3.2588 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ -\frac{16985 \sqrt{5 x+3}}{316932 \sqrt{1-2 x}}+\frac{605 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{\sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{3 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{25365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(-16985*Sqrt[3 + 5*x])/(316932*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (3*Sqrt[3
+ 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^3) - Sqrt[3 + 5*x]/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (605*Sqrt[3 + 5*x])/
(2744*Sqrt[1 - 2*x]*(2 + 3*x)) - (25365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0620954, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {99, 151, 152, 12, 93, 204} \[ -\frac{16985 \sqrt{5 x+3}}{316932 \sqrt{1-2 x}}+\frac{605 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{\sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{3 \sqrt{5 x+3}}{49 \sqrt{1-2 x} (3 x+2)^3}+\frac{2 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{25365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-16985*Sqrt[3 + 5*x])/(316932*Sqrt[1 - 2*x]) + (2*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (3*Sqrt[3
+ 5*x])/(49*Sqrt[1 - 2*x]*(2 + 3*x)^3) - Sqrt[3 + 5*x]/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (605*Sqrt[3 + 5*x])/
(2744*Sqrt[1 - 2*x]*(2 + 3*x)) - (25365*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

Rule 99

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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{\sqrt{3+5 x}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{2}{21} \int \frac{-\frac{71}{2}-60 x}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{2}{441} \int \frac{-\frac{1059}{4}-405 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{\int \frac{-\frac{14385}{8}-315 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{3087}\\ &=\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}+\frac{605 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}-\frac{\int \frac{-\frac{24465}{16}+\frac{190575 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{21609}\\ &=-\frac{16985 \sqrt{3+5 x}}{316932 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}+\frac{605 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}+\frac{2 \int \frac{17577945}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1663893}\\ &=-\frac{16985 \sqrt{3+5 x}}{316932 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}+\frac{605 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}+\frac{25365 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{38416}\\ &=-\frac{16985 \sqrt{3+5 x}}{316932 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}+\frac{605 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}+\frac{25365 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{19208}\\ &=-\frac{16985 \sqrt{3+5 x}}{316932 \sqrt{1-2 x}}+\frac{2 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{3 \sqrt{3+5 x}}{49 \sqrt{1-2 x} (2+3 x)^3}-\frac{\sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}+\frac{605 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}-\frac{25365 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{19208 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0719112, size = 100, normalized size = 0.58 \[ -\frac{-7 \sqrt{5 x+3} \left (1834380 x^4+235980 x^3-1465461 x^2-39530 x+302352\right )-837045 \sqrt{7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4437048 (1-2 x)^{3/2} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-7*Sqrt[3 + 5*x]*(302352 - 39530*x - 1465461*x^2 + 235980*x^3 + 1834380*x^4) - 837045*Sqrt[7 - 14*x]*(-1 + 2
*x)*(2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4437048*(1 - 2*x)^(3/2)*(2 + 3*x)^3)

________________________________________________________________________________________

Maple [B]  time = 0.016, size = 305, normalized size = 1.8 \begin{align*}{\frac{1}{8874096\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) ^{2}} \left ( 90400860\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-37667025\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-48548610\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+3303720\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+3348180\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-20516454\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+6696360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -553420\,x\sqrt{-10\,{x}^{2}-x+3}+4232928\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8874096*(90400860*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+90400860*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-37667025*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^3+25681320*x^4*(-10*x^2-x+3)^(1/2)-48548610*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^
2+3303720*x^3*(-10*x^2-x+3)^(1/2)+3348180*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-2051645
4*x^2*(-10*x^2-x+3)^(1/2)+6696360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-553420*x*(-10*x^2
-x+3)^(1/2)+4232928*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3/(2*x-1)^2/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 2.84503, size = 324, normalized size = 1.87 \begin{align*} \frac{25365}{268912} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{84925 \, x}{316932 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{131015}{633864 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{375 \, x}{1372 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} - \frac{1}{189 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{11}{196 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{377}{3528 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{3215}{74088 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25365/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 84925/316932*x/sqrt(-10*x^2 - x + 3)
+ 131015/633864/sqrt(-10*x^2 - x + 3) + 375/1372*x/(-10*x^2 - x + 3)^(3/2) - 1/189/(27*(-10*x^2 - x + 3)^(3/2)
*x^3 + 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 11/196/(9*
(-10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) - 377/3528/(3*(-10*x^2
- x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 3215/74088/(-10*x^2 - x + 3)^(3/2)

________________________________________________________________________________________

Fricas [A]  time = 1.55356, size = 402, normalized size = 2.32 \begin{align*} -\frac{837045 \, \sqrt{7}{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, 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 ) - 14 \,{\left (1834380 \, x^{4} + 235980 \, x^{3} - 1465461 \, x^{2} - 39530 \, x + 302352\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{8874096 \,{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8874096*(837045*sqrt(7)*(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqr
t(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1834380*x^4 + 235980*x^3 - 1465461*x^2 - 39530*x + 302352)*s
qrt(5*x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

________________________________________________________________________________________

Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [B]  time = 3.95186, size = 482, normalized size = 2.79 \begin{align*} \frac{5073}{537824} \, \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{32 \,{\left (361 \, \sqrt{5}{\left (5 \, x + 3\right )} - 2178 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{13865775 \,{\left (2 \, x - 1\right )}^{2}} - \frac{297 \,{\left (603 \, \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} - 235200 \, \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} - 37240000 \, \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 )}}{67228 \,{\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((3+5*x)^(1/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

5073/537824*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)))) - 32/13865775*(361*sqrt(5)*(5*x + 3) - 2178*sqrt(5)
)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 297/67228*(603*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 235200*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 - 37240000*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