3.2344 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=149 \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{36 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}-\frac{661 \sqrt{1-2 x} \sqrt{5 x+3}}{1512 (3 x+2)}+\frac{20}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{19573 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4536 \sqrt{7}} \]

[Out]

(-661*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1512*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + (37*
Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36*(2 + 3*x)^2) + (20*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (19573*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4536*Sqrt[7])

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Rubi [A]  time = 0.0523733, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ \frac{37 \sqrt{1-2 x} (5 x+3)^{3/2}}{36 (3 x+2)^2}-\frac{(1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^3}-\frac{661 \sqrt{1-2 x} \sqrt{5 x+3}}{1512 (3 x+2)}+\frac{20}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{19573 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4536 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-661*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1512*(2 + 3*x)) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^3) + (37*
Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36*(2 + 3*x)^2) + (20*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/81 - (19573*A
rcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4536*Sqrt[7])

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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

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

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{\left (-\frac{3}{2}-30 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}-\frac{1}{54} \int \frac{\left (-\frac{981}{4}-120 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{661 \sqrt{1-2 x} \sqrt{3+5 x}}{1512 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}-\frac{\int \frac{-\frac{41973}{8}-4200 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1134}\\ &=-\frac{661 \sqrt{1-2 x} \sqrt{3+5 x}}{1512 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac{100}{81} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{19573 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{9072}\\ &=-\frac{661 \sqrt{1-2 x} \sqrt{3+5 x}}{1512 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac{19573 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{4536}+\frac{1}{81} \left (40 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{661 \sqrt{1-2 x} \sqrt{3+5 x}}{1512 (2+3 x)}-\frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^3}+\frac{37 \sqrt{1-2 x} (3+5 x)^{3/2}}{36 (2+3 x)^2}+\frac{20}{81} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{19573 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{4536 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.150237, size = 126, normalized size = 0.85 \[ \frac{-21 \sqrt{5 x+3} \left (38082 x^3+24483 x^2-9410 x-6176\right )-7840 \sqrt{10-20 x} (3 x+2)^3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-19573 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{31752 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-21*Sqrt[3 + 5*x]*(-6176 - 9410*x + 24483*x^2 + 38082*x^3) - 7840*Sqrt[10 - 20*x]*(2 + 3*x)^3*ArcSin[Sqrt[5/1
1]*Sqrt[1 - 2*x]] - 19573*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(31752*Sqr
t[1 - 2*x]*(2 + 3*x)^3)

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Maple [B]  time = 0.01, size = 253, normalized size = 1.7 \begin{align*}{\frac{1}{63504\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 528471\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+211680\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}+1056942\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+423360\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+704628\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+282240\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+799722\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+156584\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +62720\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +914004\,x\sqrt{-10\,{x}^{2}-x+3}+259392\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^4,x)

[Out]

1/63504*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(528471*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+211
680*10^(1/2)*arcsin(20/11*x+1/11)*x^3+1056942*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+4
23360*10^(1/2)*arcsin(20/11*x+1/11)*x^2+704628*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+28
2240*10^(1/2)*arcsin(20/11*x+1/11)*x+799722*x^2*(-10*x^2-x+3)^(1/2)+156584*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))+62720*10^(1/2)*arcsin(20/11*x+1/11)+914004*x*(-10*x^2-x+3)^(1/2)+259392*(-10*x^2-x+3)^
(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.68049, size = 217, normalized size = 1.46 \begin{align*} \frac{185}{882} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{4045}{1764} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{10}{81} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{19573}{63504} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{8573}{10584} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{83 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1176 \,{\left (3 \, x + 2\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)/(2+3*x)^4,x, algorithm="maxima")

[Out]

185/882*(-10*x^2 - x + 3)^(3/2) + 1/7*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 37/196*(-10*x^2 -
x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 4045/1764*sqrt(-10*x^2 - x + 3)*x + 10/81*sqrt(10)*arcsin(20/11*x + 1/11) +
19573/63504*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 8573/10584*sqrt(-10*x^2 - x + 3) + 83
/1176*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.5546, size = 473, normalized size = 3.17 \begin{align*} -\frac{19573 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 7840 \, \sqrt{10}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \,{\left (19041 \, x^{2} + 21762 \, x + 6176\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{63504 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/63504*(19573*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) + 7840*sqrt(10)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(19041*x^2 + 21762*x + 6176)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 +
54*x^2 + 36*x + 8)

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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)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [B]  time = 2.48529, size = 520, normalized size = 3.49 \begin{align*} \frac{19573}{635040} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} + \frac{10}{81} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (661 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 499520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} - 139630400 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{756 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

19573/635040*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 10/81*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)
*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/756*(661*s
qrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(
22)))^5 + 499520*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(
-10*x + 5) - sqrt(22)))^3 - 139630400*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*
x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3