{567} \sqrt{1-2 x}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

[Out]

(2873*Sqrt[1 - 2*x])/567 + (2873*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(5/2)/(126*(2 + 3*x)^2) + (47*(1 - 2*x)^(5/
2))/(294*(2 + 3*x)) - (2873*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

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Rubi [A]  time = 0.0252962, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {89, 78, 50, 63, 206} \[ \frac{47 (1-2 x)^{5/2}}{294 (3 x+2)}-\frac{(1-2 x)^{5/2}}{126 (3 x+2)^2}+\frac{2873 (1-2 x)^{3/2}}{3969}+\frac{2873}{567} \sqrt{1-2 x}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2873*Sqrt[1 - 2*x])/567 + (2873*(1 - 2*x)^(3/2))/3969 - (1 - 2*x)^(5/2)/(126*(2 + 3*x)^2) + (47*(1 - 2*x)^(5/
2))/(294*(2 + 3*x)) - (2873*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(81*Sqrt[21])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^2}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac{1}{126} \int \frac{(1-2 x)^{3/2} (559+1050 x)}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac{47 (1-2 x)^{5/2}}{294 (2+3 x)}+\frac{2873}{882} \int \frac{(1-2 x)^{3/2}}{2+3 x} \, dx\\ &=\frac{2873 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac{47 (1-2 x)^{5/2}}{294 (2+3 x)}+\frac{2873}{378} \int \frac{\sqrt{1-2 x}}{2+3 x} \, dx\\ &=\frac{2873}{567} \sqrt{1-2 x}+\frac{2873 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac{47 (1-2 x)^{5/2}}{294 (2+3 x)}+\frac{2873}{162} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{2873}{567} \sqrt{1-2 x}+\frac{2873 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac{47 (1-2 x)^{5/2}}{294 (2+3 x)}-\frac{2873}{162} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{2873}{567} \sqrt{1-2 x}+\frac{2873 (1-2 x)^{3/2}}{3969}-\frac{(1-2 x)^{5/2}}{126 (2+3 x)^2}+\frac{47 (1-2 x)^{5/2}}{294 (2+3 x)}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0426832, size = 63, normalized size = 0.67 \[ \frac{\sqrt{1-2 x} \left (-1800 x^3+5520 x^2+10195 x+3803\right )}{162 (3 x+2)^2}-\frac{2873 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{81 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(3803 + 10195*x + 5520*x^2 - 1800*x^3))/(162*(2 + 3*x)^2) - (2873*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(81*Sqrt[21])

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Maple [A]  time = 0.009, size = 66, normalized size = 0.7 \begin{align*}{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{130}{27}\sqrt{1-2\,x}}+{\frac{2}{3\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{145}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{1001}{54}\sqrt{1-2\,x}} \right ) }-{\frac{2873\,\sqrt{21}}{1701}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/81*(1-2*x)^(3/2)+130/27*(1-2*x)^(1/2)+2/3*(-145/18*(1-2*x)^(3/2)+1001/54*(1-2*x)^(1/2))/(-6*x-4)^2-2873/170
1*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 3.36751, size = 124, normalized size = 1.32 \begin{align*} \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2873}{3402} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{130}{27} \, \sqrt{-2 \, x + 1} - \frac{435 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1001 \, \sqrt{-2 \, x + 1}}{81 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/81*(-2*x + 1)^(3/2) + 2873/3402*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1)))
+ 130/27*sqrt(-2*x + 1) - 1/81*(435*(-2*x + 1)^(3/2) - 1001*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 1.41363, size = 232, normalized size = 2.47 \begin{align*} \frac{2873 \, \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (1800 \, x^{3} - 5520 \, x^{2} - 10195 \, x - 3803\right )} \sqrt{-2 \, x + 1}}{3402 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3402*(2873*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(1800*x^3 - 5
520*x^2 - 10195*x - 3803)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.54718, size = 116, normalized size = 1.23 \begin{align*} \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{2873}{3402} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{130}{27} \, \sqrt{-2 \, x + 1} - \frac{435 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1001 \, \sqrt{-2 \, x + 1}}{324 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

50/81*(-2*x + 1)^(3/2) + 2873/3402*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*
x + 1))) + 130/27*sqrt(-2*x + 1) - 1/324*(435*(-2*x + 1)^(3/2) - 1001*sqrt(-2*x + 1))/(3*x + 2)^2