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

Optimal. Leaf size=287 \[ -\frac {(4 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2} c^3}+\frac {b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a^2 c^2 \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)^2}+\frac {b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a^3 c^2 \sqrt {a+\frac {b}{x}} (b c-a d)^3}-\frac {d^{7/2} (9 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{7/2}}+\frac {d (b c-2 a d)}{a c^2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \]

[Out]

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

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Rubi [A]  time = 0.45, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ \frac {b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a^3 c^2 \sqrt {a+\frac {b}{x}} (b c-a d)^3}+\frac {b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a^2 c^2 \left (a+\frac {b}{x}\right )^{3/2} (b c-a d)^2}-\frac {(4 a d+5 b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2} c^3}-\frac {d^{7/2} (9 b c-4 a d) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c^3 (b c-a d)^{7/2}}+\frac {d (b c-2 a d)}{a c^2 \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right ) (b c-a d)}+\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(a + b/x)^(3/2)) + (b*(b*c - 2*a*d)*(5*b^2*c^
2 - a*b*c*d + a^2*d^2))/(a^3*c^2*(b*c - a*d)^3*Sqrt[a + b/x]) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*(a + b/x)
^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)) - (d^(7/2)*(9*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x]
)/Sqrt[b*c - a*d]])/(c^3*(b*c - a*d)^(7/2)) - ((5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*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 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 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}{\left (a+\frac {b}{x}\right )^{5/2} \left (c+\frac {d}{x}\right )^2} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{5/2} (c+d x)^2} \, dx,x,\frac {1}{x}\right )\\ &=\frac {x}{a c \left (a+\frac {b}{x}\right )^{3/2} \left (c+\frac {d}{x}\right )}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{2} (5 b c+4 a d)+\frac {7 b d x}{2}}{x (a+b x)^{5/2} (c+d x)^2} \, dx,x,\frac {1}{x}\right )}{a c}\\ &=\frac {d (b c-2 a d)}{a c^2 (b c-a d) \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 )}-\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} (b c-a d) (5 b c+4 a d)-\frac {5}{2} b d (b c-2 a d) x}{x (a+b x)^{5/2} (c+d x)} \, dx,x,\frac {1}{x}\right )}{a c^2 (b c-a d)}\\ &=\frac {b \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \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 )}-\frac {2 \operatorname {Subst}\left (\int \frac {-\frac {3}{4} (b c-a d)^2 (5 b c+4 a d)-\frac {3}{4} b d \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right ) x}{x (a+b x)^{3/2} (c+d x)} \, dx,x,\frac {1}{x}\right )}{3 a^2 c^2 (b c-a d)^2}\\ &=\frac {b \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b (b c-2 a d) \left (5 b^2 c^2-a b c d+a^2 d^2\right )}{a^3 c^2 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \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 )}-\frac {4 \operatorname {Subst}\left (\int \frac {-\frac {3}{8} (b c-a d)^3 (5 b c+4 a d)-\frac {3}{8} b d (b c-2 a d) \left (5 b^2 c^2-a b c d+a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{3 a^3 c^2 (b c-a d)^3}\\ &=\frac {b \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b (b c-2 a d) \left (5 b^2 c^2-a b c d+a^2 d^2\right )}{a^3 c^2 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \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 )}-\frac {\left (d^4 (9 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)^3}+\frac {(5 b c+4 a d) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^3 c^3}\\ &=\frac {b \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b (b c-2 a d) \left (5 b^2 c^2-a b c d+a^2 d^2\right )}{a^3 c^2 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \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 )}-\frac {\left (d^4 (9 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)^3}+\frac {(5 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^3 b c^3}\\ &=\frac {b \left (5 b^2 c^2-6 a b c d+6 a^2 d^2\right )}{3 a^2 c^2 (b c-a d)^2 \left (a+\frac {b}{x}\right )^{3/2}}+\frac {b (b c-2 a d) \left (5 b^2 c^2-a b c d+a^2 d^2\right )}{a^3 c^2 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (b c-2 a d)}{a c^2 (b c-a d) \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 )}-\frac {d^{7/2} (9 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)^{7/2}}-\frac {(5 b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{7/2} c^3}\\ \end {align*}

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Mathematica [C]  time = 0.19, size = 178, normalized size = 0.62 \[ \frac {x \left ((a d-b c) \left (3 a c x (a d (c x+2 d)-b c (c x+d))-(c x+d) \left (-4 a^2 d^2-a b c d+5 b^2 c^2\right ) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {b}{a x}+1\right )\right )+a^2 d^2 (c x+d) (9 b c-4 a d) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};\frac {d \left (a+\frac {b}{x}\right )}{a d-b c}\right )\right )}{3 a^2 c^3 \sqrt {a+\frac {b}{x}} (a x+b) (c x+d) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a^2*d^2*(9*b*c - 4*a*d)*(d + c*x)*Hypergeometric2F1[-3/2, 1, -1/2, (d*(a + b/x))/(-(b*c) + a*d)] + (-(b*c)
+ a*d)*(3*a*c*x*(-(b*c*(d + c*x)) + a*d*(2*d + c*x)) - (5*b^2*c^2 - a*b*c*d - 4*a^2*d^2)*(d + c*x)*Hypergeome
tric2F1[-3/2, 1, -1/2, 1 + b/(a*x)])))/(3*a^2*c^3*(b*c - a*d)^2*Sqrt[a + b/x]*(b + a*x)*(d + c*x))

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fricas [B]  time = 5.60, size = 3887, normalized size = 13.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*(5*b^6*c^4*d - 11*a*b^5*c^3*d^2 + 3*a^2*b^4*c^2*d^3 + 7*a^3*b^3*c*d^4 - 4*a^4*b^2*d^5 + (5*a^2*b^4*c^5
- 11*a^3*b^3*c^4*d + 3*a^4*b^2*c^3*d^2 + 7*a^5*b*c^2*d^3 - 4*a^6*c*d^4)*x^3 + (10*a*b^5*c^5 - 17*a^2*b^4*c^4*
d - 5*a^3*b^3*c^3*d^2 + 17*a^4*b^2*c^2*d^3 - a^5*b*c*d^4 - 4*a^6*d^5)*x^2 + (5*b^6*c^5 - a*b^5*c^4*d - 19*a^2*
b^4*c^3*d^2 + 13*a^3*b^3*c^2*d^3 + 10*a^4*b^2*c*d^4 - 8*a^5*b*d^5)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*
x + b)/x) + b) + 3*(9*a^4*b^3*c*d^4 - 4*a^5*b^2*d^5 + (9*a^6*b*c^2*d^3 - 4*a^7*c*d^4)*x^3 + (18*a^5*b^2*c^2*d^
3 + a^6*b*c*d^4 - 4*a^7*d^5)*x^2 + (9*a^4*b^3*c^2*d^3 + 14*a^5*b^2*c*d^4 - 8*a^6*b*d^5)*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*(3*(a^3
*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3)*x^4 + (20*a^2*b^4*c^5 - 41*a^3*b^3*c^4*d + 9*a^4*b
^2*c^3*d^2 + 3*a^5*b*c^2*d^3 - 6*a^6*c*d^4)*x^3 + (15*a*b^5*c^5 - 13*a^2*b^4*c^4*d - 35*a^3*b^3*c^3*d^2 + 15*a
^4*b^2*c^2*d^3 - 12*a^5*b*c*d^4)*x^2 + 3*(5*a*b^5*c^4*d - 11*a^2*b^4*c^3*d^2 + 3*a^3*b^3*c^2*d^3 - 2*a^4*b^2*c
*d^4)*x)*sqrt((a*x + b)/x))/(a^4*b^5*c^6*d - 3*a^5*b^4*c^5*d^2 + 3*a^6*b^3*c^4*d^3 - a^7*b^2*c^3*d^4 + (a^6*b^
3*c^7 - 3*a^7*b^2*c^6*d + 3*a^8*b*c^5*d^2 - a^9*c^4*d^3)*x^3 + (2*a^5*b^4*c^7 - 5*a^6*b^3*c^6*d + 3*a^7*b^2*c^
5*d^2 + a^8*b*c^4*d^3 - a^9*c^3*d^4)*x^2 + (a^4*b^5*c^7 - a^5*b^4*c^6*d - 3*a^6*b^3*c^5*d^2 + 5*a^7*b^2*c^4*d^
3 - 2*a^8*b*c^3*d^4)*x), -1/6*(6*(9*a^4*b^3*c*d^4 - 4*a^5*b^2*d^5 + (9*a^6*b*c^2*d^3 - 4*a^7*c*d^4)*x^3 + (18*
a^5*b^2*c^2*d^3 + a^6*b*c*d^4 - 4*a^7*d^5)*x^2 + (9*a^4*b^3*c^2*d^3 + 14*a^5*b^2*c*d^4 - 8*a^6*b*d^5)*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)) - 3*(5*b^6*c^4*d - 1
1*a*b^5*c^3*d^2 + 3*a^2*b^4*c^2*d^3 + 7*a^3*b^3*c*d^4 - 4*a^4*b^2*d^5 + (5*a^2*b^4*c^5 - 11*a^3*b^3*c^4*d + 3*
a^4*b^2*c^3*d^2 + 7*a^5*b*c^2*d^3 - 4*a^6*c*d^4)*x^3 + (10*a*b^5*c^5 - 17*a^2*b^4*c^4*d - 5*a^3*b^3*c^3*d^2 +
17*a^4*b^2*c^2*d^3 - a^5*b*c*d^4 - 4*a^6*d^5)*x^2 + (5*b^6*c^5 - a*b^5*c^4*d - 19*a^2*b^4*c^3*d^2 + 13*a^3*b^3
*c^2*d^3 + 10*a^4*b^2*c*d^4 - 8*a^5*b*d^5)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 2*(3*(a
^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3)*x^4 + (20*a^2*b^4*c^5 - 41*a^3*b^3*c^4*d + 9*a^4
*b^2*c^3*d^2 + 3*a^5*b*c^2*d^3 - 6*a^6*c*d^4)*x^3 + (15*a*b^5*c^5 - 13*a^2*b^4*c^4*d - 35*a^3*b^3*c^3*d^2 + 15
*a^4*b^2*c^2*d^3 - 12*a^5*b*c*d^4)*x^2 + 3*(5*a*b^5*c^4*d - 11*a^2*b^4*c^3*d^2 + 3*a^3*b^3*c^2*d^3 - 2*a^4*b^2
*c*d^4)*x)*sqrt((a*x + b)/x))/(a^4*b^5*c^6*d - 3*a^5*b^4*c^5*d^2 + 3*a^6*b^3*c^4*d^3 - a^7*b^2*c^3*d^4 + (a^6*
b^3*c^7 - 3*a^7*b^2*c^6*d + 3*a^8*b*c^5*d^2 - a^9*c^4*d^3)*x^3 + (2*a^5*b^4*c^7 - 5*a^6*b^3*c^6*d + 3*a^7*b^2*
c^5*d^2 + a^8*b*c^4*d^3 - a^9*c^3*d^4)*x^2 + (a^4*b^5*c^7 - a^5*b^4*c^6*d - 3*a^6*b^3*c^5*d^2 + 5*a^7*b^2*c^4*
d^3 - 2*a^8*b*c^3*d^4)*x), 1/6*(6*(5*b^6*c^4*d - 11*a*b^5*c^3*d^2 + 3*a^2*b^4*c^2*d^3 + 7*a^3*b^3*c*d^4 - 4*a^
4*b^2*d^5 + (5*a^2*b^4*c^5 - 11*a^3*b^3*c^4*d + 3*a^4*b^2*c^3*d^2 + 7*a^5*b*c^2*d^3 - 4*a^6*c*d^4)*x^3 + (10*a
*b^5*c^5 - 17*a^2*b^4*c^4*d - 5*a^3*b^3*c^3*d^2 + 17*a^4*b^2*c^2*d^3 - a^5*b*c*d^4 - 4*a^6*d^5)*x^2 + (5*b^6*c
^5 - a*b^5*c^4*d - 19*a^2*b^4*c^3*d^2 + 13*a^3*b^3*c^2*d^3 + 10*a^4*b^2*c*d^4 - 8*a^5*b*d^5)*x)*sqrt(-a)*arcta
n(sqrt(-a)*sqrt((a*x + b)/x)/a) + 3*(9*a^4*b^3*c*d^4 - 4*a^5*b^2*d^5 + (9*a^6*b*c^2*d^3 - 4*a^7*c*d^4)*x^3 + (
18*a^5*b^2*c^2*d^3 + a^6*b*c*d^4 - 4*a^7*d^5)*x^2 + (9*a^4*b^3*c^2*d^3 + 14*a^5*b^2*c*d^4 - 8*a^6*b*d^5)*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*(3*(a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3)*x^4 + (20*a^2*b^4*c^5 - 41*a^3*b^
3*c^4*d + 9*a^4*b^2*c^3*d^2 + 3*a^5*b*c^2*d^3 - 6*a^6*c*d^4)*x^3 + (15*a*b^5*c^5 - 13*a^2*b^4*c^4*d - 35*a^3*b
^3*c^3*d^2 + 15*a^4*b^2*c^2*d^3 - 12*a^5*b*c*d^4)*x^2 + 3*(5*a*b^5*c^4*d - 11*a^2*b^4*c^3*d^2 + 3*a^3*b^3*c^2*
d^3 - 2*a^4*b^2*c*d^4)*x)*sqrt((a*x + b)/x))/(a^4*b^5*c^6*d - 3*a^5*b^4*c^5*d^2 + 3*a^6*b^3*c^4*d^3 - a^7*b^2*
c^3*d^4 + (a^6*b^3*c^7 - 3*a^7*b^2*c^6*d + 3*a^8*b*c^5*d^2 - a^9*c^4*d^3)*x^3 + (2*a^5*b^4*c^7 - 5*a^6*b^3*c^6
*d + 3*a^7*b^2*c^5*d^2 + a^8*b*c^4*d^3 - a^9*c^3*d^4)*x^2 + (a^4*b^5*c^7 - a^5*b^4*c^6*d - 3*a^6*b^3*c^5*d^2 +
5*a^7*b^2*c^4*d^3 - 2*a^8*b*c^3*d^4)*x), -1/3*(3*(9*a^4*b^3*c*d^4 - 4*a^5*b^2*d^5 + (9*a^6*b*c^2*d^3 - 4*a^7*
c*d^4)*x^3 + (18*a^5*b^2*c^2*d^3 + a^6*b*c*d^4 - 4*a^7*d^5)*x^2 + (9*a^4*b^3*c^2*d^3 + 14*a^5*b^2*c*d^4 - 8*a^
6*b*d^5)*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)) - 3
*(5*b^6*c^4*d - 11*a*b^5*c^3*d^2 + 3*a^2*b^4*c^2*d^3 + 7*a^3*b^3*c*d^4 - 4*a^4*b^2*d^5 + (5*a^2*b^4*c^5 - 11*a
^3*b^3*c^4*d + 3*a^4*b^2*c^3*d^2 + 7*a^5*b*c^2*d^3 - 4*a^6*c*d^4)*x^3 + (10*a*b^5*c^5 - 17*a^2*b^4*c^4*d - 5*a
^3*b^3*c^3*d^2 + 17*a^4*b^2*c^2*d^3 - a^5*b*c*d^4 - 4*a^6*d^5)*x^2 + (5*b^6*c^5 - a*b^5*c^4*d - 19*a^2*b^4*c^3
*d^2 + 13*a^3*b^3*c^2*d^3 + 10*a^4*b^2*c*d^4 - 8*a^5*b*d^5)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) -
(3*(a^3*b^3*c^5 - 3*a^4*b^2*c^4*d + 3*a^5*b*c^3*d^2 - a^6*c^2*d^3)*x^4 + (20*a^2*b^4*c^5 - 41*a^3*b^3*c^4*d +
9*a^4*b^2*c^3*d^2 + 3*a^5*b*c^2*d^3 - 6*a^6*c*d^4)*x^3 + (15*a*b^5*c^5 - 13*a^2*b^4*c^4*d - 35*a^3*b^3*c^3*d^
2 + 15*a^4*b^2*c^2*d^3 - 12*a^5*b*c*d^4)*x^2 + 3*(5*a*b^5*c^4*d - 11*a^2*b^4*c^3*d^2 + 3*a^3*b^3*c^2*d^3 - 2*a
^4*b^2*c*d^4)*x)*sqrt((a*x + b)/x))/(a^4*b^5*c^6*d - 3*a^5*b^4*c^5*d^2 + 3*a^6*b^3*c^4*d^3 - a^7*b^2*c^3*d^4 +
(a^6*b^3*c^7 - 3*a^7*b^2*c^6*d + 3*a^8*b*c^5*d^2 - a^9*c^4*d^3)*x^3 + (2*a^5*b^4*c^7 - 5*a^6*b^3*c^6*d + 3*a^
7*b^2*c^5*d^2 + a^8*b*c^4*d^3 - a^9*c^3*d^4)*x^2 + (a^4*b^5*c^7 - a^5*b^4*c^6*d - 3*a^6*b^3*c^5*d^2 + 5*a^7*b^
2*c^4*d^3 - 2*a^8*b*c^3*d^4)*x)]

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giac [B]  time = 0.33, size = 576, normalized size = 2.01 \[ -\frac {1}{3} \, b^{3} {\left (\frac {3 \, {\left (9 \, b c d^{4} - 4 \, a d^{5}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 3 \, a^{2} b^{4} c^{4} d^{2} - a^{3} b^{3} c^{3} d^{3}\right )} \sqrt {b c d - a d^{2}}} - \frac {2 \, {\left (a b c - a^{2} d + \frac {6 \, {\left (a x + b\right )} b c}{x} - \frac {12 \, {\left (a x + b\right )} a d}{x}\right )} x}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )} {\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} + \frac {3 \, {\left (b^{4} c^{4} \sqrt {\frac {a x + b}{x}} - 4 \, a b^{3} c^{3} d \sqrt {\frac {a x + b}{x}} + 6 \, a^{2} b^{2} c^{2} d^{2} \sqrt {\frac {a x + b}{x}} - 4 \, a^{3} b c d^{3} \sqrt {\frac {a x + b}{x}} + 2 \, a^{4} d^{4} \sqrt {\frac {a x + b}{x}} + \frac {{\left (a x + b\right )} b^{3} c^{3} d \sqrt {\frac {a x + b}{x}}}{x} - \frac {3 \, {\left (a x + b\right )} a b^{2} c^{2} d^{2} \sqrt {\frac {a x + b}{x}}}{x} + \frac {3 \, {\left (a x + b\right )} a^{2} b c d^{3} \sqrt {\frac {a x + b}{x}}}{x} - \frac {2 \, {\left (a x + b\right )} a^{3} d^{4} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{{\left (a^{3} b^{5} c^{5} - 3 \, a^{4} b^{4} c^{4} d + 3 \, a^{5} b^{3} c^{3} d^{2} - a^{6} b^{2} c^{2} d^{3}\right )} {\left (a b c - a^{2} d - \frac {{\left (a x + b\right )} b c}{x} + \frac {2 \, {\left (a x + b\right )} a d}{x} - \frac {{\left (a x + b\right )}^{2} d}{x^{2}}\right )}} - \frac {3 \, {\left (5 \, b c + 4 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3} b^{3} c^{3}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*b^3*(3*(9*b*c*d^4 - 4*a*d^5)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^6*c^6 - 3*a*b^5*c^5*d +
3*a^2*b^4*c^4*d^2 - a^3*b^3*c^3*d^3)*sqrt(b*c*d - a*d^2)) - 2*(a*b*c - a^2*d + 6*(a*x + b)*b*c/x - 12*(a*x + b
)*a*d/x)*x/((a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*(a*x + b)*sqrt((a*x + b)/x)) + 3*(b^4*c^
4*sqrt((a*x + b)/x) - 4*a*b^3*c^3*d*sqrt((a*x + b)/x) + 6*a^2*b^2*c^2*d^2*sqrt((a*x + b)/x) - 4*a^3*b*c*d^3*sq
rt((a*x + b)/x) + 2*a^4*d^4*sqrt((a*x + b)/x) + (a*x + b)*b^3*c^3*d*sqrt((a*x + b)/x)/x - 3*(a*x + b)*a*b^2*c^
2*d^2*sqrt((a*x + b)/x)/x + 3*(a*x + b)*a^2*b*c*d^3*sqrt((a*x + b)/x)/x - 2*(a*x + b)*a^3*d^4*sqrt((a*x + b)/x
)/x)/((a^3*b^5*c^5 - 3*a^4*b^4*c^4*d + 3*a^5*b^3*c^3*d^2 - a^6*b^2*c^2*d^3)*(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)) - 3*(5*b*c + 4*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3*
b^3*c^3))

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maple [B]  time = 0.08, size = 4644, normalized size = 16.18 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*((a*x+b)/x)^(1/2)*x*(12*a^(19/2)*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))*x^3*d^7+12*a^(13/2)*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))*b^3*
d^7+30*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*b^4*c^3*d^4-84*a^(7/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c
^2*d)^(1/2)*b^5*c^4*d^3+96*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*b^6*c^5*d^2+12*a^6*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^3*c*d^6-33*a^5*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^4*c^2*d^5+12*a^4*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^5*c^3*d^4-6*a^(17/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^4*c^3*d
^4-30*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^4*b^4*c^7-39*a^(17/2)*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))*x^4*b*c^2*d^5+27*a^(15/2)*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))*x^4*b^2*c^3*d^4+12*a^9*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^4*c^2*d^5+15*a^4*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^4*b^5*c^7+24*a^(7/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^4*c^7-12*a^(17/2)
*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*c^2*d^5-90*a^(7/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^
3*b^5*c^7-3*a^(17/2)*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))*x^3*b*c*d^
6-90*a^(15/2)*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))*x^3*b^2*c^2*d^5+8
1*a^(13/2)*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))*x^3*b^3*c^3*d^4+12*a
^9*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^3*c*d^6+45*a^3*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^3*b^6*c^7+20*a^(5/2)*((a*x+b)*x)^(3/2)*((a
*d-b*c)/c^2*d)^(1/2)*x*b^5*c^7-90*a^(5/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^6*c^7-81*a^(15/2)*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))*x^2*b^2*c*d^6-36*a^(13/2)*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))*x^2*b^3*c^2*d^5+81*a^(11/2)*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))*x^2*b^4*c^3*d^4+45*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^2*b^7*c^7+38*a^(9/2)*((a*x+b)*x)^(3/2)*((a*d-b*c
)/c^2*d)^(1/2)*b^3*c^4*d^3-64*a^(7/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*b^4*c^5*d^2+20*a^(5/2)*((a*x+b
)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*b^5*c^6*d-105*a^(13/2)*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))*x*b^3*c*d^6+42*a^(11/2)*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))*x*b^4*c^2*d^5+27*a^(9/2)*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))*x*b^5*c^3*d^4-12*a^(11/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*b^3*c^2*d^5+15*a*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^8*c^7-30*a^(3/2)*((a*x+b)*x)^(1/2)*((a*
d-b*c)/c^2*d)^(1/2)*b^7*c^6*d+15*a*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^8*c^6*d+42*a^3*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^6*c^4*d^3-
48*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^7*c^5*d^2+6*a^(17/2)*((
a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^5*c^4*d^3-6*a^(15/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*c
^4*d^3-30*a^(3/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^7*c^7-102*a^3*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^6*c^5*d^2-3*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^7*c^6*d+36*a^8*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^2*b*c*d^6-63*a^7*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^2*b^2*c^2*d^5+12*a^(15/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*b*c^3*d^4+24*a^(13/2)*((a*x+
b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*b^2*c^4*d^3-33*a^8*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^4*b*c^3*d^4+12*a^7*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^4*b^2*c^4*d^3+42*a^6*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^4*b^3*c^5*d^2-48*a^5*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^
4*b^4*c^6*d+48*a^(15/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^4*b*c^4*d^3-84*a^(13/2)*((a*x+b)*x)^(1/2)*
((a*d-b*c)/c^2*d)^(1/2)*x^4*b^2*c^5*d^2+96*a^(11/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^4*b^3*c^6*d-36
*a^(13/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^2*c^2*d^5-63*a^6*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^2*b^3*c^3*d^4+162*a^5*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^2*b^4*c^4*d^3-18*a^4*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^2*b^5*c^5*d^2-99*a^3*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^2*b^6*c^6*d+198*a^(7/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^5*c^6*d-28*a^(9/2)*((
a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^3*c^5*d^2-40*a^(7/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b
^4*c^6*d-36*a^(15/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b*c^2*d^5+72*a^(13/2)*((a*x+b)*x)^(1/2)*((a
*d-b*c)/c^2*d)^(1/2)*x^2*b^2*c^3*d^4-156*a^(11/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^3*c^4*d^3+36
*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^4*c^5*d^2+78*a^5*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^3*b^4*c^5*d^2-129*a^4*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^3*b^5*c^6*d+30*a^(11/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^2*c^
4*d^3+3*a^8*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^3*b*c^2*d^5-87*a^7
*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^3*b^2*c^3*d^4+78*a^6*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^3*b^3*c^4*d^3-156*a^(11/2)*((a*x+b)*x)
^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*b^3*c^5*d^2+258*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x^3*b^4*c
^6*d-18*a^(13/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b*c^4*d^3+48*a^(11/2)*((a*x+b)*x)^(3/2)*((a*d-b
*c)/c^2*d)^(1/2)*x^2*b^2*c^5*d^2-72*a^(9/2)*((a*x+b)*x)^(3/2)*((a*d-b*c)/c^2*d)^(1/2)*x^2*b^3*c^6*d+84*a^(11/2
)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^3*c^3*d^4-222*a^(9/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2
)*x*b^4*c^4*d^3+204*a^(7/2)*((a*x+b)*x)^(1/2)*((a*d-b*c)/c^2*d)^(1/2)*x*b^5*c^5*d^2+6*a^(5/2)*((a*x+b)*x)^(1/2
)*((a*d-b*c)/c^2*d)^(1/2)*x*b^6*c^6*d+36*a^7*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*d^6-87*a^6*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^3*c^2*d^5+3*a^5*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^4*c^3*d^
4+138*a^4*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^5*c^4*d^3+12*a^(19
/2)*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))*x^4*c*d^6+36*a^(17/2)*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))*x^2*b*d^7+36*a^(15/2)*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))*x*b^2*d^7-39*a^(11/2)*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))*b^4*c*d^6+27*a^(9/2)*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))*b^5*c^2*d^5)/a^(9/2)/c^4/((a*x+b)*x)^(1/2)/(a*d-b*c)^4/(c*x+d)/(
(a*d-b*c)/c^2*d)^(1/2)/(a*x+b)^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} {\left (c + \frac {d}{x}\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 8.73, size = 5789, normalized size = 20.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^3)/(3*(a^2*d - a*b*c)) + (10*b^3*(a + b/x)*(2*a*d - b*c))/(3*(a^2*d - a*b*c)^2) - (b*(a + b/x)^2*(6*a^4*
d^4 + 15*b^4*c^4 + 64*a^2*b^2*c^2*d^2 - 58*a*b^3*c^3*d - 12*a^3*b*c*d^3))/(3*c^2*(a^2*d - a*b*c)^3) + (b*(a +
b/x)^3*(2*a*d - b*c)*(a^2*d^3 + 5*b^2*c^2*d - a*b*c*d^2))/(c^2*(a^2*d - a*b*c)^3))/(d*(a + b/x)^(7/2) + (a + b
/x)^(3/2)*(a^2*d - a*b*c) - (a + b/x)^(5/2)*(2*a*d - b*c)) + (atan((a^15*b^19*c^19*(a + b/x)^(1/2)*125i + a^17
*b^17*c^17*d^2*(a + b/x)^(1/2)*10440i - a^18*b^16*c^16*d^3*(a + b/x)^(1/2)*37776i + a^19*b^15*c^15*d^4*(a + b/
x)^(1/2)*87276i - a^20*b^14*c^14*d^5*(a + b/x)^(1/2)*126720i + a^21*b^13*c^13*d^6*(a + b/x)^(1/2)*91560i + a^2
2*b^12*c^12*d^7*(a + b/x)^(1/2)*40965i - a^23*b^11*c^11*d^8*(a + b/x)^(1/2)*184563i + a^24*b^10*c^10*d^9*(a +
b/x)^(1/2)*212608i - a^25*b^9*c^9*d^10*(a + b/x)^(1/2)*107740i - a^26*b^8*c^8*d^11*(a + b/x)^(1/2)*19530i + a^
27*b^7*c^7*d^12*(a + b/x)^(1/2)*71070i - a^28*b^6*c^6*d^13*(a + b/x)^(1/2)*52836i + a^29*b^5*c^5*d^14*(a + b/x
)^(1/2)*20916i - a^30*b^4*c^4*d^15*(a + b/x)^(1/2)*4515i + a^31*b^3*c^3*d^16*(a + b/x)^(1/2)*420i - a^16*b^18*
c^18*d*(a + b/x)^(1/2)*1700i)/(a^7*(a^7)^(1/2)*(a^7*(a^7*(212608*b^10*c^10*d^9 - 107740*a*b^9*c^9*d^10 - 19530
*a^2*b^8*c^8*d^11 + 71070*a^3*b^7*c^7*d^12 - 52836*a^4*b^6*c^6*d^13 + 20916*a^5*b^5*c^5*d^14 - 4515*a^6*b^4*c^
4*d^15 + 420*a^7*b^3*c^3*d^16) + 10440*b^17*c^17*d^2 - 37776*a*b^16*c^16*d^3 + 87276*a^2*b^15*c^15*d^4 - 12672
0*a^3*b^14*c^14*d^5 + 91560*a^4*b^13*c^13*d^6 + 40965*a^5*b^12*c^12*d^7 - 184563*a^6*b^11*c^11*d^8) + 125*a^5*
b^19*c^19 - 1700*a^6*b^18*c^18*d)))*(4*a*d + 5*b*c)*1i)/(c^3*(a^7)^(1/2)) - (atan((((d^7*(a*d - b*c)^7)^(1/2)*
((a + b/x)^(1/2)*(670*a^10*b^18*c^22*d^4 - 50*a^9*b^19*c^23*d^3 - 4082*a^11*b^17*c^21*d^5 + 14830*a^12*b^16*c^
20*d^6 - 35210*a^13*b^15*c^19*d^7 + 55510*a^14*b^14*c^18*d^8 - 53852*a^15*b^13*c^17*d^9 + 19048*a^16*b^12*c^16
*d^10 + 25730*a^17*b^11*c^15*d^11 - 39550*a^18*b^10*c^14*d^12 + 10670*a^19*b^9*c^13*d^13 + 29414*a^20*b^8*c^12
*d^14 - 45430*a^21*b^7*c^11*d^15 + 34490*a^22*b^6*c^10*d^16 - 16240*a^23*b^5*c^9*d^17 + 4820*a^24*b^4*c^8*d^18
- 832*a^25*b^3*c^7*d^19 + 64*a^26*b^2*c^6*d^20) - ((d^7*(a*d - b*c)^7)^(1/2)*(4*a*d - 9*b*c)*(304*a^13*b^18*c
^25*d^3 - 20*a^12*b^19*c^26*d^2 - 2144*a^14*b^17*c^24*d^4 + 9280*a^15*b^16*c^23*d^5 - 27476*a^16*b^15*c^22*d^6
+ 58688*a^17*b^14*c^21*d^7 - 92840*a^18*b^13*c^20*d^8 + 109648*a^19*b^12*c^19*d^9 - 95700*a^20*b^11*c^18*d^10
+ 59312*a^21*b^10*c^17*d^11 - 23056*a^22*b^9*c^16*d^12 + 2528*a^23*b^8*c^15*d^13 + 2996*a^24*b^7*c^14*d^14 -
2080*a^25*b^6*c^13*d^15 + 664*a^26*b^5*c^12*d^16 - 112*a^27*b^4*c^11*d^17 + 8*a^28*b^3*c^10*d^18 + ((d^7*(a*d
- b*c)^7)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 9*b*c)*(8*a^15*b^18*c^28*d^2 - 136*a^16*b^17*c^27*d^3 + 1080*a^17*b^1
6*c^26*d^4 - 5320*a^18*b^15*c^25*d^5 + 18200*a^19*b^14*c^24*d^6 - 45864*a^20*b^13*c^23*d^7 + 88088*a^21*b^12*c
^22*d^8 - 131560*a^22*b^11*c^21*d^9 + 154440*a^23*b^10*c^20*d^10 - 143000*a^24*b^9*c^19*d^11 + 104104*a^25*b^8
*c^18*d^12 - 58968*a^26*b^7*c^17*d^13 + 25480*a^27*b^6*c^16*d^14 - 8120*a^28*b^5*c^15*d^15 + 1800*a^29*b^4*c^1
4*d^16 - 248*a^30*b^3*c^13*d^17 + 16*a^31*b^2*c^12*d^18))/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^
2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d))))/(2*(b^7*c^10
- a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^
5*d^5 - 7*a*b^6*c^9*d)))*(4*a*d - 9*b*c)*1i)/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2
- 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d)) + ((d^7*(a*d - b*c)^7)^(1/2)
*((a + b/x)^(1/2)*(670*a^10*b^18*c^22*d^4 - 50*a^9*b^19*c^23*d^3 - 4082*a^11*b^17*c^21*d^5 + 14830*a^12*b^16*c
^20*d^6 - 35210*a^13*b^15*c^19*d^7 + 55510*a^14*b^14*c^18*d^8 - 53852*a^15*b^13*c^17*d^9 + 19048*a^16*b^12*c^1
6*d^10 + 25730*a^17*b^11*c^15*d^11 - 39550*a^18*b^10*c^14*d^12 + 10670*a^19*b^9*c^13*d^13 + 29414*a^20*b^8*c^1
2*d^14 - 45430*a^21*b^7*c^11*d^15 + 34490*a^22*b^6*c^10*d^16 - 16240*a^23*b^5*c^9*d^17 + 4820*a^24*b^4*c^8*d^1
8 - 832*a^25*b^3*c^7*d^19 + 64*a^26*b^2*c^6*d^20) - ((d^7*(a*d - b*c)^7)^(1/2)*(4*a*d - 9*b*c)*(20*a^12*b^19*c
^26*d^2 - 304*a^13*b^18*c^25*d^3 + 2144*a^14*b^17*c^24*d^4 - 9280*a^15*b^16*c^23*d^5 + 27476*a^16*b^15*c^22*d^
6 - 58688*a^17*b^14*c^21*d^7 + 92840*a^18*b^13*c^20*d^8 - 109648*a^19*b^12*c^19*d^9 + 95700*a^20*b^11*c^18*d^1
0 - 59312*a^21*b^10*c^17*d^11 + 23056*a^22*b^9*c^16*d^12 - 2528*a^23*b^8*c^15*d^13 - 2996*a^24*b^7*c^14*d^14 +
2080*a^25*b^6*c^13*d^15 - 664*a^26*b^5*c^12*d^16 + 112*a^27*b^4*c^11*d^17 - 8*a^28*b^3*c^10*d^18 + ((d^7*(a*d
- b*c)^7)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 9*b*c)*(8*a^15*b^18*c^28*d^2 - 136*a^16*b^17*c^27*d^3 + 1080*a^17*b^
16*c^26*d^4 - 5320*a^18*b^15*c^25*d^5 + 18200*a^19*b^14*c^24*d^6 - 45864*a^20*b^13*c^23*d^7 + 88088*a^21*b^12*
c^22*d^8 - 131560*a^22*b^11*c^21*d^9 + 154440*a^23*b^10*c^20*d^10 - 143000*a^24*b^9*c^19*d^11 + 104104*a^25*b^
8*c^18*d^12 - 58968*a^26*b^7*c^17*d^13 + 25480*a^27*b^6*c^16*d^14 - 8120*a^28*b^5*c^15*d^15 + 1800*a^29*b^4*c^
14*d^16 - 248*a^30*b^3*c^13*d^17 + 16*a^31*b^2*c^12*d^18))/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a
^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d))))/(2*(b^7*c^10
- a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c
^5*d^5 - 7*a*b^6*c^9*d)))*(4*a*d - 9*b*c)*1i)/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^
2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d)))/(4880*a^10*b^16*c^17*d^7 -
450*a^9*b^17*c^18*d^6 - 23428*a^11*b^15*c^16*d^8 + 65234*a^12*b^14*c^15*d^9 - 115136*a^13*b^13*c^14*d^10 + 12
9800*a^14*b^12*c^13*d^11 - 83040*a^15*b^11*c^12*d^12 + 5916*a^16*b^10*c^11*d^13 + 45702*a^17*b^9*c^10*d^14 - 5
1528*a^18*b^8*c^9*d^15 + 32500*a^19*b^7*c^8*d^16 - 13790*a^20*b^6*c^7*d^17 + 4012*a^21*b^5*c^6*d^18 - 736*a^22
*b^4*c^5*d^19 + 64*a^23*b^3*c^4*d^20 + ((d^7*(a*d - b*c)^7)^(1/2)*((a + b/x)^(1/2)*(670*a^10*b^18*c^22*d^4 - 5
0*a^9*b^19*c^23*d^3 - 4082*a^11*b^17*c^21*d^5 + 14830*a^12*b^16*c^20*d^6 - 35210*a^13*b^15*c^19*d^7 + 55510*a^
14*b^14*c^18*d^8 - 53852*a^15*b^13*c^17*d^9 + 19048*a^16*b^12*c^16*d^10 + 25730*a^17*b^11*c^15*d^11 - 39550*a^
18*b^10*c^14*d^12 + 10670*a^19*b^9*c^13*d^13 + 29414*a^20*b^8*c^12*d^14 - 45430*a^21*b^7*c^11*d^15 + 34490*a^2
2*b^6*c^10*d^16 - 16240*a^23*b^5*c^9*d^17 + 4820*a^24*b^4*c^8*d^18 - 832*a^25*b^3*c^7*d^19 + 64*a^26*b^2*c^6*d
^20) - ((d^7*(a*d - b*c)^7)^(1/2)*(4*a*d - 9*b*c)*(304*a^13*b^18*c^25*d^3 - 20*a^12*b^19*c^26*d^2 - 2144*a^14*
b^17*c^24*d^4 + 9280*a^15*b^16*c^23*d^5 - 27476*a^16*b^15*c^22*d^6 + 58688*a^17*b^14*c^21*d^7 - 92840*a^18*b^1
3*c^20*d^8 + 109648*a^19*b^12*c^19*d^9 - 95700*a^20*b^11*c^18*d^10 + 59312*a^21*b^10*c^17*d^11 - 23056*a^22*b^
9*c^16*d^12 + 2528*a^23*b^8*c^15*d^13 + 2996*a^24*b^7*c^14*d^14 - 2080*a^25*b^6*c^13*d^15 + 664*a^26*b^5*c^12*
d^16 - 112*a^27*b^4*c^11*d^17 + 8*a^28*b^3*c^10*d^18 + ((d^7*(a*d - b*c)^7)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 9*b
*c)*(8*a^15*b^18*c^28*d^2 - 136*a^16*b^17*c^27*d^3 + 1080*a^17*b^16*c^26*d^4 - 5320*a^18*b^15*c^25*d^5 + 18200
*a^19*b^14*c^24*d^6 - 45864*a^20*b^13*c^23*d^7 + 88088*a^21*b^12*c^22*d^8 - 131560*a^22*b^11*c^21*d^9 + 154440
*a^23*b^10*c^20*d^10 - 143000*a^24*b^9*c^19*d^11 + 104104*a^25*b^8*c^18*d^12 - 58968*a^26*b^7*c^17*d^13 + 2548
0*a^27*b^6*c^16*d^14 - 8120*a^28*b^5*c^15*d^15 + 1800*a^29*b^4*c^14*d^16 - 248*a^30*b^3*c^13*d^17 + 16*a^31*b^
2*c^12*d^18))/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*
b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d))))/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5
*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d)))*(4*a*d - 9*b*c))/(2
*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21
*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d)) - ((d^7*(a*d - b*c)^7)^(1/2)*((a + b/x)^(1/2)*(670*a^10*b^18*c^22*d^4 - 50*
a^9*b^19*c^23*d^3 - 4082*a^11*b^17*c^21*d^5 + 14830*a^12*b^16*c^20*d^6 - 35210*a^13*b^15*c^19*d^7 + 55510*a^14
*b^14*c^18*d^8 - 53852*a^15*b^13*c^17*d^9 + 19048*a^16*b^12*c^16*d^10 + 25730*a^17*b^11*c^15*d^11 - 39550*a^18
*b^10*c^14*d^12 + 10670*a^19*b^9*c^13*d^13 + 29414*a^20*b^8*c^12*d^14 - 45430*a^21*b^7*c^11*d^15 + 34490*a^22*
b^6*c^10*d^16 - 16240*a^23*b^5*c^9*d^17 + 4820*a^24*b^4*c^8*d^18 - 832*a^25*b^3*c^7*d^19 + 64*a^26*b^2*c^6*d^2
0) - ((d^7*(a*d - b*c)^7)^(1/2)*(4*a*d - 9*b*c)*(20*a^12*b^19*c^26*d^2 - 304*a^13*b^18*c^25*d^3 + 2144*a^14*b^
17*c^24*d^4 - 9280*a^15*b^16*c^23*d^5 + 27476*a^16*b^15*c^22*d^6 - 58688*a^17*b^14*c^21*d^7 + 92840*a^18*b^13*
c^20*d^8 - 109648*a^19*b^12*c^19*d^9 + 95700*a^20*b^11*c^18*d^10 - 59312*a^21*b^10*c^17*d^11 + 23056*a^22*b^9*
c^16*d^12 - 2528*a^23*b^8*c^15*d^13 - 2996*a^24*b^7*c^14*d^14 + 2080*a^25*b^6*c^13*d^15 - 664*a^26*b^5*c^12*d^
16 + 112*a^27*b^4*c^11*d^17 - 8*a^28*b^3*c^10*d^18 + ((d^7*(a*d - b*c)^7)^(1/2)*(a + b/x)^(1/2)*(4*a*d - 9*b*c
)*(8*a^15*b^18*c^28*d^2 - 136*a^16*b^17*c^27*d^3 + 1080*a^17*b^16*c^26*d^4 - 5320*a^18*b^15*c^25*d^5 + 18200*a
^19*b^14*c^24*d^6 - 45864*a^20*b^13*c^23*d^7 + 88088*a^21*b^12*c^22*d^8 - 131560*a^22*b^11*c^21*d^9 + 154440*a
^23*b^10*c^20*d^10 - 143000*a^24*b^9*c^19*d^11 + 104104*a^25*b^8*c^18*d^12 - 58968*a^26*b^7*c^17*d^13 + 25480*
a^27*b^6*c^16*d^14 - 8120*a^28*b^5*c^15*d^15 + 1800*a^29*b^4*c^14*d^16 - 248*a^30*b^3*c^13*d^17 + 16*a^31*b^2*
c^12*d^18))/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^
3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d))))/(2*(b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c
^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d)))*(4*a*d - 9*b*c))/(2*(
b^7*c^10 - a^7*c^3*d^7 + 7*a^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a
^5*b^2*c^5*d^5 - 7*a*b^6*c^9*d))))*(d^7*(a*d - b*c)^7)^(1/2)*(4*a*d - 9*b*c)*1i)/(b^7*c^10 - a^7*c^3*d^7 + 7*a
^6*b*c^4*d^6 + 21*a^2*b^5*c^8*d^2 - 35*a^3*b^4*c^7*d^3 + 35*a^4*b^3*c^6*d^4 - 21*a^5*b^2*c^5*d^5 - 7*a*b^6*c^9
*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/(a+b/x)**(5/2)/(c+d/x)**2,x)

[Out]

Timed out

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