3.258 \(\int \frac{1}{(a+\frac{b}{x})^{3/2} (c+\frac{d}{x})^3} \, dx\)
Optimal. Leaf size=320 \[ \frac{d \left (12 a^2 d^2-21 a b c d+4 b^2 c^2\right )}{4 a c^3 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right ) (b c-a d)^2}+\frac{3 b (2 b c-a d) \left (4 a^2 d^2-a b c d+2 b^2 c^2\right )}{4 a^2 c^3 \sqrt{a+\frac{b}{x}} (b c-a d)^3}+\frac{3 d^{5/2} \left (8 a^2 d^2-24 a b c d+21 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 (b c-a d)^{7/2}}-\frac{3 (2 a d+b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{5/2} c^4}+\frac{d (2 b c-3 a d)}{2 a c^2 \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2 (b c-a d)}+\frac{x}{a c \sqrt{a+\frac{b}{x}} \left (c+\frac{d}{x}\right )^2} \]
[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)
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Rubi [A] time = 0.524976, antiderivative size = 320, normalized size of antiderivative = 1.,
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 375
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Rule 103
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])
Rule 151
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]
Rule 152
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]
Rule 156
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Rule 63
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]
Rule 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rule 205
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x}\right )^{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*}
Mathematica [C] time = 0.212408, 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)*(-(a*c*d*(-(b*c) + a*d)
*(4*b^2*c^2 - 21*a*b*c*d + 12*a^2*d^2)*x)/2 + 2*(d + c*x)*((3*a^2*d^2*(21*b^2*c^2 - 24*a*b*c*d + 8*a^2*d^2)*Hy
pergeometric2F1[-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)*Hypergeometric2
F1[-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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Maple [B] time = 0.016, size = 5158, normalized size = 16.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+b/x)^(3/2)/(c+d/x)^3,x)
[Out]
result too large to display
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x}\right )}^{\frac{3}{2}}{\left (c + \frac{d}{x}\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(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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Fricas [B] time = 10.4814, size = 8142, normalized size = 25.44 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)^(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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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+b/x)**(3/2)/(c+d/x)**3,x)
[Out]
Exception raised: ValueError
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Giac [A] time = 1.20721, size = 678, normalized size = 2.12 \begin{align*} \frac{1}{4} \, b{\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^{4} c^{7} - 3 \, a b^{3} c^{6} d + 3 \, a^{2} b^{2} c^{5} d^{2} - a^{3} b 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^{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}\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^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - a^{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 c^{4}}\right )} \end{align*}
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*(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^4*c^7
- 3*a*b^3*c^6*d + 3*a^2*b^2*c^5*d^2 - a^3*b*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^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)*(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*sqr
t((a*x + b)/x)/x - 8*(a*x + b)*a*d^5*sqrt((a*x + b)/x)/x)/((b^3*c^6 - 3*a*b^2*c^5*d + 3*a^2*b*c^4*d^2 - a^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))/(sqrt(-a)*a^2*b*c^
4))