3.2455 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{5 x+3}}+\frac{7843 \sqrt{1-2 x}}{24 (3 x+2) (5 x+3)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{196735 \sqrt{1-2 x}}{72 (5 x+3)^{3/2}}-\frac{1361195 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

[Out]

(-196735*Sqrt[1 - 2*x])/(72*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (77*Sqrt[
1 - 2*x])/(4*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (7843*Sqrt[1 - 2*x])/(24*(2 + 3*x)*(3 + 5*x)^(3/2)) + (1784635*Sqr
t[1 - 2*x])/(72*Sqrt[3 + 5*x]) - (1361195*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Rubi [A]  time = 0.0595878, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {98, 149, 151, 152, 12, 93, 204} \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{5 x+3}}+\frac{7843 \sqrt{1-2 x}}{24 (3 x+2) (5 x+3)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (3 x+2)^2 (5 x+3)^{3/2}}-\frac{196735 \sqrt{1-2 x}}{72 (5 x+3)^{3/2}}-\frac{1361195 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-196735*Sqrt[1 - 2*x])/(72*(3 + 5*x)^(3/2)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (77*Sqrt[
1 - 2*x])/(4*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (7843*Sqrt[1 - 2*x])/(24*(2 + 3*x)*(3 + 5*x)^(3/2)) + (1784635*Sqr
t[1 - 2*x])/(72*Sqrt[3 + 5*x]) - (1361195*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*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 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 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{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{1}{9} \int \frac{\left (\frac{429}{2}-198 x\right ) \sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}-\frac{1}{54} \int \frac{-\frac{83655}{4}+30393 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{7843 \sqrt{1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}-\frac{1}{378} \int \frac{-\frac{15408855}{8}+2470545 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{196735 \sqrt{1-2 x}}{72 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{7843 \sqrt{1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac{\int \frac{-\frac{1739225565}{16}+\frac{409012065 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{6237}\\ &=-\frac{196735 \sqrt{1-2 x}}{72 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{7843 \sqrt{1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{3+5 x}}-\frac{2 \int -\frac{93387505365}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{68607}\\ &=-\frac{196735 \sqrt{1-2 x}}{72 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{7843 \sqrt{1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{3+5 x}}+\frac{1361195}{16} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{196735 \sqrt{1-2 x}}{72 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{7843 \sqrt{1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{3+5 x}}+\frac{1361195}{8} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{196735 \sqrt{1-2 x}}{72 (3+5 x)^{3/2}}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{77 \sqrt{1-2 x}}{4 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{7843 \sqrt{1-2 x}}{24 (2+3 x) (3+5 x)^{3/2}}+\frac{1784635 \sqrt{1-2 x}}{72 \sqrt{3+5 x}}-\frac{1361195 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{8 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0706463, size = 84, normalized size = 0.51 \[ \frac{\sqrt{1-2 x} \left (80308575 x^4+207031680 x^3+199977747 x^2+85776638 x+13784768\right )}{24 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{1361195 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(13784768 + 85776638*x + 199977747*x^2 + 207031680*x^3 + 80308575*x^4))/(24*(2 + 3*x)^3*(3 + 5*
x)^(3/2)) - (1361195*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(8*Sqrt[7])

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Maple [B]  time = 0.013, size = 298, normalized size = 1.8 \begin{align*}{\frac{1}{336\, \left ( 2+3\,x \right ) ^{3}} \left ( 2756419875\,\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]\,\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}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \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+2799688458\,{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 ) \sqrt{1-2\,x}{\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((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^(5/2),x)

[Out]

1/336*(2756419875*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+8820543600*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+11282945355*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^3+1124320050*x^4*(-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^2+2898443520*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+2799688458*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))+
1200872932*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(2+3*x)^3/(-10*x^2-x+3)^(1/2)/(3
+5*x)^(3/2)

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Maxima [A]  time = 2.11552, size = 324, normalized size = 1.95 \begin{align*} \frac{1361195}{112} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{1784635 \, x}{36 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1863329}{72 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{149501 \, x}{12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{2401}{243 \,{\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{31213}{324 \,{\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{1115681}{648 \,{\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{13081615}{1944 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1361195/112*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1784635/36*x/sqrt(-10*x^2 - x + 3) + 1
863329/72/sqrt(-10*x^2 - x + 3) + 149501/12*x/(-10*x^2 - x + 3)^(3/2) + 2401/243/(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)) + 31213/324/(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)) + 1115681/648/(3*(-10
*x^2 - x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 13081615/1944/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.90492, size = 435, normalized size = 2.62 \begin{align*} -\frac{4083585 \, \sqrt{7}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 (80308575 \, x^{4} + 207031680 \, x^{3} + 199977747 \, x^{2} + 85776638 \, x + 13784768\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{336 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/336*(4083585*sqrt(7)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37*x + 20
)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(80308575*x^4 + 207031680*x^3 + 199977747*x^2 + 85776638
*x + 13784768)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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

[Out]

Timed out

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Giac [B]  time = 3.48099, size = 591, normalized size = 3.56 \begin{align*} -\frac{11}{48} \, \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{272239}{224} \, \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 )} + 748 \, \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{11 \,{\left (63359 \, \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} + 30251200 \, \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} + 3730664000 \, \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 )}}{4 \,{\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)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-11/48*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 + 272239/224*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)))) + 748*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))) + 11/4*(63359*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 + 30251200*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 + 3730664000*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