3.250 \(\int \frac {1}{\sqrt {a+\frac {b}{x}} (c+\frac {d}{x})^2} \, dx\)

Optimal. Leaf size=172 \[ -\frac {(4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^3}-\frac {d^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{3/2}}+\frac {d \sqrt {a+\frac {b}{x}} (b c-2 a d)}{a c^2 \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )} \]

[Out]

-d^(3/2)*(-4*a*d+5*b*c)*arctan(d^(1/2)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^3/(-a*d+b*c)^(3/2)-(4*a*d+b*c)*arctan
h((a+b/x)^(1/2)/a^(1/2))/a^(3/2)/c^3+d*(-2*a*d+b*c)*(a+b/x)^(1/2)/a/c^2/(-a*d+b*c)/(c+d/x)+x*(a+b/x)^(1/2)/a/c
/(c+d/x)

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Rubi [A]  time = 0.22, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {375, 103, 151, 156, 63, 208, 205} \[ -\frac {(4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^3}-\frac {d^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{3/2}}+\frac {d \sqrt {a+\frac {b}{x}} (b c-2 a d)}{a c^2 \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {x \sqrt {a+\frac {b}{x}}}{a c \left (c+\frac {d}{x}\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b/x]*(c + d/x)^2),x]

[Out]

(d*(b*c - 2*a*d)*Sqrt[a + b/x])/(a*c^2*(b*c - a*d)*(c + d/x)) + (Sqrt[a + b/x]*x)/(a*c*(c + d/x)) - (d^(3/2)*(
5*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(c^3*(b*c - a*d)^(3/2)) - ((b*c + 4*a*d)*ArcTa
nh[Sqrt[a + b/x]/Sqrt[a]])/(a^(3/2)*c^3)

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 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 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 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]

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 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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {a+b x} (c+d x)^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (b c+4 a d)+\frac {3 b d x}{2}}{x \sqrt {a+b x} (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {d (b c-2 a d) \sqrt {a+\frac {b}{x}}}{a c^2 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} (b c-a d) (b c+4 a d)-\frac {1}{2} b d (b c-2 a d) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{a c^2 (b c-a d)}\\ &=\frac {d (b c-2 a d) \sqrt {a+\frac {b}{x}}}{a c^2 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )}-\frac {\left (d^2 (5 b c-4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 c^3 (b c-a d)}+\frac {(b c+4 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a c^3}\\ &=\frac {d (b c-2 a d) \sqrt {a+\frac {b}{x}}}{a c^2 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )}-\frac {\left (d^2 (5 b c-4 a d)\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 )}{b c^3 (b c-a d)}+\frac {(b c+4 a d) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a b c^3}\\ &=\frac {d (b c-2 a d) \sqrt {a+\frac {b}{x}}}{a c^2 (b c-a d) \left (c+\frac {d}{x}\right )}+\frac {\sqrt {a+\frac {b}{x}} x}{a c \left (c+\frac {d}{x}\right )}-\frac {d^{3/2} (5 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{3/2}}-\frac {(b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{3/2} c^3}\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 150, normalized size = 0.87 \[ \frac {\frac {a d^{3/2} (4 a d-5 b c) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{3/2}}+\frac {c x \sqrt {a+\frac {b}{x}} (b c (c x+d)-a d (c x+2 d))}{(c x+d) (b c-a d)}-\frac {(4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{\sqrt {a}}}{a c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c*Sqrt[a + b/x]*x*(b*c*(d + c*x) - a*d*(2*d + c*x)))/((b*c - a*d)*(d + c*x)) + (a*d^(3/2)*(-5*b*c + 4*a*d)*A
rcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(b*c - a*d)^(3/2) - ((b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[
a]])/Sqrt[a])/(a*c^3)

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fricas [A]  time = 1.07, size = 1163, normalized size = 6.76 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*((b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*log(2*a*x - 2*sq
rt(a)*x*sqrt((a*x + b)/x) + b) + (5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a^2*b*c^2*d - 4*a^3*c*d^2)*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*((a
*b*c^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(a^2*b*c^4*d - a^3*c^3*d^2 + (a^2*b*
c^5 - a^3*c^4*d)*x), -1/2*(2*(5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a^2*b*c^2*d - 4*a^3*c*d^2)*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)) - (b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2
*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*((a
*b*c^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(a^2*b*c^4*d - a^3*c^3*d^2 + (a^2*b*
c^5 - a^3*c^4*d)*x), 1/2*(2*(b^2*c^2*d + 3*a*b*c*d^2 - 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*sq
rt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a^2*b*c^2*d - 4*a^3*c*d^2)*x)*sq
rt(-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*((a*b*c^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(a^2*b*c^4*d - a^3*c^3*
d^2 + (a^2*b*c^5 - a^3*c^4*d)*x), -((5*a^2*b*c*d^2 - 4*a^3*d^3 + (5*a^2*b*c^2*d - 4*a^3*c*d^2)*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)) - (b^2*c^2*d + 3*a*b*c*d^2
- 4*a^2*d^3 + (b^2*c^3 + 3*a*b*c^2*d - 4*a^2*c*d^2)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - ((a*b*c
^3 - a^2*c^2*d)*x^2 + (a*b*c^2*d - 2*a^2*c*d^2)*x)*sqrt((a*x + b)/x))/(a^2*b*c^4*d - a^3*c^3*d^2 + (a^2*b*c^5
- a^3*c^4*d)*x)]

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giac [A]  time = 0.22, size = 300, normalized size = 1.74 \[ -b^{3} {\left (\frac {{\left (5 \, b c d^{2} - 4 \, a d^{3}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{4} c^{4} - a b^{3} c^{3} d\right )} \sqrt {b c d - a d^{2}}} + \frac {b^{2} c^{2} \sqrt {\frac {a x + b}{x}} - 2 \, a b c d \sqrt {\frac {a x + b}{x}} + 2 \, a^{2} d^{2} \sqrt {\frac {a x + b}{x}} + \frac {{\left (a x + b\right )} b c d \sqrt {\frac {a x + b}{x}}}{x} - \frac {2 \, {\left (a x + b\right )} a d^{2} \sqrt {\frac {a x + b}{x}}}{x}}{{\left (a b^{3} c^{3} - a^{2} b^{2} c^{2} d\right )} {\left (a b c - a^{2} d - \frac {{\left (a x + b\right )} b c}{x} + \frac {2 \, {\left (a x + b\right )} a d}{x} - \frac {{\left (a x + b\right )}^{2} d}{x^{2}}\right )}} - \frac {{\left (b c + 4 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a b^{3} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^3*((5*b*c*d^2 - 4*a*d^3)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^4*c^4 - a*b^3*c^3*d)*sqrt(b*c*
d - a*d^2)) + (b^2*c^2*sqrt((a*x + b)/x) - 2*a*b*c*d*sqrt((a*x + b)/x) + 2*a^2*d^2*sqrt((a*x + b)/x) + (a*x +
b)*b*c*d*sqrt((a*x + b)/x)/x - 2*(a*x + b)*a*d^2*sqrt((a*x + b)/x)/x)/((a*b^3*c^3 - a^2*b^2*c^2*d)*(a*b*c - a^
2*d - (a*x + b)*b*c/x + 2*(a*x + b)*a*d/x - (a*x + b)^2*d/x^2)) - (b*c + 4*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(
-a))/(sqrt(-a)*a*b^3*c^3))

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maple [B]  time = 0.07, size = 1135, normalized size = 6.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a*x+b)/x)^(1/2)*x*(4*a^(9/2)*c*d^4*x*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)
*c)/(c*x+d))+2*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(7/2)*c^4*d*x^2+4*a^(9/2)*d^5*ln((-2*a*d*x+b*c*x-b*
d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))-2*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(7/2)*
c^3*d^2*x-9*a^(7/2)*b*c^2*d^3*x*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))
-4*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(7/2)*c^2*d^3-9*a^(7/2)*b*c*d^4*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-
b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))-2*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*c^4*d+6*((
a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*b*c^4*d*x+5*a^(5/2)*b^2*c^3*d^2*x*ln((-2*a*d*x+b*c*x-b*d+2*((a
*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))+6*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*a^(5/2)*b*c^3*d
^2+5*a^(5/2)*b^2*c^2*d^3*ln((-2*a*d*x+b*c*x-b*d+2*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*c)/(c*x+d))-2*a^(3
/2)*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*x*b^2*c^5-2*a^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*((a*x+b)*x)^(1/2)*b^
2*c^4*d+4*((a*d-b*c)/c^2*d)^(1/2)*a^4*c^2*d^3*x*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))-7*((a*d-
b*c)/c^2*d)^(1/2)*a^3*b*c^3*d^2*x*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))+2*a^2*ln(1/2*(2*a*x+b+
2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*x*b^2*c^4*d+((a*d-b*c)/c^2*d)^(1/2)*a*b^3*c^5*x*
ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))+4*((a*d-b*c)/c^2*d)^(1/2)*a^4*c*d^4*ln(1/2*(2*a*x+b+2*((
a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))-7*((a*d-b*c)/c^2*d)^(1/2)*a^3*b*c^2*d^3*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*
a^(1/2))/a^(1/2))+2*a^2*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2))*((a*d-b*c)/c^2*d)^(1/2)*b^2*c^3*
d^2+((a*d-b*c)/c^2*d)^(1/2)*a*b^3*c^4*d*ln(1/2*(2*a*x+b+2*((a*x+b)*x)^(1/2)*a^(1/2))/a^(1/2)))/c^4/((a*x+b)*x)
^(1/2)/(a*d-b*c)^2/(c*x+d)/a^(5/2)/((a*d-b*c)/c^2*d)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a + \frac {b}{x}} {\left (c + \frac {d}{x}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 3.54, size = 3813, normalized size = 22.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((a + b/x)^(1/2)*(b^3*c^2 + 2*a^2*b*d^2 - 2*a*b^2*c*d))/(c^2*(a^2*d - a*b*c)) + (d*(a + b/x)^(3/2)*(b^2*c - 2
*a*b*d))/(c^2*(a^2*d - a*b*c)))/((a + b/x)*(2*a*d - b*c) - d*(a + b/x)^2 - a^2*d + a*b*c) - (atan(((((2*(a + b
/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c
^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d) + (((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^
6*d^5)/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) + ((a + b/x)^(1/2)*(4*a*d + b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^
3*b^4*c^8*d^3 + 20*a^4*b^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/(c^3*(a^3)^(1/2)*(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b
*c^5*d)))*(4*a*d + b*c))/(2*c^3*(a^3)^(1/2)))*(4*a*d + b*c)*1i)/(2*c^3*(a^3)^(1/2)) + (((2*(a + b/x)^(1/2)*(32
*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*
d^2 - 2*a^3*b*c^5*d) - (((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b
^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) - ((a + b/x)^(1/2)*(4*a*d + b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3
+ 20*a^4*b^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/(c^3*(a^3)^(1/2)*(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)))*(4*
a*d + b*c))/(2*c^3*(a^3)^(1/2)))*(4*a*d + b*c)*1i)/(2*c^3*(a^3)^(1/2)))/((2*(32*a^3*b^3*d^7 + 5*b^6*c^3*d^4 +
6*a*b^5*c^2*d^5 - 48*a^2*b^4*c*d^6))/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) - (((2*(a + b/x)^(1/2)*(32*a^
4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2
- 2*a^3*b*c^5*d) + (((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*
c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) + ((a + b/x)^(1/2)*(4*a*d + b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 +
20*a^4*b^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/(c^3*(a^3)^(1/2)*(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)))*(4*a*d
+ b*c))/(2*c^3*(a^3)^(1/2)))*(4*a*d + b*c))/(2*c^3*(a^3)^(1/2)) + (((2*(a + b/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*
c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d
) - (((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*c^8 + a^4*c^6*d^
2 - 2*a^3*b*c^7*d) - ((a + b/x)^(1/2)*(4*a*d + b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 + 20*a^4*b^3*c^7*d
^4 - 8*a^5*b^2*c^6*d^5))/(c^3*(a^3)^(1/2)*(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)))*(4*a*d + b*c))/(2*c^3*
(a^3)^(1/2)))*(4*a*d + b*c))/(2*c^3*(a^3)^(1/2))))*(4*a*d + b*c)*1i)/(c^3*(a^3)^(1/2)) - (atan((((d^3*(a*d - b
*c)^3)^(1/2)*(4*a*d - 5*b*c)*((2*(a + b/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*
c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d) + ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d -
5*b*c)*((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*c^8 + a^4*c^6
*d^2 - 2*a^3*b*c^7*d) + ((d^3*(a*d - b*c)^3)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 5*b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3
*b^4*c^8*d^3 + 20*a^4*b^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/((a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)*(b^3*c^6
- a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d))))/(2*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*
d)))*1i)/(2*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d)) + ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5
*b*c)*((2*(a + b/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*
d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d) - ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5*b*c)*((4*a*b^6*c^9*d
^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) -
((d^3*(a*d - b*c)^3)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 5*b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 + 20*a^4*b
^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/((a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b
*c^4*d^2 - 3*a*b^2*c^5*d))))/(2*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d)))*1i)/(2*(b^3*c^6 -
a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d)))/((2*(32*a^3*b^3*d^7 + 5*b^6*c^3*d^4 + 6*a*b^5*c^2*d^5 - 48*a^
2*b^4*c*d^6))/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) - ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5*b*c)*((2*(a
+ b/x)^(1/2)*(32*a^4*b^2*d^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^
2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d) + ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5*b*c)*((4*a*b^6*c^9*d^2 + 4*a^2*b^
5*c^8*d^3 - 16*a^3*b^4*c^7*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) + ((d^3*(a*d -
b*c)^3)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 5*b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 + 20*a^4*b^3*c^7*d^4 -
8*a^5*b^2*c^6*d^5))/((a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*
a*b^2*c^5*d))))/(2*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d))))/(2*(b^3*c^6 - a^3*c^3*d^3 + 3*
a^2*b*c^4*d^2 - 3*a*b^2*c^5*d)) + ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5*b*c)*((2*(a + b/x)^(1/2)*(32*a^4*b^2*d
^7 + b^6*c^4*d^3 + 6*a*b^5*c^3*d^4 - 64*a^3*b^3*c*d^6 + 26*a^2*b^4*c^2*d^5))/(a^2*b^2*c^6 + a^4*c^4*d^2 - 2*a^
3*b*c^5*d) - ((d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5*b*c)*((4*a*b^6*c^9*d^2 + 4*a^2*b^5*c^8*d^3 - 16*a^3*b^4*c^7
*d^4 + 8*a^4*b^3*c^6*d^5)/(a^2*b^2*c^8 + a^4*c^6*d^2 - 2*a^3*b*c^7*d) - ((d^3*(a*d - b*c)^3)^(1/2)*(a + b/x)^(
1/2)*(4*a*d - 5*b*c)*(4*a^2*b^5*c^9*d^2 - 16*a^3*b^4*c^8*d^3 + 20*a^4*b^3*c^7*d^4 - 8*a^5*b^2*c^6*d^5))/((a^2*
b^2*c^6 + a^4*c^4*d^2 - 2*a^3*b*c^5*d)*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d))))/(2*(b^3*c^
6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d))))/(2*(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^
5*d))))*(d^3*(a*d - b*c)^3)^(1/2)*(4*a*d - 5*b*c)*1i)/(b^3*c^6 - a^3*c^3*d^3 + 3*a^2*b*c^4*d^2 - 3*a*b^2*c^5*d
)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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