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

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

[Out]

1/12*b*(-36*a^3*d^3+87*a^2*b*c*d^2-36*a*b^2*c^2*d+20*b^3*c^3)/a^2/c^3/(-a*d+b*c)^3/(a+b/x)^(3/2)+1/2*d*(-3*a*d
+2*b*c)/a/c^2/(-a*d+b*c)/(a+b/x)^(3/2)/(c+d/x)^2+1/4*d*(12*a^2*d^2-23*a*b*c*d+4*b^2*c^2)/a/c^3/(-a*d+b*c)^2/(a
+b/x)^(3/2)/(c+d/x)+x/a/c/(a+b/x)^(3/2)/(c+d/x)^2-1/4*d^(7/2)*(24*a^2*d^2-88*a*b*c*d+99*b^2*c^2)*arctan(d^(1/2
)*(a+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^4/(-a*d+b*c)^(9/2)-(6*a*d+5*b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(7/2)/c^
4+1/4*b*(12*a^4*d^4-35*a^3*b*c*d^3+24*a^2*b^2*c^2*d^2-56*a*b^3*c^3*d+20*b^4*c^4)/a^3/c^3/(-a*d+b*c)^4/(a+b/x)^
(1/2)

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Rubi [A]  time = 0.70, antiderivative size = 409, normalized size of antiderivative = 1.00, 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 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 )^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*}

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Mathematica [C]  time = 0.39, 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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fricas [B]  time = 14.87, size = 6171, normalized size = 15.09 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x)^(5/2)/(c+d/x)^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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giac [A]  time = 0.29, size = 523, normalized size = 1.28 \[ -\frac {1}{12} \, b^{4} {\left (\frac {3 \, {\left (99 \, b^{2} c^{2} d^{4} - 88 \, a b c d^{5} + 24 \, a^{2} d^{6}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{8} c^{8} - 4 \, a b^{7} c^{7} d + 6 \, a^{2} b^{6} c^{6} d^{2} - 4 \, a^{3} b^{5} c^{5} d^{3} + a^{4} b^{4} c^{4} d^{4}\right )} \sqrt {b c d - a d^{2}}} - \frac {8 \, {\left (a b c - a^{2} d + \frac {6 \, {\left (a x + b\right )} b c}{x} - \frac {15 \, {\left (a x + b\right )} a d}{x}\right )} x}{{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} {\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}} + \frac {3 \, {\left (21 \, b^{2} c^{2} d^{4} \sqrt {\frac {a x + b}{x}} - 29 \, a b c d^{5} \sqrt {\frac {a x + b}{x}} + 8 \, a^{2} d^{6} \sqrt {\frac {a x + b}{x}} + \frac {19 \, {\left (a x + b\right )} b c d^{5} \sqrt {\frac {a x + b}{x}}}{x} - \frac {8 \, {\left (a x + b\right )} a d^{6} \sqrt {\frac {a x + b}{x}}}{x}\right )}}{{\left (b^{7} c^{7} - 4 \, a b^{6} c^{6} d + 6 \, a^{2} b^{5} c^{5} d^{2} - 4 \, a^{3} b^{4} c^{4} d^{3} + a^{4} b^{3} c^{3} d^{4}\right )} {\left (b c - a d + \frac {{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac {12 \, \sqrt {\frac {a x + b}{x}}}{{\left (a - \frac {a x + b}{x}\right )} a^{3} b^{3} c^{3}} - \frac {12 \, {\left (5 \, b c + 6 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{3} b^{4} c^{4}}\right )} \]

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^4*(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^
8*c^8 - 4*a*b^7*c^7*d + 6*a^2*b^6*c^6*d^2 - 4*a^3*b^5*c^5*d^3 + a^4*b^4*c^4*d^4)*sqrt(b*c*d - a*d^2)) - 8*(a*b
*c - a^2*d + 6*(a*x + b)*b*c/x - 15*(a*x + b)*a*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*sqr
t((a*x + b)/x)/x)/((b^7*c^7 - 4*a*b^6*c^6*d + 6*a^2*b^5*c^5*d^2 - 4*a^3*b^4*c^4*d^3 + a^4*b^3*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*b^3*c^3) - 12*(5*b*c + 6*a*d)*arctan(sqr
t((a*x + b)/x)/sqrt(-a))/(sqrt(-a)*a^3*b^4*c^4))

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maple [B]  time = 0.08, size = 7300, normalized size = 17.85 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}} {\left (c + \frac {d}{x}\right )}^{3}}\,{d x} \]

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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mupad [B]  time = 8.23, size = 4284, normalized size = 10.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*b^4)/(3*(a^2*d - a*b*c)) + (2*b^4*(a + b/x)*(12*a*d - 5*b*c))/(3*(a^2*d - a*b*c)^2) + (b*(a + b/x)^2*(36*a
^5*d^5 - 60*b^5*c^5 - 456*a^2*b^3*c^3*d^2 + 120*a^3*b^2*c^2*d^3 + 308*a*b^4*c^4*d - 123*a^4*b*c*d^4))/(12*a^2*
c^3*(a^2*d - a*b*c)*(a*d - b*c)^2) + (b*(a + b/x)^4*(12*a^4*d^6 + 20*b^4*c^4*d^2 - 56*a*b^3*c^3*d^3 + 24*a^2*b
^2*c^2*d^4 - 35*a^3*b*c*d^5))/(4*a^2*c^3*(a^2*d - a*b*c)*(a*d - b*c)^3) - (b*(a + b/x)^3*(72*a^5*d^6 - 120*b^5
*c^5*d + 496*a*b^4*c^4*d^2 - 592*a^2*b^3*c^3*d^3 + 303*a^3*b^2*c^2*d^4 - 264*a^4*b*c*d^5))/(12*a^2*c^3*(a^2*d
- a*b*c)*(a*d - b*c)^3))/((a + b/x)^(5/2)*(3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d) - (a + b/x)^(7/2)*(3*a*d^2 - 2*b*c
*d) + d^2*(a + b/x)^(9/2) - (a + b/x)^(3/2)*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)) + (atan((a^15*b^24*c^24*(a +
b/x)^(1/2)*2000i + a^17*b^22*c^22*d^2*(a + b/x)^(1/2)*277440i - a^18*b^21*c^21*d^3*(a + b/x)^(1/2)*1325984i +
a^19*b^20*c^20*d^4*(a + b/x)^(1/2)*4135824i - a^20*b^19*c^19*d^5*(a + b/x)^(1/2)*8371440i + a^21*b^18*c^18*d^6
*(a + b/x)^(1/2)*9129120i + a^22*b^17*c^17*d^7*(a + b/x)^(1/2)*3058605i - a^23*b^16*c^16*d^8*(a + b/x)^(1/2)*3
2337558i + a^24*b^15*c^15*d^9*(a + b/x)^(1/2)*63677218i - a^25*b^14*c^14*d^10*(a + b/x)^(1/2)*66665280i + a^26
*b^13*c^13*d^11*(a + b/x)^(1/2)*24871035i + a^27*b^12*c^12*d^12*(a + b/x)^(1/2)*40203170i - a^28*b^11*c^11*d^1
3*(a + b/x)^(1/2)*85652532i + a^29*b^10*c^10*d^14*(a + b/x)^(1/2)*88170192i - a^30*b^9*c^9*d^15*(a + b/x)^(1/2
)*60362445i + a^31*b^8*c^8*d^16*(a + b/x)^(1/2)*29178270i - a^32*b^7*c^7*d^17*(a + b/x)^(1/2)*9940590i + a^33*
b^6*c^6*d^18*(a + b/x)^(1/2)*2287824i - a^34*b^5*c^5*d^19*(a + b/x)^(1/2)*320859i + a^35*b^4*c^4*d^20*(a + b/x
)^(1/2)*20790i - a^16*b^23*c^23*d*(a + b/x)^(1/2)*34800i)/(a^7*(a^7)^(1/2)*(a^7*(a^7*(a^7*(29178270*b^8*c^8*d^
16 - 9940590*a*b^7*c^7*d^17 + 2287824*a^2*b^6*c^6*d^18 - 320859*a^3*b^5*c^5*d^19 + 20790*a^4*b^4*c^4*d^20) + 6
3677218*b^15*c^15*d^9 - 66665280*a*b^14*c^14*d^10 + 24871035*a^2*b^13*c^13*d^11 + 40203170*a^3*b^12*c^12*d^12
- 85652532*a^4*b^11*c^11*d^13 + 88170192*a^5*b^10*c^10*d^14 - 60362445*a^6*b^9*c^9*d^15) + 277440*b^22*c^22*d^
2 - 1325984*a*b^21*c^21*d^3 + 4135824*a^2*b^20*c^20*d^4 - 8371440*a^3*b^19*c^19*d^5 + 9129120*a^4*b^18*c^18*d^
6 + 3058605*a^5*b^17*c^17*d^7 - 32337558*a^6*b^16*c^16*d^8) + 2000*a^5*b^24*c^24 - 34800*a^6*b^23*c^23*d)))*(6
*a*d + 5*b*c)*1i)/(c^4*(a^7)^(1/2)) + (log(400*b^25*c^25*d^4 - 8240*a*b^24*c^24*d^5 - 1152*a^11*d^5*(d^7*(a*d
- b*c)^9)^(3/2)*(a + b/x)^(1/2) + 1152*a^20*d^21*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 79696*a^2*b^23*c^
23*d^6 - 478768*a^3*b^22*c^22*d^7 + 1987568*a^4*b^21*c^21*d^8 - 5978896*a^5*b^20*c^20*d^9 + 13176240*a^6*b^19*
c^19*d^10 - 20525703*a^7*b^18*c^18*d^11 + 18765714*a^8*b^17*c^17*d^12 + 3763331*a^9*b^16*c^16*d^13 - 49787452*
a^10*b^15*c^15*d^14 + 104120705*a^11*b^14*c^14*d^15 - 140185682*a^12*b^13*c^13*d^16 + 139985251*a^13*b^12*c^12
*d^17 - 108046616*a^14*b^11*c^11*d^18 + 65184867*a^15*b^10*c^10*d^19 - 30607170*a^16*b^9*c^9*d^20 + 10996689*a
^17*b^8*c^8*d^21 - 2926572*a^18*b^7*c^7*d^22 + 544467*a^19*b^6*c^6*d^23 - 63294*a^20*b^5*c^5*d^24 + 3465*a^21*
b^4*c^4*d^25 + 400*b^20*c^20*d*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 9801*a^6*b^5*c^5*(d^7*(a*d - b*c)^9
)^(3/2)*(a + b/x)^(1/2) - 37026*a^7*b^4*c^4*d*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) - 6240*a*b^19*c^19*d^2
*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 47344*a^8*b^3*c^3*d^2*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) -
29216*a^9*b^2*c^2*d^3*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) + 44496*a^2*b^18*c^18*d^3*(d^7*(a*d - b*c)^9)
^(1/2)*(a + b/x)^(1/2) - 189888*a^3*b^17*c^17*d^4*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 528768*a^4*b^16*
c^16*d^5*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 959616*a^5*b^15*c^15*d^6*(d^7*(a*d - b*c)^9)^(1/2)*(a + b
/x)^(1/2) + 972681*a^6*b^14*c^14*d^7*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 41238*a^7*b^13*c^13*d^8*(d^7*
(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 1727195*a^8*b^12*c^12*d^9*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 2
139672*a^9*b^11*c^11*d^10*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 786834*a^10*b^10*c^10*d^11*(d^7*(a*d - b
*c)^9)^(1/2)*(a + b/x)^(1/2) - 6551292*a^11*b^9*c^9*d^12*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 11685186*
a^12*b^8*c^8*d^13*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 12876696*a^13*b^7*c^7*d^14*(d^7*(a*d - b*c)^9)^(
1/2)*(a + b/x)^(1/2) + 10033077*a^14*b^6*c^6*d^15*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 5737770*a^15*b^5
*c^5*d^16*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 2414601*a^16*b^4*c^4*d^17*(d^7*(a*d - b*c)^9)^(1/2)*(a +
b/x)^(1/2) - 731920*a^17*b^3*c^3*d^18*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 151904*a^18*b^2*c^2*d^19*(d
^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 9024*a^10*b*c*d^4*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) - 19392*
a^19*b*c*d^20*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2))*(d^7*(a*d - b*c)^9)^(1/2)*(3*a^2*d^2 + (99*b^2*c^2)/8
- 11*a*b*c*d))/(b^9*c^13 - a^9*c^4*d^9 + 9*a^8*b*c^5*d^8 + 36*a^2*b^7*c^11*d^2 - 84*a^3*b^6*c^10*d^3 + 126*a^
4*b^5*c^9*d^4 - 126*a^5*b^4*c^8*d^5 + 84*a^6*b^3*c^7*d^6 - 36*a^7*b^2*c^6*d^7 - 9*a*b^8*c^12*d) - (log(8240*a*
b^24*c^24*d^5 - 400*b^25*c^25*d^4 - 1152*a^11*d^5*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) + 1152*a^20*d^21*(
d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 79696*a^2*b^23*c^23*d^6 + 478768*a^3*b^22*c^22*d^7 - 1987568*a^4*b^
21*c^21*d^8 + 5978896*a^5*b^20*c^20*d^9 - 13176240*a^6*b^19*c^19*d^10 + 20525703*a^7*b^18*c^18*d^11 - 18765714
*a^8*b^17*c^17*d^12 - 3763331*a^9*b^16*c^16*d^13 + 49787452*a^10*b^15*c^15*d^14 - 104120705*a^11*b^14*c^14*d^1
5 + 140185682*a^12*b^13*c^13*d^16 - 139985251*a^13*b^12*c^12*d^17 + 108046616*a^14*b^11*c^11*d^18 - 65184867*a
^15*b^10*c^10*d^19 + 30607170*a^16*b^9*c^9*d^20 - 10996689*a^17*b^8*c^8*d^21 + 2926572*a^18*b^7*c^7*d^22 - 544
467*a^19*b^6*c^6*d^23 + 63294*a^20*b^5*c^5*d^24 - 3465*a^21*b^4*c^4*d^25 + 400*b^20*c^20*d*(d^7*(a*d - b*c)^9)
^(1/2)*(a + b/x)^(1/2) + 9801*a^6*b^5*c^5*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) - 37026*a^7*b^4*c^4*d*(d^7
*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) - 6240*a*b^19*c^19*d^2*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 47344
*a^8*b^3*c^3*d^2*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) - 29216*a^9*b^2*c^2*d^3*(d^7*(a*d - b*c)^9)^(3/2)*(
a + b/x)^(1/2) + 44496*a^2*b^18*c^18*d^3*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 189888*a^3*b^17*c^17*d^4*
(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 528768*a^4*b^16*c^16*d^5*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2)
- 959616*a^5*b^15*c^15*d^6*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 972681*a^6*b^14*c^14*d^7*(d^7*(a*d - b
*c)^9)^(1/2)*(a + b/x)^(1/2) + 41238*a^7*b^13*c^13*d^8*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 1727195*a^8
*b^12*c^12*d^9*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 2139672*a^9*b^11*c^11*d^10*(d^7*(a*d - b*c)^9)^(1/2
)*(a + b/x)^(1/2) + 786834*a^10*b^10*c^10*d^11*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 6551292*a^11*b^9*c^
9*d^12*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 11685186*a^12*b^8*c^8*d^13*(d^7*(a*d - b*c)^9)^(1/2)*(a + b
/x)^(1/2) - 12876696*a^13*b^7*c^7*d^14*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 10033077*a^14*b^6*c^6*d^15*
(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 5737770*a^15*b^5*c^5*d^16*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2
) + 2414601*a^16*b^4*c^4*d^17*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) - 731920*a^17*b^3*c^3*d^18*(d^7*(a*d -
b*c)^9)^(1/2)*(a + b/x)^(1/2) + 151904*a^18*b^2*c^2*d^19*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^(1/2) + 9024*a^1
0*b*c*d^4*(d^7*(a*d - b*c)^9)^(3/2)*(a + b/x)^(1/2) - 19392*a^19*b*c*d^20*(d^7*(a*d - b*c)^9)^(1/2)*(a + b/x)^
(1/2))*(d^7*(a*d - b*c)^9)^(1/2)*(24*a^2*d^2 + 99*b^2*c^2 - 88*a*b*c*d))/(8*(b^9*c^13 - a^9*c^4*d^9 + 9*a^8*b*
c^5*d^8 + 36*a^2*b^7*c^11*d^2 - 84*a^3*b^6*c^10*d^3 + 126*a^4*b^5*c^9*d^4 - 126*a^5*b^4*c^8*d^5 + 84*a^6*b^3*c
^7*d^6 - 36*a^7*b^2*c^6*d^7 - 9*a*b^8*c^12*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)**3,x)

[Out]

Timed out

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