3.2407 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=178 \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac{2675 (5 x+3)^{3/2} \sqrt{1-2 x}}{864 (3 x+2)^2}-\frac{97235 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108864 \sqrt{7}} \]

[Out]

(-97235*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(12*(2 + 3*x)^4) +
(115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(216*(2 + 3*x)^3) + (2675*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(864*(2 + 3*x)^
2) - (40*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (3244595*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(108864*Sqrt[7])

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Rubi [A]  time = 0.0684561, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{(5 x+3)^{3/2} (1-2 x)^{5/2}}{12 (3 x+2)^4}+\frac{115 (5 x+3)^{3/2} (1-2 x)^{3/2}}{216 (3 x+2)^3}+\frac{2675 (5 x+3)^{3/2} \sqrt{1-2 x}}{864 (3 x+2)^2}-\frac{97235 \sqrt{5 x+3} \sqrt{1-2 x}}{36288 (3 x+2)}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{108864 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-97235*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36288*(2 + 3*x)) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/(12*(2 + 3*x)^4) +
(115*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(216*(2 + 3*x)^3) + (2675*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(864*(2 + 3*x)^
2) - (40*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (3244595*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(108864*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)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}-\frac{1}{108} \int \frac{\left (-\frac{3315}{4}-240 x\right ) \sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac{1}{648} \int \frac{\left (\frac{92115}{8}-960 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac{\int \frac{\frac{2886195}{16}-33600 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{13608}\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac{200}{243} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{3244595 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{217728}\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}+\frac{3244595 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{108864}-\frac{1}{243} \left (80 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{97235 \sqrt{1-2 x} \sqrt{3+5 x}}{36288 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{12 (2+3 x)^4}+\frac{115 (1-2 x)^{3/2} (3+5 x)^{3/2}}{216 (2+3 x)^3}+\frac{2675 \sqrt{1-2 x} (3+5 x)^{3/2}}{864 (2+3 x)^2}-\frac{40}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{3244595 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{108864 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.187706, size = 131, normalized size = 0.74 \[ \frac{-21 \sqrt{5 x+3} \left (3580650 x^4+6416859 x^3+1791504 x^2-1593212 x-677168\right )+125440 \sqrt{10-20 x} (3 x+2)^4 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-3244595 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*Sqrt[3 + 5*x]*(-677168 - 1593212*x + 1791504*x^2 + 6416859*x^3 + 3580650*x^4) + 125440*Sqrt[10 - 20*x]*(2
+ 3*x)^4*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] - 3244595*Sqrt[7 - 14*x]*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*
Sqrt[3 + 5*x])])/(762048*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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Maple [B]  time = 0.013, size = 315, normalized size = 1.8 \begin{align*}{\frac{1}{1524096\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 262812195\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-10160640\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-27095040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-27095040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-12042240\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+172350864\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2007040\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\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)^(3/2)/(2+3*x)^5,x)

[Out]

1/1524096*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(262812195*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
4-10160640*10^(1/2)*arcsin(20/11*x+1/11)*x^4+700832520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/
2))*x^3-27095040*10^(1/2)*arcsin(20/11*x+1/11)*x^3+700832520*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))*x^2-27095040*10^(1/2)*arcsin(20/11*x+1/11)*x^2+75193650*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*ar
ctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-12042240*10^(1/2)*arcsin(20/11*x+1/11)*x+172350864*x^2*(-10
*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-2007040*10^(1/2)*arcsin(20
/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^4

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Maxima [A]  time = 3.97522, size = 266, normalized size = 1.49 \begin{align*} \frac{21775}{21168} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{95 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{168 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4355 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{4704 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{539675}{42336} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{20}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3244595}{1524096} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1460395}{254016} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{18245 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{28224 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

21775/21168*(-10*x^2 - x + 3)^(3/2) + 1/4*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 9
5/168*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4355/4704*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x +
4) + 539675/42336*sqrt(-10*x^2 - x + 3)*x - 20/243*sqrt(10)*arcsin(20/11*x + 1/11) + 3244595/1524096*sqrt(7)*
arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1460395/254016*sqrt(-10*x^2 - x + 3) + 18245/28224*(-10*x^
2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.62934, size = 556, normalized size = 3.12 \begin{align*} -\frac{3244595 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 ) - 125440 \, \sqrt{10}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (1790325 \, x^{3} + 4103592 \, x^{2} + 2947548 \, x + 677168\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1524096 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1524096*(3244595*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x
+ 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 125440*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20*s
qrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(1790325*x^3 + 4103592*x^2 + 2947548*x
+ 677168)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

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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)**(3/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [B]  time = 3.45323, size = 602, normalized size = 3.38 \begin{align*} \frac{648919}{3048192} \, \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{20}{243} \, \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{55 \,{\left (19447 \, \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 )}^{7} + 19946472 \, \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} - 6199166400 \, \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} - 348224576000 \, \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 )}}{18144 \,{\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 )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

648919/3048192*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*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)))) - 20/243*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)))) - 55/18144*(
19447*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)))^7 + 19946472*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 - 6199166400*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 - 348224576000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4