3.258 \(\int \frac {1}{(a+\frac {b}{x})^{3/2} (c+\frac {d}{x})^3} \, dx\)
Optimal. Leaf size=320 \[ -\frac {3 (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2} c^4}+\frac {3 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{7/2}}+\frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{4 a c^3 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)^2}+\frac {3 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{4 a^2 c^3 \sqrt {a+\frac {b}{x}} (b c-a d)^3}+\frac {d (2 b c-3 a d)}{2 a c^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \]
[Out]
3/4*d^(5/2)*(8*a^2*d^2-24*a*b*c*d+21*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)^(7
/2)-3*(2*a*d+b*c)*arctanh((a+b/x)^(1/2)/a^(1/2))/a^(5/2)/c^4+3/4*b*(-a*d+2*b*c)*(4*a^2*d^2-a*b*c*d+2*b^2*c^2)/
a^2/c^3/(-a*d+b*c)^3/(a+b/x)^(1/2)+1/2*d*(-3*a*d+2*b*c)/a/c^2/(-a*d+b*c)/(c+d/x)^2/(a+b/x)^(1/2)+1/4*d*(12*a^2
*d^2-21*a*b*c*d+4*b^2*c^2)/a/c^3/(-a*d+b*c)^2/(c+d/x)/(a+b/x)^(1/2)+x/a/c/(c+d/x)^2/(a+b/x)^(1/2)
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Rubi [A] time = 0.52, antiderivative size = 320, 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} \[ \frac {d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{4 a c^3 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right ) (b c-a d)^2}+\frac {3 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{4 a^2 c^3 \sqrt {a+\frac {b}{x}} (b c-a d)^3}+\frac {3 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 (b c-a d)^{7/2}}-\frac {3 (2 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2} c^4}+\frac {d (2 b c-3 a d)}{2 a c^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2 (b c-a d)}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[1/((a + b/x)^(3/2)*(c + d/x)^3),x]
[Out]
(3*b*(2*b*c - a*d)*(2*b^2*c^2 - a*b*c*d + 4*a^2*d^2))/(4*a^2*c^3*(b*c - a*d)^3*Sqrt[a + b/x]) + (d*(2*b*c - 3*
a*d))/(2*a*c^2*(b*c - a*d)*Sqrt[a + b/x]*(c + d/x)^2) + (d*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2))/(4*a*c^3*(b*
c - a*d)^2*Sqrt[a + b/x]*(c + d/x)) + x/(a*c*Sqrt[a + b/x]*(c + d/x)^2) + (3*d^(5/2)*(21*b^2*c^2 - 24*a*b*c*d
+ 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*(b*c - a*d)^(7/2)) - (3*(b*c + 2*a*d)*Arc
Tanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(5/2)*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 )^{3/2} \left (c+\frac {d}{x}\right )^3} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^{3/2} (c+d x)^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{2} (b c+2 a d)+\frac {7 b d x}{2}}{x (a+b x)^{3/2} (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) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-3 (b c-a d) (b c+2 a d)-\frac {5}{2} b d (2 b c-3 a d) x}{x (a+b x)^{3/2} (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) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {3 (b c-a d)^2 (b c+2 a d)+\frac {3}{4} b d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right ) x}{x (a+b x)^{3/2} (c+d x)} \, dx,x,\frac {1}{x}\right )}{2 a c^3 (b c-a d)^2}\\ &=\frac {3 b (2 b c-a d) \left (2 b^2 c^2-a b c d+4 a^2 d^2\right )}{4 a^2 c^3 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (2 b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{2} (b c-a d)^3 (b c+2 a d)+\frac {3}{8} b d (2 b c-a d) \left (2 b^2 c^2-a b c d+4 a^2 d^2\right ) x}{x \sqrt {a+b x} (c+d x)} \, dx,x,\frac {1}{x}\right )}{a^2 c^3 (b c-a d)^3}\\ &=\frac {3 b (2 b c-a d) \left (2 b^2 c^2-a b c d+4 a^2 d^2\right )}{4 a^2 c^3 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (2 b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {(3 (b c+2 a d)) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\frac {1}{x}\right )}{2 a^2 c^4}+\frac {\left (3 d^3 \left (21 b^2 c^2-24 a b c d+8 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)^3}\\ &=\frac {3 b (2 b c-a d) \left (2 b^2 c^2-a b c d+4 a^2 d^2\right )}{4 a^2 c^3 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (2 b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {(3 (b c+2 a d)) \operatorname {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+\frac {b}{x}}\right )}{a^2 b c^4}+\frac {\left (3 d^3 \left (21 b^2 c^2-24 a b c d+8 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)^3}\\ &=\frac {3 b (2 b c-a d) \left (2 b^2 c^2-a b c d+4 a^2 d^2\right )}{4 a^2 c^3 (b c-a d)^3 \sqrt {a+\frac {b}{x}}}+\frac {d (2 b c-3 a d)}{2 a c^2 (b c-a d) \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {d \left (4 b^2 c^2-21 a b c d+12 a^2 d^2\right )}{4 a c^3 (b c-a d)^2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}+\frac {x}{a c \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}+\frac {3 d^{5/2} \left (21 b^2 c^2-24 a b c d+8 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)^{7/2}}-\frac {3 (b c+2 a d) \tanh ^{-1}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{a^{5/2} c^4}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 239, normalized size = 0.75 \[ \frac {(c x+d) \left (2 (c x+d) \left (\frac {3}{4} a^2 d^2 \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {d \left (a+\frac {b}{x}\right )}{a d-b c}\right )+3 (2 a d+b c) (b c-a d)^3 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {b}{a x}+1\right )\right )-\frac {1}{2} a c d x (a d-b c) \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )\right )+2 a c^3 x^3 (b c-a d)^3-a c^2 d x^2 (b c-a d)^2 (3 a d-2 b c)}{2 a^2 c^4 \sqrt {a+\frac {b}{x}} (c x+d)^2 (b c-a d)^3} \]
Antiderivative was successfully verified.
[In]
Integrate[1/((a + b/x)^(3/2)*(c + d/x)^3),x]
[Out]
(-(a*c^2*d*(b*c - a*d)^2*(-2*b*c + 3*a*d)*x^2) + 2*a*c^3*(b*c - a*d)^3*x^3 + (d + c*x)*(-1/2*(a*c*d*(-(b*c) +
a*d)*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2)*x) + 2*(d + c*x)*((3*a^2*d^2*(21*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*
Hypergeometric2F1[-1/2, 1, 1/2, (d*(a + b/x))/(-(b*c) + a*d)])/4 + 3*(b*c - a*d)^3*(b*c + 2*a*d)*Hypergeometri
c2F1[-1/2, 1, 1/2, 1 + b/(a*x)])))/(2*a^2*c^4*(b*c - a*d)^3*Sqrt[a + b/x]*(d + c*x)^2)
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fricas [B] time = 5.03, size = 4093, normalized size = 12.79 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/(c+d/x)^3,x, algorithm="fricas")
[Out]
[1/8*(12*(b^5*c^4*d^2 - a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 5*a^3*b^2*c*d^5 - 2*a^4*b*d^6 + (a*b^4*c^6 - a^2*b
^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 5*a^4*b*c^3*d^3 - 2*a^5*c^2*d^4)*x^3 + (b^5*c^6 + a*b^4*c^5*d - 5*a^2*b^3*c^4*d
^2 - a^3*b^2*c^3*d^3 + 8*a^4*b*c^2*d^4 - 4*a^5*c*d^5)*x^2 + (2*b^5*c^5*d - a*b^4*c^4*d^2 - 7*a^2*b^3*c^3*d^3 +
7*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 2*a^5*d^6)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) - 3*(2
1*a^3*b^3*c^2*d^4 - 24*a^4*b^2*c*d^5 + 8*a^5*b*d^6 + (21*a^4*b^2*c^4*d^2 - 24*a^5*b*c^3*d^3 + 8*a^6*c^2*d^4)*x
^3 + (21*a^3*b^3*c^4*d^2 + 18*a^4*b^2*c^3*d^3 - 40*a^5*b*c^2*d^4 + 16*a^6*c*d^5)*x^2 + (42*a^3*b^3*c^3*d^3 - 2
7*a^4*b^2*c^2*d^4 - 8*a^5*b*c*d^5 + 8*a^6*d^6)*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*(4*(a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*
d^2 - a^5*c^3*d^3)*x^4 + (12*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 12*a^3*b^2*c^4*d^2 + 37*a^4*b*c^3*d^3 - 18*a^5*c^2*
d^4)*x^3 + (24*a*b^4*c^5*d - 20*a^2*b^3*c^4*d^2 + 29*a^3*b^2*c^3*d^3 + 9*a^4*b*c^2*d^4 - 12*a^5*c*d^5)*x^2 + 3
*(4*a*b^4*c^4*d^2 - 4*a^2*b^3*c^3*d^3 + 9*a^3*b^2*c^2*d^4 - 4*a^4*b*c*d^5)*x)*sqrt((a*x + b)/x))/(a^3*b^4*c^7*
d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2
- a^7*c^6*d^3)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2
+ (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x), 1/8*(24*(b^5*c^4
*d^2 - a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 5*a^3*b^2*c*d^5 - 2*a^4*b*d^6 + (a*b^4*c^6 - a^2*b^3*c^5*d - 3*a^3*
b^2*c^4*d^2 + 5*a^4*b*c^3*d^3 - 2*a^5*c^2*d^4)*x^3 + (b^5*c^6 + a*b^4*c^5*d - 5*a^2*b^3*c^4*d^2 - a^3*b^2*c^3*
d^3 + 8*a^4*b*c^2*d^4 - 4*a^5*c*d^5)*x^2 + (2*b^5*c^5*d - a*b^4*c^4*d^2 - 7*a^2*b^3*c^3*d^3 + 7*a^3*b^2*c^2*d^
4 + a^4*b*c*d^5 - 2*a^5*d^6)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) - 3*(21*a^3*b^3*c^2*d^4 - 24*a^4
*b^2*c*d^5 + 8*a^5*b*d^6 + (21*a^4*b^2*c^4*d^2 - 24*a^5*b*c^3*d^3 + 8*a^6*c^2*d^4)*x^3 + (21*a^3*b^3*c^4*d^2 +
18*a^4*b^2*c^3*d^3 - 40*a^5*b*c^2*d^4 + 16*a^6*c*d^5)*x^2 + (42*a^3*b^3*c^3*d^3 - 27*a^4*b^2*c^2*d^4 - 8*a^5*
b*c*d^5 + 8*a^6*d^6)*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*(4*(a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*x^4 + (
12*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 12*a^3*b^2*c^4*d^2 + 37*a^4*b*c^3*d^3 - 18*a^5*c^2*d^4)*x^3 + (24*a*b^4*c^5*d
- 20*a^2*b^3*c^4*d^2 + 29*a^3*b^2*c^3*d^3 + 9*a^4*b*c^2*d^4 - 12*a^5*c*d^5)*x^2 + 3*(4*a*b^4*c^4*d^2 - 4*a^2*
b^3*c^3*d^3 + 9*a^3*b^2*c^2*d^4 - 4*a^4*b*c*d^5)*x)*sqrt((a*x + b)/x))/(a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 +
3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^3 + (a^3
*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (2*a^3*b^4*c^8*d - 5*a^4
*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x), 1/4*(3*(21*a^3*b^3*c^2*d^4 - 24*a^4*b^2*c*
d^5 + 8*a^5*b*d^6 + (21*a^4*b^2*c^4*d^2 - 24*a^5*b*c^3*d^3 + 8*a^6*c^2*d^4)*x^3 + (21*a^3*b^3*c^4*d^2 + 18*a^4
*b^2*c^3*d^3 - 40*a^5*b*c^2*d^4 + 16*a^6*c*d^5)*x^2 + (42*a^3*b^3*c^3*d^3 - 27*a^4*b^2*c^2*d^4 - 8*a^5*b*c*d^5
+ 8*a^6*d^6)*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*(b^5*c^4*d^2 - a*b^4*c^3*d^3 - 3*a^2*b^3*c^2*d^4 + 5*a^3*b^2*c*d^5 - 2*a^4*b*d^6 + (a*b^4*c^6 - a^2*b^3*
c^5*d - 3*a^3*b^2*c^4*d^2 + 5*a^4*b*c^3*d^3 - 2*a^5*c^2*d^4)*x^3 + (b^5*c^6 + a*b^4*c^5*d - 5*a^2*b^3*c^4*d^2
- a^3*b^2*c^3*d^3 + 8*a^4*b*c^2*d^4 - 4*a^5*c*d^5)*x^2 + (2*b^5*c^5*d - a*b^4*c^4*d^2 - 7*a^2*b^3*c^3*d^3 + 7*
a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 2*a^5*d^6)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (4*(a^2
*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*c^3*d^3)*x^4 + (12*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 12*a^3*b^2
*c^4*d^2 + 37*a^4*b*c^3*d^3 - 18*a^5*c^2*d^4)*x^3 + (24*a*b^4*c^5*d - 20*a^2*b^3*c^4*d^2 + 29*a^3*b^2*c^3*d^3
+ 9*a^4*b*c^2*d^4 - 12*a^5*c*d^5)*x^2 + 3*(4*a*b^4*c^4*d^2 - 4*a^2*b^3*c^3*d^3 + 9*a^3*b^2*c^2*d^4 - 4*a^4*b*c
*d^5)*x)*sqrt((a*x + b)/x))/(a^3*b^4*c^7*d^2 - 3*a^4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5 + (a^4*b^
3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6*d^3)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^
2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (2*a^3*b^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^
5*d^4 - a^7*c^4*d^5)*x), 1/4*(3*(21*a^3*b^3*c^2*d^4 - 24*a^4*b^2*c*d^5 + 8*a^5*b*d^6 + (21*a^4*b^2*c^4*d^2 - 2
4*a^5*b*c^3*d^3 + 8*a^6*c^2*d^4)*x^3 + (21*a^3*b^3*c^4*d^2 + 18*a^4*b^2*c^3*d^3 - 40*a^5*b*c^2*d^4 + 16*a^6*c*
d^5)*x^2 + (42*a^3*b^3*c^3*d^3 - 27*a^4*b^2*c^2*d^4 - 8*a^5*b*c*d^5 + 8*a^6*d^6)*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)) + 12*(b^5*c^4*d^2 - a*b^4*c^3*d^3 - 3*a^2
*b^3*c^2*d^4 + 5*a^3*b^2*c*d^5 - 2*a^4*b*d^6 + (a*b^4*c^6 - a^2*b^3*c^5*d - 3*a^3*b^2*c^4*d^2 + 5*a^4*b*c^3*d^
3 - 2*a^5*c^2*d^4)*x^3 + (b^5*c^6 + a*b^4*c^5*d - 5*a^2*b^3*c^4*d^2 - a^3*b^2*c^3*d^3 + 8*a^4*b*c^2*d^4 - 4*a^
5*c*d^5)*x^2 + (2*b^5*c^5*d - a*b^4*c^4*d^2 - 7*a^2*b^3*c^3*d^3 + 7*a^3*b^2*c^2*d^4 + a^4*b*c*d^5 - 2*a^5*d^6)
*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (4*(a^2*b^3*c^6 - 3*a^3*b^2*c^5*d + 3*a^4*b*c^4*d^2 - a^5*
c^3*d^3)*x^4 + (12*a*b^4*c^6 - 4*a^2*b^3*c^5*d - 12*a^3*b^2*c^4*d^2 + 37*a^4*b*c^3*d^3 - 18*a^5*c^2*d^4)*x^3 +
(24*a*b^4*c^5*d - 20*a^2*b^3*c^4*d^2 + 29*a^3*b^2*c^3*d^3 + 9*a^4*b*c^2*d^4 - 12*a^5*c*d^5)*x^2 + 3*(4*a*b^4*
c^4*d^2 - 4*a^2*b^3*c^3*d^3 + 9*a^3*b^2*c^2*d^4 - 4*a^4*b*c*d^5)*x)*sqrt((a*x + b)/x))/(a^3*b^4*c^7*d^2 - 3*a^
4*b^3*c^6*d^3 + 3*a^5*b^2*c^5*d^4 - a^6*b*c^4*d^5 + (a^4*b^3*c^9 - 3*a^5*b^2*c^8*d + 3*a^6*b*c^7*d^2 - a^7*c^6
*d^3)*x^3 + (a^3*b^4*c^9 - a^4*b^3*c^8*d - 3*a^5*b^2*c^7*d^2 + 5*a^6*b*c^6*d^3 - 2*a^7*c^5*d^4)*x^2 + (2*a^3*b
^4*c^8*d - 5*a^4*b^3*c^7*d^2 + 3*a^5*b^2*c^6*d^3 + a^6*b*c^5*d^4 - a^7*c^4*d^5)*x)]
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giac [A] time = 0.31, size = 516, normalized size = 1.61 \[ \frac {1}{4} \, b^{4} {\left (\frac {3 \, {\left (21 \, b^{2} c^{2} d^{3} - 24 \, a b c d^{4} + 8 \, a^{2} d^{5}\right )} \arctan \left (\frac {d \sqrt {\frac {a x + b}{x}}}{\sqrt {b c d - a d^{2}}}\right )}{{\left (b^{7} c^{7} - 3 \, a b^{6} c^{6} d + 3 \, a^{2} b^{5} c^{5} d^{2} - a^{3} b^{4} c^{4} d^{3}\right )} \sqrt {b c d - a d^{2}}} + \frac {4 \, {\left (2 \, a b^{3} c^{3} - \frac {3 \, {\left (a x + b\right )} b^{3} c^{3}}{x} + \frac {3 \, {\left (a x + b\right )} a b^{2} c^{2} d}{x} - \frac {3 \, {\left (a x + b\right )} a^{2} b c d^{2}}{x} + \frac {{\left (a x + b\right )} a^{3} d^{3}}{x}\right )}}{{\left (a^{2} b^{6} c^{6} - 3 \, a^{3} b^{5} c^{5} d + 3 \, a^{4} b^{4} c^{4} d^{2} - a^{5} b^{3} c^{3} d^{3}\right )} {\left (a \sqrt {\frac {a x + b}{x}} - \frac {{\left (a x + b\right )} \sqrt {\frac {a x + b}{x}}}{x}\right )}} + \frac {17 \, b^{2} c^{2} d^{3} \sqrt {\frac {a x + b}{x}} - 25 \, a b c d^{4} \sqrt {\frac {a x + b}{x}} + 8 \, a^{2} d^{5} \sqrt {\frac {a x + b}{x}} + \frac {15 \, {\left (a x + b\right )} b c d^{4} \sqrt {\frac {a x + b}{x}}}{x} - \frac {8 \, {\left (a x + b\right )} a d^{5} \sqrt {\frac {a x + b}{x}}}{x}}{{\left (b^{6} c^{6} - 3 \, a b^{5} c^{5} d + 3 \, a^{2} b^{4} c^{4} d^{2} - a^{3} b^{3} c^{3} d^{3}\right )} {\left (b c - a d + \frac {{\left (a x + b\right )} d}{x}\right )}^{2}} + \frac {12 \, {\left (b c + 2 \, a d\right )} \arctan \left (\frac {\sqrt {\frac {a x + b}{x}}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2} b^{4} c^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/(c+d/x)^3,x, algorithm="giac")
[Out]
1/4*b^4*(3*(21*b^2*c^2*d^3 - 24*a*b*c*d^4 + 8*a^2*d^5)*arctan(d*sqrt((a*x + b)/x)/sqrt(b*c*d - a*d^2))/((b^7*c
^7 - 3*a*b^6*c^6*d + 3*a^2*b^5*c^5*d^2 - a^3*b^4*c^4*d^3)*sqrt(b*c*d - a*d^2)) + 4*(2*a*b^3*c^3 - 3*(a*x + b)*
b^3*c^3/x + 3*(a*x + b)*a*b^2*c^2*d/x - 3*(a*x + b)*a^2*b*c*d^2/x + (a*x + b)*a^3*d^3/x)/((a^2*b^6*c^6 - 3*a^3
*b^5*c^5*d + 3*a^4*b^4*c^4*d^2 - a^5*b^3*c^3*d^3)*(a*sqrt((a*x + b)/x) - (a*x + b)*sqrt((a*x + b)/x)/x)) + (17
*b^2*c^2*d^3*sqrt((a*x + b)/x) - 25*a*b*c*d^4*sqrt((a*x + b)/x) + 8*a^2*d^5*sqrt((a*x + b)/x) + 15*(a*x + b)*b
*c*d^4*sqrt((a*x + b)/x)/x - 8*(a*x + b)*a*d^5*sqrt((a*x + b)/x)/x)/((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)*(b*c - a*d + (a*x + b)*d/x)^2) + 12*(b*c + 2*a*d)*arctan(sqrt((a*x + b)/x)/sqrt(-a))/(s
qrt(-a)*a^2*b^4*c^4))
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maple [B] time = 0.08, size = 5158, normalized size = 16.12 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b/x)^(3/2)/(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 \[ \int \frac {1}{{\left (a + \frac {b}{x}\right )}^{\frac {3}{2}} {\left (c + \frac {d}{x}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(3/2)/(c+d/x)^3,x, algorithm="maxima")
[Out]
integrate(1/((a + b/x)^(3/2)*(c + d/x)^3), x)
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mupad [B] time = 9.49, size = 8936, normalized size = 27.92 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/((a + b/x)^(3/2)*(c + d/x)^3),x)
[Out]
((2*b^4)/(a^2*d - a*b*c) + (b*(a + b/x)*(12*a^4*d^4 + 12*b^4*c^4 + 24*a^2*b^2*c^2*d^2 - 40*a*b^3*c^3*d - 33*a^
3*b*c*d^3))/(4*a*c^3*(a^2*d - a*b*c)*(a*d - b*c)) + (3*b*(a + b/x)^3*(4*a^3*d^5 - 4*b^3*c^3*d^2 + 4*a*b^2*c^2*
d^3 - 9*a^2*b*c*d^4))/(4*a*c^3*(a^2*d - a*b*c)*(a*d - b*c)^2) - (b*(a + b/x)^2*(24*a^4*d^5 + 24*b^4*c^4*d - 56
*a*b^3*c^3*d^2 + 65*a^2*b^2*c^2*d^3 - 72*a^3*b*c*d^4))/(4*a*c^3*(a^2*d - a*b*c)*(a*d - b*c)^2))/((a + b/x)^(3/
2)*(3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d) - (a + b/x)^(5/2)*(3*a*d^2 - 2*b*c*d) + d^2*(a + b/x)^(7/2) - (a + b/x)^(
1/2)*(a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d)) + (atan(((((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b^1
8*c^25*d^4 + 903168*a^8*b^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 6007680*a^11*b
^14*c^21*d^8 + 1276416*a^12*b^13*c^20*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 407
418624*a^15*b^10*c^17*d^12 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 328809600*a^18*b^7*
c^14*d^15 - 164257920*a^19*b^6*c^13*d^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 2138112
*a^22*b^3*c^10*d^19 - 147456*a^23*b^2*c^9*d^20) - (3*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*
b*c*d)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^18*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^
27*d^5 + 9449472*a^12*b^15*c^26*d^6 - 14112768*a^13*b^14*c^25*d^7 + 10407936*a^14*b^13*c^24*d^8 + 6454272*a^15
*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 + 45551616*a^17*b^10*c^21*d^11 - 44064768*a^18*b^9*c^20*d^12 + 3
0096384*a^19*b^8*c^19*d^13 - 14831616*a^20*b^7*c^18*d^14 + 5203968*a^21*b^6*c^17*d^15 - 1241088*a^22*b^5*c^16*
d^16 + 181248*a^23*b^4*c^15*d^17 - 12288*a^24*b^3*c^14*d^18 - (3*(d^5*(a*d - b*c)^7)^(1/2)*(a + b/x)^(1/2)*(8*
a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(8192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*a^12*b^16*c^
31*d^4 - 5447680*a^13*b^15*c^30*d^5 + 18636800*a^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 + 90202112*a^1
6*b^12*c^27*d^8 - 134717440*a^17*b^11*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^9*c^24*d^11
+ 106602496*a^20*b^8*c^23*d^12 - 60383232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 8314880*a^23*b^5*
c^20*d^15 + 1843200*a^24*b^4*c^19*d^16 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(8*(b^7*c^11 -
a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6
*d^5 - 7*a*b^6*c^10*d))))/(8*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d
^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d)))*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2
*c^2 - 24*a*b*c*d)*3i)/(8*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3
+ 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d)) + (((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 20
2752*a^7*b^18*c^25*d^4 + 903168*a^8*b^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 60
07680*a^11*b^14*c^21*d^8 + 1276416*a^12*b^13*c^20*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^1
8*d^11 - 407418624*a^15*b^10*c^17*d^12 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 3288096
00*a^18*b^7*c^14*d^15 - 164257920*a^19*b^6*c^13*d^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^
18 + 2138112*a^22*b^3*c^10*d^19 - 147456*a^23*b^2*c^9*d^20) + (3*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2
*c^2 - 24*a*b*c*d)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^18*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*
a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26*d^6 - 14112768*a^13*b^14*c^25*d^7 + 10407936*a^14*b^13*c^24*d^8 +
6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 + 45551616*a^17*b^10*c^21*d^11 - 44064768*a^18*b^9*c
^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14831616*a^20*b^7*c^18*d^14 + 5203968*a^21*b^6*c^17*d^15 - 1241088*a^
22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^17 - 12288*a^24*b^3*c^14*d^18 + (3*(d^5*(a*d - b*c)^7)^(1/2)*(a + b/
x)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(8192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*
a^12*b^16*c^31*d^4 - 5447680*a^13*b^15*c^30*d^5 + 18636800*a^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 +
90202112*a^16*b^12*c^27*d^8 - 134717440*a^17*b^11*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^
9*c^24*d^11 + 106602496*a^20*b^8*c^23*d^12 - 60383232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 83148
80*a^23*b^5*c^20*d^15 + 1843200*a^24*b^4*c^19*d^16 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(8
*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21
*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d))))/(8*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a
^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d)))*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*
d^2 + 21*b^2*c^2 - 24*a*b*c*d)*3i)/(8*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*
b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d)))/(290304*a^6*b^18*c^21*d^5 - 2654208*
a^7*b^17*c^20*d^6 + 10675584*a^8*b^16*c^19*d^7 - 23497344*a^9*b^15*c^18*d^8 + 23604480*a^10*b^14*c^17*d^9 + 24
731136*a^11*b^13*c^16*d^10 - 148172544*a^12*b^12*c^15*d^11 + 320101632*a^13*b^11*c^14*d^12 - 452086272*a^14*b^
10*c^13*d^13 + 459302400*a^15*b^9*c^12*d^14 - 343108224*a^16*b^8*c^11*d^15 + 187373952*a^17*b^7*c^10*d^16 - 72
873216*a^18*b^6*c^9*d^17 + 19132416*a^19*b^5*c^8*d^18 - 3041280*a^20*b^4*c^7*d^19 + 221184*a^21*b^3*c^6*d^20 -
(3*((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b^18*c^25*d^4 + 903168*a^8*b^17*c^24*d^5 - 1751040*
a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 6007680*a^11*b^14*c^21*d^8 + 1276416*a^12*b^13*c^20*d^9 - 6538
2912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 407418624*a^15*b^10*c^17*d^12 + 521961984*a^16*b^9*
c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 328809600*a^18*b^7*c^14*d^15 - 164257920*a^19*b^6*c^13*d^16 + 58816
512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 2138112*a^22*b^3*c^10*d^19 - 147456*a^23*b^2*c^9*d^20)
- (3*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^1
8*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26*d^6 - 14112768*a
^13*b^14*c^25*d^7 + 10407936*a^14*b^13*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 +
45551616*a^17*b^10*c^21*d^11 - 44064768*a^18*b^9*c^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14831616*a^20*b^7*c
^18*d^14 + 5203968*a^21*b^6*c^17*d^15 - 1241088*a^22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^17 - 12288*a^24*b^
3*c^14*d^18 - (3*(d^5*(a*d - b*c)^7)^(1/2)*(a + b/x)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(8192*a^10*b^
18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*a^12*b^16*c^31*d^4 - 5447680*a^13*b^15*c^30*d^5 + 18636800*a
^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 + 90202112*a^16*b^12*c^27*d^8 - 134717440*a^17*b^11*c^26*d^9 +
158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^9*c^24*d^11 + 106602496*a^20*b^8*c^23*d^12 - 60383232*a^21*b
^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 8314880*a^23*b^5*c^20*d^15 + 1843200*a^24*b^4*c^19*d^16 - 253952*
a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(8*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*
d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d))))/(8*(b^7*c^11 - a^7*c^4
*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7
*a*b^6*c^10*d)))*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d))/(8*(b^7*c^11 - a^7*c^4*d^7 +
7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6
*c^10*d)) + (3*((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b^18*c^25*d^4 + 903168*a^8*b^17*c^24*d^5
- 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 6007680*a^11*b^14*c^21*d^8 + 1276416*a^12*b^13*c^20
*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 407418624*a^15*b^10*c^17*d^12 + 52196198
4*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 328809600*a^18*b^7*c^14*d^15 - 164257920*a^19*b^6*c^13*d
^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 2138112*a^22*b^3*c^10*d^19 - 147456*a^23*b^2
*c^9*d^20) + (3*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(12288*a^8*b^19*c^30*d^2 - 172
032*a^9*b^18*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26*d^6 -
14112768*a^13*b^14*c^25*d^7 + 10407936*a^14*b^13*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c
^22*d^10 + 45551616*a^17*b^10*c^21*d^11 - 44064768*a^18*b^9*c^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14831616
*a^20*b^7*c^18*d^14 + 5203968*a^21*b^6*c^17*d^15 - 1241088*a^22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^17 - 12
288*a^24*b^3*c^14*d^18 + (3*(d^5*(a*d - b*c)^7)^(1/2)*(a + b/x)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*(8
192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*a^12*b^16*c^31*d^4 - 5447680*a^13*b^15*c^30*d^5 +
18636800*a^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 + 90202112*a^16*b^12*c^27*d^8 - 134717440*a^17*b^11
*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^9*c^24*d^11 + 106602496*a^20*b^8*c^23*d^12 - 6038
3232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 8314880*a^23*b^5*c^20*d^15 + 1843200*a^24*b^4*c^19*d^1
6 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(8*(b^7*c^11 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a
^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5 - 7*a*b^6*c^10*d))))/(8*(b^7*c^1
1 - a^7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*
c^6*d^5 - 7*a*b^6*c^10*d)))*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d))/(8*(b^7*c^11 - a^
7*c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^
5 - 7*a*b^6*c^10*d))))*(d^5*(a*d - b*c)^7)^(1/2)*(8*a^2*d^2 + 21*b^2*c^2 - 24*a*b*c*d)*3i)/(4*(b^7*c^11 - a^7*
c^4*d^7 + 7*a^6*b*c^5*d^6 + 21*a^2*b^5*c^9*d^2 - 35*a^3*b^4*c^8*d^3 + 35*a^4*b^3*c^7*d^4 - 21*a^5*b^2*c^6*d^5
- 7*a*b^6*c^10*d)) + (atan((((2*a*d + b*c)*((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b^18*c^25*d^
4 + 903168*a^8*b^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 6007680*a^11*b^14*c^21*
d^8 + 1276416*a^12*b^13*c^20*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 407418624*a^
15*b^10*c^17*d^12 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 328809600*a^18*b^7*c^14*d^15
- 164257920*a^19*b^6*c^13*d^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 2138112*a^22*b^3
*c^10*d^19 - 147456*a^23*b^2*c^9*d^20) - (3*(2*a*d + b*c)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^18*c^29*d^3
+ 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26*d^6 - 14112768*a^13*b^14*c^
25*d^7 + 10407936*a^14*b^13*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 + 45551616*a^
17*b^10*c^21*d^11 - 44064768*a^18*b^9*c^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14831616*a^20*b^7*c^18*d^14 +
5203968*a^21*b^6*c^17*d^15 - 1241088*a^22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^17 - 12288*a^24*b^3*c^14*d^18
- (3*(a + b/x)^(1/2)*(2*a*d + b*c)*(8192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*a^12*b^16*c
^31*d^4 - 5447680*a^13*b^15*c^30*d^5 + 18636800*a^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 + 90202112*a^
16*b^12*c^27*d^8 - 134717440*a^17*b^11*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^9*c^24*d^11
+ 106602496*a^20*b^8*c^23*d^12 - 60383232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 8314880*a^23*b^5
*c^20*d^15 + 1843200*a^24*b^4*c^19*d^16 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(2*c^4*(a^5)^
(1/2))))/(2*c^4*(a^5)^(1/2)))*3i)/(2*c^4*(a^5)^(1/2)) + ((2*a*d + b*c)*((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d
^3 - 202752*a^7*b^18*c^25*d^4 + 903168*a^8*b^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d
^7 + 6007680*a^11*b^14*c^21*d^8 + 1276416*a^12*b^13*c^20*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b
^11*c^18*d^11 - 407418624*a^15*b^10*c^17*d^12 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 +
328809600*a^18*b^7*c^14*d^15 - 164257920*a^19*b^6*c^13*d^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*
c^11*d^18 + 2138112*a^22*b^3*c^10*d^19 - 147456*a^23*b^2*c^9*d^20) + (3*(2*a*d + b*c)*(12288*a^8*b^19*c^30*d^2
- 172032*a^9*b^18*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26
*d^6 - 14112768*a^13*b^14*c^25*d^7 + 10407936*a^14*b^13*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*
b^11*c^22*d^10 + 45551616*a^17*b^10*c^21*d^11 - 44064768*a^18*b^9*c^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14
831616*a^20*b^7*c^18*d^14 + 5203968*a^21*b^6*c^17*d^15 - 1241088*a^22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^1
7 - 12288*a^24*b^3*c^14*d^18 + (3*(a + b/x)^(1/2)*(2*a*d + b*c)*(8192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^
32*d^3 + 1105920*a^12*b^16*c^31*d^4 - 5447680*a^13*b^15*c^30*d^5 + 18636800*a^14*b^14*c^29*d^6 - 46964736*a^15
*b^13*c^28*d^7 + 90202112*a^16*b^12*c^27*d^8 - 134717440*a^17*b^11*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 -
146432000*a^19*b^9*c^24*d^11 + 106602496*a^20*b^8*c^23*d^12 - 60383232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*
c^21*d^14 - 8314880*a^23*b^5*c^20*d^15 + 1843200*a^24*b^4*c^19*d^16 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b
^2*c^17*d^18))/(2*c^4*(a^5)^(1/2))))/(2*c^4*(a^5)^(1/2)))*3i)/(2*c^4*(a^5)^(1/2)))/(290304*a^6*b^18*c^21*d^5 -
2654208*a^7*b^17*c^20*d^6 + 10675584*a^8*b^16*c^19*d^7 - 23497344*a^9*b^15*c^18*d^8 + 23604480*a^10*b^14*c^17
*d^9 + 24731136*a^11*b^13*c^16*d^10 - 148172544*a^12*b^12*c^15*d^11 + 320101632*a^13*b^11*c^14*d^12 - 45208627
2*a^14*b^10*c^13*d^13 + 459302400*a^15*b^9*c^12*d^14 - 343108224*a^16*b^8*c^11*d^15 + 187373952*a^17*b^7*c^10*
d^16 - 72873216*a^18*b^6*c^9*d^17 + 19132416*a^19*b^5*c^8*d^18 - 3041280*a^20*b^4*c^7*d^19 + 221184*a^21*b^3*c
^6*d^20 - (3*(2*a*d + b*c)*((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b^18*c^25*d^4 + 903168*a^8*b
^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 6007680*a^11*b^14*c^21*d^8 + 1276416*a^
12*b^13*c^20*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 407418624*a^15*b^10*c^17*d^1
2 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 328809600*a^18*b^7*c^14*d^15 - 164257920*a^1
9*b^6*c^13*d^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 2138112*a^22*b^3*c^10*d^19 - 147
456*a^23*b^2*c^9*d^20) - (3*(2*a*d + b*c)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^18*c^29*d^3 + 1081344*a^10*b
^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26*d^6 - 14112768*a^13*b^14*c^25*d^7 + 1040793
6*a^14*b^13*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 + 45551616*a^17*b^10*c^21*d^1
1 - 44064768*a^18*b^9*c^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14831616*a^20*b^7*c^18*d^14 + 5203968*a^21*b^6
*c^17*d^15 - 1241088*a^22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^17 - 12288*a^24*b^3*c^14*d^18 - (3*(a + b/x)^
(1/2)*(2*a*d + b*c)*(8192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*a^12*b^16*c^31*d^4 - 544768
0*a^13*b^15*c^30*d^5 + 18636800*a^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 + 90202112*a^16*b^12*c^27*d^8
- 134717440*a^17*b^11*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^9*c^24*d^11 + 106602496*a^2
0*b^8*c^23*d^12 - 60383232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 8314880*a^23*b^5*c^20*d^15 + 184
3200*a^24*b^4*c^19*d^16 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(2*c^4*(a^5)^(1/2))))/(2*c^4*
(a^5)^(1/2))))/(2*c^4*(a^5)^(1/2)) + (3*(2*a*d + b*c)*((a + b/x)^(1/2)*(18432*a^6*b^19*c^26*d^3 - 202752*a^7*b
^18*c^25*d^4 + 903168*a^8*b^17*c^24*d^5 - 1751040*a^9*b^16*c^23*d^6 - 137088*a^10*b^15*c^22*d^7 + 6007680*a^11
*b^14*c^21*d^8 + 1276416*a^12*b^13*c^20*d^9 - 65382912*a^13*b^12*c^19*d^10 + 216610560*a^14*b^11*c^18*d^11 - 4
07418624*a^15*b^10*c^17*d^12 + 521961984*a^16*b^9*c^16*d^13 - 482904576*a^17*b^8*c^15*d^14 + 328809600*a^18*b^
7*c^14*d^15 - 164257920*a^19*b^6*c^13*d^16 + 58816512*a^20*b^5*c^12*d^17 - 14340096*a^21*b^4*c^11*d^18 + 21381
12*a^22*b^3*c^10*d^19 - 147456*a^23*b^2*c^9*d^20) + (3*(2*a*d + b*c)*(12288*a^8*b^19*c^30*d^2 - 172032*a^9*b^1
8*c^29*d^3 + 1081344*a^10*b^17*c^28*d^4 - 3996672*a^11*b^16*c^27*d^5 + 9449472*a^12*b^15*c^26*d^6 - 14112768*a
^13*b^14*c^25*d^7 + 10407936*a^14*b^13*c^24*d^8 + 6454272*a^15*b^12*c^23*d^9 - 30007296*a^16*b^11*c^22*d^10 +
45551616*a^17*b^10*c^21*d^11 - 44064768*a^18*b^9*c^20*d^12 + 30096384*a^19*b^8*c^19*d^13 - 14831616*a^20*b^7*c
^18*d^14 + 5203968*a^21*b^6*c^17*d^15 - 1241088*a^22*b^5*c^16*d^16 + 181248*a^23*b^4*c^15*d^17 - 12288*a^24*b^
3*c^14*d^18 + (3*(a + b/x)^(1/2)*(2*a*d + b*c)*(8192*a^10*b^18*c^33*d^2 - 139264*a^11*b^17*c^32*d^3 + 1105920*
a^12*b^16*c^31*d^4 - 5447680*a^13*b^15*c^30*d^5 + 18636800*a^14*b^14*c^29*d^6 - 46964736*a^15*b^13*c^28*d^7 +
90202112*a^16*b^12*c^27*d^8 - 134717440*a^17*b^11*c^26*d^9 + 158146560*a^18*b^10*c^25*d^10 - 146432000*a^19*b^
9*c^24*d^11 + 106602496*a^20*b^8*c^23*d^12 - 60383232*a^21*b^7*c^22*d^13 + 26091520*a^22*b^6*c^21*d^14 - 83148
80*a^23*b^5*c^20*d^15 + 1843200*a^24*b^4*c^19*d^16 - 253952*a^25*b^3*c^18*d^17 + 16384*a^26*b^2*c^17*d^18))/(2
*c^4*(a^5)^(1/2))))/(2*c^4*(a^5)^(1/2))))/(2*c^4*(a^5)^(1/2))))*(2*a*d + b*c)*3i)/(c^4*(a^5)^(1/2))
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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)**(3/2)/(c+d/x)**3,x)
[Out]
Timed out
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