3.2597 \(\int \frac{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx\)

Optimal. Leaf size=173 \[ \frac{465 \sqrt{5 x+3}}{9604 \sqrt{1-2 x}}-\frac{85 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{23 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{32 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{9395 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

[Out]

(465*Sqrt[3 + 5*x])/(9604*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (32*Sqrt[3 +
5*x])/(147*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (23*Sqrt[3 + 5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (85*Sqrt[3 + 5*x]
)/(2744*Sqrt[1 - 2*x]*(2 + 3*x)) - (9395*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

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Rubi [A]  time = 0.0621775, 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 = {98, 151, 152, 12, 93, 204} \[ \frac{465 \sqrt{5 x+3}}{9604 \sqrt{1-2 x}}-\frac{85 \sqrt{5 x+3}}{2744 \sqrt{1-2 x} (3 x+2)}-\frac{23 \sqrt{5 x+3}}{196 \sqrt{1-2 x} (3 x+2)^2}-\frac{32 \sqrt{5 x+3}}{147 \sqrt{1-2 x} (3 x+2)^3}+\frac{11 \sqrt{5 x+3}}{21 (1-2 x)^{3/2} (3 x+2)^3}-\frac{9395 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{19208 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(465*Sqrt[3 + 5*x])/(9604*Sqrt[1 - 2*x]) + (11*Sqrt[3 + 5*x])/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^3) - (32*Sqrt[3 +
5*x])/(147*Sqrt[1 - 2*x]*(2 + 3*x)^3) - (23*Sqrt[3 + 5*x])/(196*Sqrt[1 - 2*x]*(2 + 3*x)^2) - (85*Sqrt[3 + 5*x]
)/(2744*Sqrt[1 - 2*x]*(2 + 3*x)) - (9395*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(19208*Sqrt[7])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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{(3+5 x)^{3/2}}{(1-2 x)^{5/2} (2+3 x)^4} \, dx &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{1}{21} \int \frac{-233-\frac{795 x}{2}}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{1}{441} \int \frac{-\frac{3357}{2}-2880 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{23 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{\int \frac{-\frac{36855}{4}-14490 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{6174}\\ &=\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{23 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{85 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}-\frac{\int \frac{-\frac{268695}{8}-\frac{26775 x}{2}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{43218}\\ &=\frac{465 \sqrt{3+5 x}}{9604 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{23 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{85 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}+\frac{\int \frac{6510735}{16 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1663893}\\ &=\frac{465 \sqrt{3+5 x}}{9604 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{23 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{85 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}+\frac{9395 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{38416}\\ &=\frac{465 \sqrt{3+5 x}}{9604 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{23 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{85 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}+\frac{9395 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{19208}\\ &=\frac{465 \sqrt{3+5 x}}{9604 \sqrt{1-2 x}}+\frac{11 \sqrt{3+5 x}}{21 (1-2 x)^{3/2} (2+3 x)^3}-\frac{32 \sqrt{3+5 x}}{147 \sqrt{1-2 x} (2+3 x)^3}-\frac{23 \sqrt{3+5 x}}{196 \sqrt{1-2 x} (2+3 x)^2}-\frac{85 \sqrt{3+5 x}}{2744 \sqrt{1-2 x} (2+3 x)}-\frac{9395 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{19208 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.069942, size = 100, normalized size = 0.58 \[ -\frac{7 \sqrt{5 x+3} \left (150660 x^4+193860 x^3-17127 x^2-80510 x-19296\right )-28185 \sqrt{7-14 x} (2 x-1) (3 x+2)^3 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{403368 (1-2 x)^{3/2} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(7*Sqrt[3 + 5*x]*(-19296 - 80510*x - 17127*x^2 + 193860*x^3 + 150660*x^4) - 28185*Sqrt[7 - 14*x]*(-1 + 2*x)*(
2 + 3*x)^3*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(403368*(1 - 2*x)^(3/2)*(2 + 3*x)^3)

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Maple [B]  time = 0.016, size = 305, normalized size = 1.8 \begin{align*}{\frac{1}{806736\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) ^{2}} \left ( 3043980\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+3043980\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-1268325\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-2109240\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-1634730\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-2714040\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+112740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+239778\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+225480\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1127140\,x\sqrt{-10\,{x}^{2}-x+3}+270144\,\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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x)

[Out]

1/806736*(3043980*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+3043980*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-1268325*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x
^3-2109240*x^4*(-10*x^2-x+3)^(1/2)-1634730*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-2714
040*x^3*(-10*x^2-x+3)^(1/2)+112740*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+239778*x^2*(-1
0*x^2-x+3)^(1/2)+225480*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+1127140*x*(-10*x^2-x+3)^(1/
2)+270144*(-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)

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Maxima [A]  time = 3.26314, size = 324, normalized size = 1.87 \begin{align*} \frac{9395}{268912} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{2325 \, x}{9604 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{5395}{57624 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{7625 \, x}{12348 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{1}{567 \,{\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{169}{5292 \,{\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{1987}{10584 \,{\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{2165}{222264 \,{\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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

9395/268912*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2325/9604*x/sqrt(-10*x^2 - x + 3) + 53
95/57624/sqrt(-10*x^2 - x + 3) + 7625/12348*x/(-10*x^2 - x + 3)^(3/2) + 1/567/(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)) - 169/5292/(9*(-1
0*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)) + 1987/10584/(3*(-10*x^2
- x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) + 2165/222264/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.56737, size = 394, normalized size = 2.28 \begin{align*} -\frac{28185 \, \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 (150660 \, x^{4} + 193860 \, x^{3} - 17127 \, x^{2} - 80510 \, x - 19296\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{806736 \,{\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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/806736*(28185*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)*sqrt(
5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 14*(150660*x^4 + 193860*x^3 - 17127*x^2 - 80510*x - 19296)*sqrt(5*
x + 3)*sqrt(-2*x + 1))/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 5.24495, size = 482, normalized size = 2.79 \begin{align*} \frac{1879}{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{8 \,{\left (512 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3201 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1260525 \,{\left (2 \, x - 1\right )}^{2}} - \frac{99 \,{\left (727 \, \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} + 548800 \, \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} + 20776000 \, \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)^(3/2)/(1-2*x)^(5/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

1879/537824*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)))) - 8/1260525*(512*sqrt(5)*(5*x + 3) - 3201*sqrt(5))*
sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 99/67228*(727*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 + 548800*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 + 20776000*sqrt(10)*((sqr
t(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqr
t(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