3.265 \(\int \frac{1}{(a+\frac{b}{x})^{5/2} (c+\frac{d}{x})^3} \, dx\)

Optimal. Leaf size=409 \[ \frac{d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{4 a c^3 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{b \left (24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4-56 a b^3 c^3 d+20 b^4 c^4\right )}{4 a^3 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^4}+\frac{b \left (87 a^2 b c d^2-36 a^3 d^3-36 a b^2 c^2 d+20 b^3 c^3\right )}{12 a^2 c^3 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^3}-\frac{d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{9/2}}-\frac{(6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^4}+\frac{d (2 b c-3 a d)}{2 a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2} \]

[Out]

(b*(20*b^3*c^3 - 36*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 36*a^3*d^3))/(12*a^2*c^3*(b*c - a*d)^3*(a + b/x)^(3/2)) + (
b*(20*b^4*c^4 - 56*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 - 35*a^3*b*c*d^3 + 12*a^4*d^4))/(4*a^3*c^3*(b*c - a*d)^4*S
qrt[a + b/x]) + (d*(2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)^2) + (d*(4*b^2*c^2 - 23*a*b
*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*(a + b/x)^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)^2) -
(d^(7/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c -
a*d)^(9/2)) - ((5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^4)

________________________________________________________________________________________

Rubi [A]  time = 0.702414, antiderivative size = 409, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {375, 103, 151, 152, 156, 63, 208, 205} \[ \frac{d \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )}{4 a c^3 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{b \left (24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4-56 a b^3 c^3 d+20 b^4 c^4\right )}{4 a^3 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^4}+\frac{b \left (87 a^2 b c d^2-36 a^3 d^3-36 a b^2 c^2 d+20 b^3 c^3\right )}{12 a^2 c^3 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^3}-\frac{d^{7/2} \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{9/2}}-\frac{(6 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^4}+\frac{d (2 b c-3 a d)}{2 a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x)^(5/2)*(c + d/x)^3),x]

[Out]

(b*(20*b^3*c^3 - 36*a*b^2*c^2*d + 87*a^2*b*c*d^2 - 36*a^3*d^3))/(12*a^2*c^3*(b*c - a*d)^3*(a + b/x)^(3/2)) + (
b*(20*b^4*c^4 - 56*a*b^3*c^3*d + 24*a^2*b^2*c^2*d^2 - 35*a^3*b*c*d^3 + 12*a^4*d^4))/(4*a^3*c^3*(b*c - a*d)^4*S
qrt[a + b/x]) + (d*(2*b*c - 3*a*d))/(2*a*c^2*(b*c - a*d)*(a + b/x)^(3/2)*(c + d/x)^2) + (d*(4*b^2*c^2 - 23*a*b
*c*d + 12*a^2*d^2))/(4*a*c^3*(b*c - a*d)^2*(a + b/x)^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)^2) -
(d^(7/2)*(99*b^2*c^2 - 88*a*b*c*d + 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c -
a*d)^(9/2)) - ((5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^4)

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

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 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{5/2} \left (c+\frac{d}{x}\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^{5/2} (c+d x)^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{\frac{1}{2} (5 b c+6 a d)+\frac{9 b d x}{2}}{x (a+b x)^{5/2} (c+d x)^3} \, dx,x,\frac{1}{x}\right )}{a c}\\ &=\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-(b c-a d) (5 b c+6 a d)-\frac{7}{2} b d (2 b c-3 a d) x}{x (a+b x)^{5/2} (c+d x)^2} \, dx,x,\frac{1}{x}\right )}{2 a c^2 (b c-a d)}\\ &=\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{(b c-a d)^2 (5 b c+6 a d)+\frac{5}{4} b d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right ) x}{x (a+b x)^{5/2} (c+d x)} \, dx,x,\frac{1}{x}\right )}{2 a c^3 (b c-a d)^2}\\ &=\frac{b \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right )}{12 a^2 c^3 (b c-a d)^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{\frac{3}{2} (b c-a d)^3 (5 b c+6 a d)+\frac{3}{8} b d \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right ) x}{x (a+b x)^{3/2} (c+d x)} \, dx,x,\frac{1}{x}\right )}{3 a^2 c^3 (b c-a d)^3}\\ &=\frac{b \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right )}{12 a^2 c^3 (b c-a d)^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{b \left (20 b^4 c^4-56 a b^3 c^3 d+24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4\right )}{4 a^3 c^3 (b c-a d)^4 \sqrt{a+\frac{b}{x}}}+\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{3}{4} (b c-a d)^4 (5 b c+6 a d)+\frac{3}{16} b d \left (20 b^4 c^4-56 a b^3 c^3 d+24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4\right ) x}{x \sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{3 a^3 c^3 (b c-a d)^4}\\ &=\frac{b \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right )}{12 a^2 c^3 (b c-a d)^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{b \left (20 b^4 c^4-56 a b^3 c^3 d+24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4\right )}{4 a^3 c^3 (b c-a d)^4 \sqrt{a+\frac{b}{x}}}+\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{(5 b c+6 a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a^3 c^4}-\frac{\left (d^4 \left (99 b^2 c^2-88 a b c d+24 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{8 c^4 (b c-a d)^4}\\ &=\frac{b \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right )}{12 a^2 c^3 (b c-a d)^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{b \left (20 b^4 c^4-56 a b^3 c^3 d+24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4\right )}{4 a^3 c^3 (b c-a d)^4 \sqrt{a+\frac{b}{x}}}+\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{(5 b c+6 a d) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{a^3 b c^4}-\frac{\left (d^4 \left (99 b^2 c^2-88 a b c d+24 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{a d}{b}+\frac{d x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 b c^4 (b c-a d)^4}\\ &=\frac{b \left (20 b^3 c^3-36 a b^2 c^2 d+87 a^2 b c d^2-36 a^3 d^3\right )}{12 a^2 c^3 (b c-a d)^3 \left (a+\frac{b}{x}\right )^{3/2}}+\frac{b \left (20 b^4 c^4-56 a b^3 c^3 d+24 a^2 b^2 c^2 d^2-35 a^3 b c d^3+12 a^4 d^4\right )}{4 a^3 c^3 (b c-a d)^4 \sqrt{a+\frac{b}{x}}}+\frac{d (2 b c-3 a d)}{2 a c^2 (b c-a d) \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}+\frac{d \left (4 b^2 c^2-23 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )^2}-\frac{d^{7/2} \left (99 b^2 c^2-88 a b c d+24 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{9/2}}-\frac{(5 b c+6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^4}\\ \end{align*}

Mathematica [C]  time = 0.214204, size = 239, normalized size = 0.58 \[ \frac{(c x+d) \left (2 (c x+d) \left (\frac{1}{4} a^2 d^2 \left (24 a^2 d^2-88 a b c d+99 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{d \left (a+\frac{b}{x}\right )}{a d-b c}\right )+(6 a d+5 b c) (b c-a d)^3 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right )\right )-\frac{3}{2} a c d x (a d-b c) \left (12 a^2 d^2-23 a b c d+4 b^2 c^2\right )\right )+6 a c^3 x^3 (b c-a d)^3-3 a c^2 d x^2 (b c-a d)^2 (3 a d-2 b c)}{6 a^2 c^4 \left (a+\frac{b}{x}\right )^{3/2} (c x+d)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x)^(5/2)*(c + d/x)^3),x]

[Out]

(-3*a*c^2*d*(b*c - a*d)^2*(-2*b*c + 3*a*d)*x^2 + 6*a*c^3*(b*c - a*d)^3*x^3 + (d + c*x)*((-3*a*c*d*(-(b*c) + a*
d)*(4*b^2*c^2 - 23*a*b*c*d + 12*a^2*d^2)*x)/2 + 2*(d + c*x)*((a^2*d^2*(99*b^2*c^2 - 88*a*b*c*d + 24*a^2*d^2)*H
ypergeometric2F1[-3/2, 1, -1/2, (d*(a + b/x))/(-(b*c) + a*d)])/4 + (b*c - a*d)^3*(5*b*c + 6*a*d)*Hypergeometri
c2F1[-3/2, 1, -1/2, 1 + b/(a*x)])))/(6*a^2*c^4*(b*c - a*d)^3*(a + b/x)^(3/2)*(d + c*x)^2)

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Maple [B]  time = 0.019, size = 7300, normalized size = 17.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x)^(5/2)/(c+d/x)^3,x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x}\right )}^{\frac{5}{2}}{\left (c + \frac{d}{x}\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="maxima")

[Out]

integrate(1/((a + b/x)^(5/2)*(c + d/x)^3), x)

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Fricas [B]  time = 31.4349, size = 12556, normalized size = 30.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="fricas")

[Out]

[1/24*(12*(5*b^7*c^5*d^2 - 14*a*b^6*c^4*d^3 + 6*a^2*b^5*c^3*d^4 + 16*a^3*b^4*c^2*d^5 - 19*a^4*b^3*c*d^6 + 6*a^
5*b^2*d^7 + (5*a^2*b^5*c^7 - 14*a^3*b^4*c^6*d + 6*a^4*b^3*c^5*d^2 + 16*a^5*b^2*c^4*d^3 - 19*a^6*b*c^3*d^4 + 6*
a^7*c^2*d^5)*x^4 + 2*(5*a*b^6*c^7 - 9*a^2*b^5*c^6*d - 8*a^3*b^4*c^5*d^2 + 22*a^4*b^3*c^4*d^3 - 3*a^5*b^2*c^3*d
^4 - 13*a^6*b*c^2*d^5 + 6*a^7*c*d^6)*x^3 + (5*b^7*c^7 + 6*a*b^6*c^6*d - 45*a^2*b^5*c^5*d^2 + 26*a^3*b^4*c^4*d^
3 + 51*a^4*b^3*c^3*d^4 - 54*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + 6*a^7*d^7)*x^2 + 2*(5*b^7*c^6*d - 9*a*b^6*c^5*d^
2 - 8*a^2*b^5*c^4*d^3 + 22*a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 - 13*a^5*b^2*c*d^6 + 6*a^6*b*d^7)*x)*sqrt(a)*lo
g(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 3*(99*a^4*b^4*c^2*d^5 - 88*a^5*b^3*c*d^6 + 24*a^6*b^2*d^7 + (99
*a^6*b^2*c^4*d^3 - 88*a^7*b*c^3*d^4 + 24*a^8*c^2*d^5)*x^4 + 2*(99*a^5*b^3*c^4*d^3 + 11*a^6*b^2*c^3*d^4 - 64*a^
7*b*c^2*d^5 + 24*a^8*c*d^6)*x^3 + (99*a^4*b^4*c^4*d^3 + 308*a^5*b^3*c^3*d^4 - 229*a^6*b^2*c^2*d^5 + 8*a^7*b*c*
d^6 + 24*a^8*d^7)*x^2 + 2*(99*a^4*b^4*c^3*d^4 + 11*a^5*b^3*c^2*d^5 - 64*a^6*b^2*c*d^6 + 24*a^7*b*d^7)*x)*sqrt(
-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c - 2*a*d)*x)/(c*x + d
)) + 2*(12*(a^3*b^4*c^7 - 4*a^4*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d^3 + a^7*c^3*d^4)*x^5 + (80*a^2*b
^5*c^7 - 200*a^3*b^4*c^6*d + 48*a^4*b^3*c^5*d^2 + 48*a^5*b^2*c^4*d^3 - 135*a^6*b*c^3*d^4 + 54*a^7*c^2*d^5)*x^4
+ (60*a*b^6*c^7 - 8*a^2*b^5*c^6*d - 364*a^3*b^4*c^5*d^2 + 192*a^4*b^3*c^4*d^3 - 234*a^5*b^2*c^3*d^4 + 3*a^6*b
*c^2*d^5 + 36*a^7*c*d^6)*x^3 + (120*a*b^6*c^6*d - 256*a^2*b^5*c^5*d^2 - 80*a^3*b^4*c^4*d^3 - 15*a^4*b^3*c^3*d^
4 - 156*a^5*b^2*c^2*d^5 + 72*a^6*b*c*d^6)*x^2 + 3*(20*a*b^6*c^5*d^2 - 56*a^2*b^5*c^4*d^3 + 24*a^3*b^4*c^3*d^4
- 35*a^4*b^3*c^2*d^5 + 12*a^5*b^2*c*d^6)*x)*sqrt((a*x + b)/x))/(a^4*b^6*c^8*d^2 - 4*a^5*b^5*c^7*d^3 + 6*a^6*b^
4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4*d^6 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d + 6*a^8*b^2*c^8*d^2 - 4*a^9*
b*c^7*d^3 + a^10*c^6*d^4)*x^4 + 2*(a^5*b^5*c^10 - 3*a^6*b^4*c^9*d + 2*a^7*b^3*c^8*d^2 + 2*a^8*b^2*c^7*d^3 - 3*
a^9*b*c^6*d^4 + a^10*c^5*d^5)*x^3 + (a^4*b^6*c^10 - 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*c^7*d^3 - 9*a^8*b^2*c^6*d^4
+ a^10*c^4*d^6)*x^2 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7*b^3*c^6*d^4 - 3*a^8*b^
2*c^5*d^5 + a^9*b*c^4*d^6)*x), 1/24*(24*(5*b^7*c^5*d^2 - 14*a*b^6*c^4*d^3 + 6*a^2*b^5*c^3*d^4 + 16*a^3*b^4*c^2
*d^5 - 19*a^4*b^3*c*d^6 + 6*a^5*b^2*d^7 + (5*a^2*b^5*c^7 - 14*a^3*b^4*c^6*d + 6*a^4*b^3*c^5*d^2 + 16*a^5*b^2*c
^4*d^3 - 19*a^6*b*c^3*d^4 + 6*a^7*c^2*d^5)*x^4 + 2*(5*a*b^6*c^7 - 9*a^2*b^5*c^6*d - 8*a^3*b^4*c^5*d^2 + 22*a^4
*b^3*c^4*d^3 - 3*a^5*b^2*c^3*d^4 - 13*a^6*b*c^2*d^5 + 6*a^7*c*d^6)*x^3 + (5*b^7*c^7 + 6*a*b^6*c^6*d - 45*a^2*b
^5*c^5*d^2 + 26*a^3*b^4*c^4*d^3 + 51*a^4*b^3*c^3*d^4 - 54*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + 6*a^7*d^7)*x^2 + 2
*(5*b^7*c^6*d - 9*a*b^6*c^5*d^2 - 8*a^2*b^5*c^4*d^3 + 22*a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 - 13*a^5*b^2*c*d^
6 + 6*a^6*b*d^7)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + 3*(99*a^4*b^4*c^2*d^5 - 88*a^5*b^3*c*d^6 +
24*a^6*b^2*d^7 + (99*a^6*b^2*c^4*d^3 - 88*a^7*b*c^3*d^4 + 24*a^8*c^2*d^5)*x^4 + 2*(99*a^5*b^3*c^4*d^3 + 11*a^
6*b^2*c^3*d^4 - 64*a^7*b*c^2*d^5 + 24*a^8*c*d^6)*x^3 + (99*a^4*b^4*c^4*d^3 + 308*a^5*b^3*c^3*d^4 - 229*a^6*b^2
*c^2*d^5 + 8*a^7*b*c*d^6 + 24*a^8*d^7)*x^2 + 2*(99*a^4*b^4*c^3*d^4 + 11*a^5*b^3*c^2*d^5 - 64*a^6*b^2*c*d^6 + 2
4*a^7*b*d^7)*x)*sqrt(-d/(b*c - a*d))*log(-(2*(b*c - a*d)*x*sqrt(-d/(b*c - a*d))*sqrt((a*x + b)/x) - b*d + (b*c
- 2*a*d)*x)/(c*x + d)) + 2*(12*(a^3*b^4*c^7 - 4*a^4*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d^3 + a^7*c^3
*d^4)*x^5 + (80*a^2*b^5*c^7 - 200*a^3*b^4*c^6*d + 48*a^4*b^3*c^5*d^2 + 48*a^5*b^2*c^4*d^3 - 135*a^6*b*c^3*d^4
+ 54*a^7*c^2*d^5)*x^4 + (60*a*b^6*c^7 - 8*a^2*b^5*c^6*d - 364*a^3*b^4*c^5*d^2 + 192*a^4*b^3*c^4*d^3 - 234*a^5*
b^2*c^3*d^4 + 3*a^6*b*c^2*d^5 + 36*a^7*c*d^6)*x^3 + (120*a*b^6*c^6*d - 256*a^2*b^5*c^5*d^2 - 80*a^3*b^4*c^4*d^
3 - 15*a^4*b^3*c^3*d^4 - 156*a^5*b^2*c^2*d^5 + 72*a^6*b*c*d^6)*x^2 + 3*(20*a*b^6*c^5*d^2 - 56*a^2*b^5*c^4*d^3
+ 24*a^3*b^4*c^3*d^4 - 35*a^4*b^3*c^2*d^5 + 12*a^5*b^2*c*d^6)*x)*sqrt((a*x + b)/x))/(a^4*b^6*c^8*d^2 - 4*a^5*b
^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4*d^6 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d + 6*a^8
*b^2*c^8*d^2 - 4*a^9*b*c^7*d^3 + a^10*c^6*d^4)*x^4 + 2*(a^5*b^5*c^10 - 3*a^6*b^4*c^9*d + 2*a^7*b^3*c^8*d^2 + 2
*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*d^4 + a^10*c^5*d^5)*x^3 + (a^4*b^6*c^10 - 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*c^7*d^
3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^6)*x^2 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7*b
^3*c^6*d^4 - 3*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6)*x), -1/12*(3*(99*a^4*b^4*c^2*d^5 - 88*a^5*b^3*c*d^6 + 24*a^6*b
^2*d^7 + (99*a^6*b^2*c^4*d^3 - 88*a^7*b*c^3*d^4 + 24*a^8*c^2*d^5)*x^4 + 2*(99*a^5*b^3*c^4*d^3 + 11*a^6*b^2*c^3
*d^4 - 64*a^7*b*c^2*d^5 + 24*a^8*c*d^6)*x^3 + (99*a^4*b^4*c^4*d^3 + 308*a^5*b^3*c^3*d^4 - 229*a^6*b^2*c^2*d^5
+ 8*a^7*b*c*d^6 + 24*a^8*d^7)*x^2 + 2*(99*a^4*b^4*c^3*d^4 + 11*a^5*b^3*c^2*d^5 - 64*a^6*b^2*c*d^6 + 24*a^7*b*d
^7)*x)*sqrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 6*(5*b
^7*c^5*d^2 - 14*a*b^6*c^4*d^3 + 6*a^2*b^5*c^3*d^4 + 16*a^3*b^4*c^2*d^5 - 19*a^4*b^3*c*d^6 + 6*a^5*b^2*d^7 + (5
*a^2*b^5*c^7 - 14*a^3*b^4*c^6*d + 6*a^4*b^3*c^5*d^2 + 16*a^5*b^2*c^4*d^3 - 19*a^6*b*c^3*d^4 + 6*a^7*c^2*d^5)*x
^4 + 2*(5*a*b^6*c^7 - 9*a^2*b^5*c^6*d - 8*a^3*b^4*c^5*d^2 + 22*a^4*b^3*c^4*d^3 - 3*a^5*b^2*c^3*d^4 - 13*a^6*b*
c^2*d^5 + 6*a^7*c*d^6)*x^3 + (5*b^7*c^7 + 6*a*b^6*c^6*d - 45*a^2*b^5*c^5*d^2 + 26*a^3*b^4*c^4*d^3 + 51*a^4*b^3
*c^3*d^4 - 54*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + 6*a^7*d^7)*x^2 + 2*(5*b^7*c^6*d - 9*a*b^6*c^5*d^2 - 8*a^2*b^5*
c^4*d^3 + 22*a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 - 13*a^5*b^2*c*d^6 + 6*a^6*b*d^7)*x)*sqrt(a)*log(2*a*x - 2*sq
rt(a)*x*sqrt((a*x + b)/x) + b) - (12*(a^3*b^4*c^7 - 4*a^4*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d^3 + a^
7*c^3*d^4)*x^5 + (80*a^2*b^5*c^7 - 200*a^3*b^4*c^6*d + 48*a^4*b^3*c^5*d^2 + 48*a^5*b^2*c^4*d^3 - 135*a^6*b*c^3
*d^4 + 54*a^7*c^2*d^5)*x^4 + (60*a*b^6*c^7 - 8*a^2*b^5*c^6*d - 364*a^3*b^4*c^5*d^2 + 192*a^4*b^3*c^4*d^3 - 234
*a^5*b^2*c^3*d^4 + 3*a^6*b*c^2*d^5 + 36*a^7*c*d^6)*x^3 + (120*a*b^6*c^6*d - 256*a^2*b^5*c^5*d^2 - 80*a^3*b^4*c
^4*d^3 - 15*a^4*b^3*c^3*d^4 - 156*a^5*b^2*c^2*d^5 + 72*a^6*b*c*d^6)*x^2 + 3*(20*a*b^6*c^5*d^2 - 56*a^2*b^5*c^4
*d^3 + 24*a^3*b^4*c^3*d^4 - 35*a^4*b^3*c^2*d^5 + 12*a^5*b^2*c*d^6)*x)*sqrt((a*x + b)/x))/(a^4*b^6*c^8*d^2 - 4*
a^5*b^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4*d^6 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d +
6*a^8*b^2*c^8*d^2 - 4*a^9*b*c^7*d^3 + a^10*c^6*d^4)*x^4 + 2*(a^5*b^5*c^10 - 3*a^6*b^4*c^9*d + 2*a^7*b^3*c^8*d^
2 + 2*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*d^4 + a^10*c^5*d^5)*x^3 + (a^4*b^6*c^10 - 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*c
^7*d^3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^6)*x^2 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*
a^7*b^3*c^6*d^4 - 3*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6)*x), -1/12*(3*(99*a^4*b^4*c^2*d^5 - 88*a^5*b^3*c*d^6 + 24*
a^6*b^2*d^7 + (99*a^6*b^2*c^4*d^3 - 88*a^7*b*c^3*d^4 + 24*a^8*c^2*d^5)*x^4 + 2*(99*a^5*b^3*c^4*d^3 + 11*a^6*b^
2*c^3*d^4 - 64*a^7*b*c^2*d^5 + 24*a^8*c*d^6)*x^3 + (99*a^4*b^4*c^4*d^3 + 308*a^5*b^3*c^3*d^4 - 229*a^6*b^2*c^2
*d^5 + 8*a^7*b*c*d^6 + 24*a^8*d^7)*x^2 + 2*(99*a^4*b^4*c^3*d^4 + 11*a^5*b^3*c^2*d^5 - 64*a^6*b^2*c*d^6 + 24*a^
7*b*d^7)*x)*sqrt(d/(b*c - a*d))*arctan(-(b*c - a*d)*x*sqrt(d/(b*c - a*d))*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 1
2*(5*b^7*c^5*d^2 - 14*a*b^6*c^4*d^3 + 6*a^2*b^5*c^3*d^4 + 16*a^3*b^4*c^2*d^5 - 19*a^4*b^3*c*d^6 + 6*a^5*b^2*d^
7 + (5*a^2*b^5*c^7 - 14*a^3*b^4*c^6*d + 6*a^4*b^3*c^5*d^2 + 16*a^5*b^2*c^4*d^3 - 19*a^6*b*c^3*d^4 + 6*a^7*c^2*
d^5)*x^4 + 2*(5*a*b^6*c^7 - 9*a^2*b^5*c^6*d - 8*a^3*b^4*c^5*d^2 + 22*a^4*b^3*c^4*d^3 - 3*a^5*b^2*c^3*d^4 - 13*
a^6*b*c^2*d^5 + 6*a^7*c*d^6)*x^3 + (5*b^7*c^7 + 6*a*b^6*c^6*d - 45*a^2*b^5*c^5*d^2 + 26*a^3*b^4*c^4*d^3 + 51*a
^4*b^3*c^3*d^4 - 54*a^5*b^2*c^2*d^5 + 5*a^6*b*c*d^6 + 6*a^7*d^7)*x^2 + 2*(5*b^7*c^6*d - 9*a*b^6*c^5*d^2 - 8*a^
2*b^5*c^4*d^3 + 22*a^3*b^4*c^3*d^4 - 3*a^4*b^3*c^2*d^5 - 13*a^5*b^2*c*d^6 + 6*a^6*b*d^7)*x)*sqrt(-a)*arctan(sq
rt(-a)*sqrt((a*x + b)/x)/a) - (12*(a^3*b^4*c^7 - 4*a^4*b^3*c^6*d + 6*a^5*b^2*c^5*d^2 - 4*a^6*b*c^4*d^3 + a^7*c
^3*d^4)*x^5 + (80*a^2*b^5*c^7 - 200*a^3*b^4*c^6*d + 48*a^4*b^3*c^5*d^2 + 48*a^5*b^2*c^4*d^3 - 135*a^6*b*c^3*d^
4 + 54*a^7*c^2*d^5)*x^4 + (60*a*b^6*c^7 - 8*a^2*b^5*c^6*d - 364*a^3*b^4*c^5*d^2 + 192*a^4*b^3*c^4*d^3 - 234*a^
5*b^2*c^3*d^4 + 3*a^6*b*c^2*d^5 + 36*a^7*c*d^6)*x^3 + (120*a*b^6*c^6*d - 256*a^2*b^5*c^5*d^2 - 80*a^3*b^4*c^4*
d^3 - 15*a^4*b^3*c^3*d^4 - 156*a^5*b^2*c^2*d^5 + 72*a^6*b*c*d^6)*x^2 + 3*(20*a*b^6*c^5*d^2 - 56*a^2*b^5*c^4*d^
3 + 24*a^3*b^4*c^3*d^4 - 35*a^4*b^3*c^2*d^5 + 12*a^5*b^2*c*d^6)*x)*sqrt((a*x + b)/x))/(a^4*b^6*c^8*d^2 - 4*a^5
*b^5*c^7*d^3 + 6*a^6*b^4*c^6*d^4 - 4*a^7*b^3*c^5*d^5 + a^8*b^2*c^4*d^6 + (a^6*b^4*c^10 - 4*a^7*b^3*c^9*d + 6*a
^8*b^2*c^8*d^2 - 4*a^9*b*c^7*d^3 + a^10*c^6*d^4)*x^4 + 2*(a^5*b^5*c^10 - 3*a^6*b^4*c^9*d + 2*a^7*b^3*c^8*d^2 +
2*a^8*b^2*c^7*d^3 - 3*a^9*b*c^6*d^4 + a^10*c^5*d^5)*x^3 + (a^4*b^6*c^10 - 9*a^6*b^4*c^8*d^2 + 16*a^7*b^3*c^7*
d^3 - 9*a^8*b^2*c^6*d^4 + a^10*c^4*d^6)*x^2 + 2*(a^4*b^6*c^9*d - 3*a^5*b^5*c^8*d^2 + 2*a^6*b^4*c^7*d^3 + 2*a^7
*b^3*c^6*d^4 - 3*a^8*b^2*c^5*d^5 + a^9*b*c^4*d^6)*x)]

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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(1/(a+b/x)**(5/2)/(c+d/x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.22993, size = 703, normalized size = 1.72 \begin{align*} -\frac{1}{12} \, b{\left (\frac{3 \,{\left (99 \, b^{2} c^{2} d^{4} - 88 \, a b c d^{5} + 24 \, a^{2} d^{6}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{5} c^{8} - 4 \, a b^{4} c^{7} d + 6 \, a^{2} b^{3} c^{6} d^{2} - 4 \, a^{3} b^{2} c^{5} d^{3} + a^{4} b c^{4} d^{4}\right )} \sqrt{b c d - a d^{2}}} - \frac{8 \,{\left (a b^{4} c - a^{2} b^{3} d + \frac{6 \,{\left (a x + b\right )} b^{4} c}{x} - \frac{15 \,{\left (a x + b\right )} a b^{3} d}{x}\right )} x}{{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )}{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}} + \frac{3 \,{\left (21 \, b^{2} c^{2} d^{4} \sqrt{\frac{a x + b}{x}} - 29 \, a b c d^{5} \sqrt{\frac{a x + b}{x}} + 8 \, a^{2} d^{6} \sqrt{\frac{a x + b}{x}} + \frac{19 \,{\left (a x + b\right )} b c d^{5} \sqrt{\frac{a x + b}{x}}}{x} - \frac{8 \,{\left (a x + b\right )} a d^{6} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (b^{4} c^{7} - 4 \, a b^{3} c^{6} d + 6 \, a^{2} b^{2} c^{5} d^{2} - 4 \, a^{3} b c^{4} d^{3} + a^{4} c^{3} d^{4}\right )}{\left (b c - a d + \frac{{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac{12 \, \sqrt{\frac{a x + b}{x}}}{{\left (a - \frac{a x + b}{x}\right )} a^{3} c^{3}} - \frac{12 \,{\left (5 \, b c + 6 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b c^{4}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="giac")

[Out]

-1/12*b*(3*(99*b^2*c^2*d^4 - 88*a*b*c*d^5 + 24*a^2*d^6)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^5*
c^8 - 4*a*b^4*c^7*d + 6*a^2*b^3*c^6*d^2 - 4*a^3*b^2*c^5*d^3 + a^4*b*c^4*d^4)*sqrt(b*c*d - a*d^2)) - 8*(a*b^4*c
- a^2*b^3*d + 6*(a*x + b)*b^4*c/x - 15*(a*x + b)*a*b^3*d/x)*x/((a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2
*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*(a*x + b)*sqrt((a*x + b)/x)) + 3*(21*b^2*c^2*d^4*sqrt((a*x + b)/x) - 29*a*b*c*
d^5*sqrt((a*x + b)/x) + 8*a^2*d^6*sqrt((a*x + b)/x) + 19*(a*x + b)*b*c*d^5*sqrt((a*x + b)/x)/x - 8*(a*x + b)*a
*d^6*sqrt((a*x + b)/x)/x)/((b^4*c^7 - 4*a*b^3*c^6*d + 6*a^2*b^2*c^5*d^2 - 4*a^3*b*c^4*d^3 + a^4*c^3*d^4)*(b*c
- a*d + (a*x + b)*d/x)^2) + 12*sqrt((a*x + b)/x)/((a - (a*x + b)/x)*a^3*c^3) - 12*(5*b*c + 6*a*d)*arctan(sqrt(
(a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3*b*c^4))