3.2636 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=181 \[ \frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{5 x+3}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{538245 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-3715/(3234*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 40765/(83006*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (34551425*Sqrt[1
- 2*x])/(5478396*(3 + 5*x)^(3/2)) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 111/(28*(1 - 2*x)^(3/
2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + (3443814775*Sqrt[1 - 2*x])/(60262356*Sqrt[3 + 5*x]) - (538245*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.0715641, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ \frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{5 x+3}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (5 x+3)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (3 x+2) (5 x+3)^{3/2}}-\frac{3715}{3234 (1-2 x)^{3/2} (5 x+3)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{538245 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-3715/(3234*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - 40765/(83006*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)) - (34551425*Sqrt[1
- 2*x])/(5478396*(3 + 5*x)^(3/2)) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + 111/(28*(1 - 2*x)^(3/
2)*(2 + 3*x)*(3 + 5*x)^(3/2)) + (3443814775*Sqrt[1 - 2*x])/(60262356*Sqrt[3 + 5*x]) - (538245*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2}} \, dx &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{1}{14} \int \frac{\frac{59}{2}-150 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac{1}{98} \int \frac{\frac{5075}{4}-15540 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac{\int \frac{-\frac{1484385}{8}+\frac{1170225 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}} \, dx}{11319}\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac{2 \int \frac{\frac{83479305}{16}-\frac{12840975 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx}{871563}\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}-\frac{4 \int \frac{\frac{9239846595}{32}-\frac{2176739775 x}{8}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{28761579}\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{3+5 x}}+\frac{8 \int \frac{496468037835}{64 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{316377369}\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{3+5 x}}+\frac{538245 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{3+5 x}}+\frac{538245 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=-\frac{3715}{3234 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{40765}{83006 \sqrt{1-2 x} (3+5 x)^{3/2}}-\frac{34551425 \sqrt{1-2 x}}{5478396 (3+5 x)^{3/2}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}+\frac{111}{28 (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}}+\frac{3443814775 \sqrt{1-2 x}}{60262356 \sqrt{3+5 x}}-\frac{538245 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.109298, size = 89, normalized size = 0.49 \[ \frac{619886659500 x^5+564878517900 x^4-276089438305 x^3-297937101390 x^2+28838387211 x+39900939556}{60262356 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{3/2}}-\frac{538245 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(39900939556 + 28838387211*x - 297937101390*x^2 - 276089438305*x^3 + 564878517900*x^4 + 619886659500*x^5)/(602
62356*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2)) - (538245*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1
372*Sqrt[7])

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Maple [B]  time = 0.017, size = 353, normalized size = 2. \begin{align*}{\frac{1}{843672984\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 21277201621500\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\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}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-16123390562070\,\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}-5366583075645\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-3865252136270\,{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-4171119419460\,{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 ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/843672984*(1-2*x)^(1/2)*(21277201621500*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^6+32625
042486300*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+2576905529715*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+8678413233000*x^5*(-10*x^2-x+3)^(1/2)-16123390562070*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+7908299250600*x^4*(-10*x^2-x+3)^(1/2)-5366583075645*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2-3865252136270*x^3*(-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))*x-4171119419460*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))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2
-x+3)^(1/2))/(2+3*x)^2/(2*x-1)^2/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [A]  time = 1.92062, size = 232, normalized size = 1.28 \begin{align*} \frac{538245}{19208} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{3443814775 \, x}{30131178 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3595841045}{60262356 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1022125 \, x}{35574 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{3}{14 \,{\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{111}{28 \,{\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{1103855}{71148 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

538245/19208*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 3443814775/30131178*x/sqrt(-10*x^2 -
x + 3) + 3595841045/60262356/sqrt(-10*x^2 - x + 3) + 1022125/35574*x/(-10*x^2 - x + 3)^(3/2) + 3/14/(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)) + 111/28/(3*(-10*x^2 - x + 3)
^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1103855/71148/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.81896, size = 514, normalized size = 2.84 \begin{align*} -\frac{23641335135 \, \sqrt{7}{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\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 (619886659500 \, x^{5} + 564878517900 \, x^{4} - 276089438305 \, x^{3} - 297937101390 \, x^{2} + 28838387211 \, x + 39900939556\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{843672984 \,{\left (900 \, x^{6} + 1380 \, x^{5} + 109 \, x^{4} - 682 \, x^{3} - 227 \, x^{2} + 84 \, x + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/843672984*(23641335135*sqrt(7)*(900*x^6 + 1380*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)*arctan(1/14*s
qrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(619886659500*x^5 + 564878517900*x^4 -
276089438305*x^3 - 297937101390*x^2 + 28838387211*x + 39900939556)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(900*x^6 + 13
80*x^5 + 109*x^4 - 682*x^3 - 227*x^2 + 84*x + 36)

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

[Out]

Timed out

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Giac [B]  time = 3.76342, size = 562, normalized size = 3.1 \begin{align*} -\frac{625}{702768} \, \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} + \frac{107649}{38416} \, \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{58125}{29282} \, \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 )} - \frac{128 \,{\left (577 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3366 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{2636478075 \,{\left (2 \, x - 1\right )}^{2}} + \frac{8019 \,{\left (159 \, \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} + 38360 \, \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 )}}{4802 \,{\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-625/702768*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 + 107649/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*s
qrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 58125/29282*sqrt(10)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 128/
2636478075*(577*sqrt(5)*(5*x + 3) - 3366*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 + 8019/4802*(159*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)))^3 + 38360*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)^2