3.1889 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^3}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=118 \[ \frac{6 \sqrt{1-2 x} (5 x+3)^3}{3 x+2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{31}{3} \sqrt{1-2 x} (5 x+3)^2+\frac{1}{54} \sqrt{1-2 x} (1715 x+367)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

[Out]

(-31*Sqrt[1 - 2*x]*(3 + 5*x)^2)/3 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(6*(2 + 3*x)^2) + (6*Sqrt[1 - 2*x]*(3 + 5*x)
^3)/(2 + 3*x) + (Sqrt[1 - 2*x]*(367 + 1715*x))/54 + (887*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

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Rubi [A]  time = 0.0397846, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {97, 149, 153, 147, 63, 206} \[ \frac{6 \sqrt{1-2 x} (5 x+3)^3}{3 x+2}-\frac{(1-2 x)^{3/2} (5 x+3)^3}{6 (3 x+2)^2}-\frac{31}{3} \sqrt{1-2 x} (5 x+3)^2+\frac{1}{54} \sqrt{1-2 x} (1715 x+367)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-31*Sqrt[1 - 2*x]*(3 + 5*x)^2)/3 - ((1 - 2*x)^(3/2)*(3 + 5*x)^3)/(6*(2 + 3*x)^2) + (6*Sqrt[1 - 2*x]*(3 + 5*x)
^3)/(2 + 3*x) + (Sqrt[1 - 2*x]*(367 + 1715*x))/54 + (887*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(27*Sqrt[21])

Rule 97

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

Rule 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

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)^3}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{(6-45 x) \sqrt{1-2 x} (3+5 x)^2}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{6 \sqrt{1-2 x} (3+5 x)^3}{2+3 x}-\frac{1}{18} \int \frac{(801-2790 x) (3+5 x)^2}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{31}{3} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{6 \sqrt{1-2 x} (3+5 x)^3}{2+3 x}+\frac{1}{270} \int \frac{(3015-15435 x) (3+5 x)}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{31}{3} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{6 \sqrt{1-2 x} (3+5 x)^3}{2+3 x}+\frac{1}{54} \sqrt{1-2 x} (367+1715 x)-\frac{887}{54} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=-\frac{31}{3} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{6 \sqrt{1-2 x} (3+5 x)^3}{2+3 x}+\frac{1}{54} \sqrt{1-2 x} (367+1715 x)+\frac{887}{54} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{31}{3} \sqrt{1-2 x} (3+5 x)^2-\frac{(1-2 x)^{3/2} (3+5 x)^3}{6 (2+3 x)^2}+\frac{6 \sqrt{1-2 x} (3+5 x)^3}{2+3 x}+\frac{1}{54} \sqrt{1-2 x} (367+1715 x)+\frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0435685, size = 63, normalized size = 0.53 \[ \frac{887 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{27 \sqrt{21}}-\frac{\sqrt{1-2 x} \left (1800 x^4+570 x^2+2965 x+1367\right )}{54 (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(1367 + 2965*x + 570*x^2 + 1800*x^4))/(54*(2 + 3*x)^2) + (887*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
)/(27*Sqrt[21])

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Maple [A]  time = 0.01, size = 75, normalized size = 0.6 \begin{align*} -{\frac{25}{27} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{50}{81} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{370}{81}\sqrt{1-2\,x}}-{\frac{2}{9\, \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{215}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{497}{18}\sqrt{1-2\,x}} \right ) }+{\frac{887\,\sqrt{21}}{567}{\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)^3/(2+3*x)^3,x)

[Out]

-25/27*(1-2*x)^(5/2)-50/81*(1-2*x)^(3/2)-370/81*(1-2*x)^(1/2)-2/9*(-215/18*(1-2*x)^(3/2)+497/18*(1-2*x)^(1/2))
/(-6*x-4)^2+887/567*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 1.54242, size = 136, normalized size = 1.15 \begin{align*} -\frac{25}{27} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{887}{1134} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{370}{81} \, \sqrt{-2 \, x + 1} + \frac{215 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \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)^3/(2+3*x)^3,x, algorithm="maxima")

[Out]

-25/27*(-2*x + 1)^(5/2) - 50/81*(-2*x + 1)^(3/2) - 887/1134*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(
21) + 3*sqrt(-2*x + 1))) - 370/81*sqrt(-2*x + 1) + 1/81*(215*(-2*x + 1)^(3/2) - 497*sqrt(-2*x + 1))/(9*(2*x -
1)^2 + 84*x + 7)

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Fricas [A]  time = 1.38502, size = 228, normalized size = 1.93 \begin{align*} \frac{887 \, \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^{4} + 570 \, x^{2} + 2965 \, x + 1367\right )} \sqrt{-2 \, x + 1}}{1134 \,{\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)^3/(2+3*x)^3,x, algorithm="fricas")

[Out]

1/1134*(887*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(1800*x^4 + 57
0*x^2 + 2965*x + 1367)*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)**3/(2+3*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.49272, size = 138, normalized size = 1.17 \begin{align*} -\frac{25}{27} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{50}{81} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - \frac{887}{1134} \, \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{370}{81} \, \sqrt{-2 \, x + 1} + \frac{215 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \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)^3/(2+3*x)^3,x, algorithm="giac")

[Out]

-25/27*(2*x - 1)^2*sqrt(-2*x + 1) - 50/81*(-2*x + 1)^(3/2) - 887/1134*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqr
t(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 370/81*sqrt(-2*x + 1) + 1/324*(215*(-2*x + 1)^(3/2) - 497*sqrt(-
2*x + 1))/(3*x + 2)^2