3.2418 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=195 \[ -\frac{10385 \sqrt{1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{2075}{72} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{48625 \sqrt{1-2 x} \sqrt{5 x+3}}{1944}-\frac{21935 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1458}-\frac{408665 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5832 \sqrt{7}} \]

[Out]

(-48625*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 + (2075*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/72 - ((1 - 2*x)^(5/2)*(3 + 5*
x)^(5/2))/(9*(2 + 3*x)^3) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (10385*Sqrt[1 - 2*x]*(3
+ 5*x)^(5/2))/(648*(2 + 3*x)) - (21935*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/1458 - (408665*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5832*Sqrt[7])

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Rubi [A]  time = 0.079504, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac{10385 \sqrt{1-2 x} (5 x+3)^{5/2}}{648 (3 x+2)}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{108 (3 x+2)^2}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^3}+\frac{2075}{72} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{48625 \sqrt{1-2 x} \sqrt{5 x+3}}{1944}-\frac{21935 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1458}-\frac{408665 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5832 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-48625*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1944 + (2075*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/72 - ((1 - 2*x)^(5/2)*(3 + 5*
x)^(5/2))/(9*(2 + 3*x)^3) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(108*(2 + 3*x)^2) - (10385*Sqrt[1 - 2*x]*(3
+ 5*x)^(5/2))/(648*(2 + 3*x)) - (21935*Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/1458 - (408665*ArcTan[Sqrt[
1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(5832*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 154

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] && IntegersQ[2*m, 2
*n, 2*p]

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)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^4} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{1}{9} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{1}{54} \int \frac{\left (-\frac{2005}{4}-2050 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^2} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{10385 \sqrt{1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac{1}{162} \int \frac{\left (\frac{109865}{8}-56025 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{2075}{72} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{10385 \sqrt{1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac{\int \frac{\left (\frac{19665}{2}-291750 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{1944}\\ &=-\frac{48625 \sqrt{1-2 x} \sqrt{3+5 x}}{1944}+\frac{2075}{72} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{10385 \sqrt{1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac{\int \frac{-468735-1316100 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{11664}\\ &=-\frac{48625 \sqrt{1-2 x} \sqrt{3+5 x}}{1944}+\frac{2075}{72} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{10385 \sqrt{1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac{408665 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{11664}-\frac{109675 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2916}\\ &=-\frac{48625 \sqrt{1-2 x} \sqrt{3+5 x}}{1944}+\frac{2075}{72} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{10385 \sqrt{1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}+\frac{408665 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{5832}-\frac{\left (21935 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1458}\\ &=-\frac{48625 \sqrt{1-2 x} \sqrt{3+5 x}}{1944}+\frac{2075}{72} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^3}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{108 (2+3 x)^2}-\frac{10385 \sqrt{1-2 x} (3+5 x)^{5/2}}{648 (2+3 x)}-\frac{21935 \sqrt{\frac{5}{2}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1458}-\frac{408665 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{5832 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.180657, size = 136, normalized size = 0.7 \[ \frac{-21 \sqrt{5 x+3} \left (64800 x^5-219240 x^4-747642 x^3-361497 x^2+175046 x+107984\right )+307090 \sqrt{10-20 x} (3 x+2)^3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-408665 \sqrt{7-14 x} (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{40824 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*Sqrt[3 + 5*x]*(107984 + 175046*x - 361497*x^2 - 747642*x^3 - 219240*x^4 + 64800*x^5) + 307090*Sqrt[10 - 2
0*x]*(2 + 3*x)^3*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] - 408665*Sqrt[7 - 14*x]*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sq
rt[7]*Sqrt[3 + 5*x])])/(40824*Sqrt[1 - 2*x]*(2 + 3*x)^3)

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Maple [A]  time = 0.012, size = 287, normalized size = 1.5 \begin{align*} -{\frac{1}{81648\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 8291430\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-11033955\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-1360800\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-22067910\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+3923640\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-14711940\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+17662302\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+2456720\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3269320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}+4535328\,\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)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^4,x)

[Out]

-1/81648*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(8291430*10^(1/2)*arcsin(20/11*x+1/11)*x^3-11033955*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-1360800*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/1
1)*x^2-22067910*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3923640*x^3*(-10*x^2-x+3)^(1/2)
[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-14711940*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x+17662302*x^2*(-10*x^2-x+3)^(1/2)+2456720*10^(1/2)*arcsin(20/11*x+1/11)-3269320*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)+4535328*(-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 = 2.94126, size = 257, normalized size = 1.32 \begin{align*} -\frac{185}{882} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{7 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} - \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{16075}{1764} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{189865}{31752} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{6347 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{3528 \,{\left (3 \, x + 2\right )}} + \frac{41225}{2268} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{21935}{5832} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{408665}{81648} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{191965}{13608} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-185/882*(-10*x^2 - x + 3)^(5/2) + 1/7*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) - 37/196*(-10*x^2
- x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 16075/1764*(-10*x^2 - x + 3)^(3/2)*x + 189865/31752*(-10*x^2 - x + 3)^(3/2
) - 6347/3528*(-10*x^2 - x + 3)^(5/2)/(3*x + 2) + 41225/2268*sqrt(-10*x^2 - x + 3)*x - 21935/5832*sqrt(10)*arc
sin(20/11*x + 1/11) + 408665/81648*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 191965/13608*sq
rt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.86191, size = 532, normalized size = 2.73 \begin{align*} \frac{307090 \, \sqrt{5} \sqrt{2}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 408665 \, \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 ) + 42 \,{\left (32400 \, x^{4} - 93420 \, x^{3} - 420531 \, x^{2} - 391014 \, x - 107984\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{81648 \,{\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)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

1/81648*(307090*sqrt(5)*sqrt(2)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 408665*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)) + 42*(32400*x^4 - 93420*x^3 - 420531*x^2 - 391014*x - 1079
84)*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)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [B]  time = 4.59102, size = 563, normalized size = 2.89 \begin{align*} \frac{81733}{163296} \, \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{1}{486} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 329 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{21935}{5832} \, \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 (2803 \, \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} + 1982400 \, \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} + 411208000 \, \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 )}}{324 \,{\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)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

81733/163296*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)))) + 1/486*(12*sqrt(5)*(5*x + 3) - 329*sqrt(5))*sqrt(
5*x + 3)*sqrt(-10*x + 5) - 21935/5832*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/324*(2803*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)))^5 + 1982400*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 +
411208000*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