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

Optimal. Leaf size=209 \[ -\frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 \sqrt{d} \sqrt{b c-a d}}-\frac{3 \sqrt{a+\frac{b}{x}} (b c-4 a d)}{4 c^3 \left (c+\frac{d}{x}\right )}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{3 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )^2} \]

[Out]

-((b*c - 3*a*d)*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) - (3*(b*c - 4*a*d)*Sqrt[a + b/x])/(4*c^3*(c + d/x)) + (a*Sq
rt[a + b/x]*x)/(c*(c + d/x)^2) - (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c
- a*d]])/(4*c^4*Sqrt[d]*Sqrt[b*c - a*d]) + (3*Sqrt[a]*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/c^4

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Rubi [A]  time = 0.344981, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {375, 98, 151, 156, 63, 208, 205} \[ -\frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 \sqrt{d} \sqrt{b c-a d}}-\frac{3 \sqrt{a+\frac{b}{x}} (b c-4 a d)}{4 c^3 \left (c+\frac{d}{x}\right )}-\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d)}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{3 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}+\frac{a x \sqrt{a+\frac{b}{x}}}{c \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(3/2)/(c + d/x)^3,x]

[Out]

-((b*c - 3*a*d)*Sqrt[a + b/x])/(2*c^2*(c + d/x)^2) - (3*(b*c - 4*a*d)*Sqrt[a + b/x])/(4*c^3*(c + d/x)) + (a*Sq
rt[a + b/x]*x)/(c*(c + d/x)^2) - (3*(b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c
- a*d]])/(4*c^4*Sqrt[d]*Sqrt[b*c - a*d]) + (3*Sqrt[a]*(b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 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{\left (a+\frac{b}{x}\right )^{3/2}}{\left (c+\frac{d}{x}\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{3/2}}{x^2 (c+d x)^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a \sqrt{a+\frac{b}{x}} x}{c \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{2} a (b c-2 a d)-\frac{1}{2} b (2 b c-5 a d) x}{x \sqrt{a+b x} (c+d x)^3} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}+\frac{a \sqrt{a+\frac{b}{x}} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{3 a (b c-2 a d) (b c-a d)+\frac{3}{2} b (b c-3 a d) (b c-a d) x}{x \sqrt{a+b x} (c+d x)^2} \, dx,x,\frac{1}{x}\right )}{2 c^2 (b c-a d)}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}-\frac{3 (b c-4 a d) \sqrt{a+\frac{b}{x}}}{4 c^3 \left (c+\frac{d}{x}\right )}+\frac{a \sqrt{a+\frac{b}{x}} x}{c \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-3 a (b c-2 a d) (b c-a d)^2-\frac{3}{4} b (b c-4 a d) (b c-a d)^2 x}{x \sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{2 c^3 (b c-a d)^2}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}-\frac{3 (b c-4 a d) \sqrt{a+\frac{b}{x}}}{4 c^3 \left (c+\frac{d}{x}\right )}+\frac{a \sqrt{a+\frac{b}{x}} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{(3 a (b c-2 a d)) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 c^4}-\frac{\left (3 \left (b^2 c^2-8 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}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}-\frac{3 (b c-4 a d) \sqrt{a+\frac{b}{x}}}{4 c^3 \left (c+\frac{d}{x}\right )}+\frac{a \sqrt{a+\frac{b}{x}} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{(3 a (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 )}{b c^4}-\frac{\left (3 \left (b^2 c^2-8 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}\\ &=-\frac{(b c-3 a d) \sqrt{a+\frac{b}{x}}}{2 c^2 \left (c+\frac{d}{x}\right )^2}-\frac{3 (b c-4 a d) \sqrt{a+\frac{b}{x}}}{4 c^3 \left (c+\frac{d}{x}\right )}+\frac{a \sqrt{a+\frac{b}{x}} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{3 \left (b^2 c^2-8 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 \sqrt{d} \sqrt{b c-a d}}+\frac{3 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}\\ \end{align*}

Mathematica [A]  time = 0.448489, size = 168, normalized size = 0.8 \[ \frac{-\frac{3 \left (8 a^2 d^2-8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{\sqrt{d} \sqrt{b c-a d}}+\frac{c x \sqrt{a+\frac{b}{x}} \left (2 a \left (2 c^2 x^2+9 c d x+6 d^2\right )-b c (5 c x+3 d)\right )}{(c x+d)^2}+12 \sqrt{a} (b c-2 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(3/2)/(c + d/x)^3,x]

[Out]

((c*Sqrt[a + b/x]*x*(-(b*c*(3*d + 5*c*x)) + 2*a*(6*d^2 + 9*c*d*x + 2*c^2*x^2)))/(d + c*x)^2 - (3*(b^2*c^2 - 8*
a*b*c*d + 8*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(Sqrt[d]*Sqrt[b*c - a*d]) + 12*Sqrt[a]*(
b*c - 2*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*c^4)

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Maple [B]  time = 0.011, size = 1817, normalized size = 8.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(24*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*x*b^2*c^4*d^2+54*a^
(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*b*c^4*d^2+12*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^
(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*x^2*b^2*c^5*d+18*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x^2*b*c^
5*d-72*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*x*b*c^3*d^3-36*ln(1/2
*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*x^2*b*c^4*d^2-36*ln(1/2*(2*((a*x+b
)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^3*((a*d-b*c)*d/c^2)^(1/2)*b*c^2*d^4+12*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1
/2)+2*a*x+b)/a^(1/2))*a^2*((a*d-b*c)*d/c^2)^(1/2)*b^2*c^3*d^3+30*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(
1/2)*b*c^3*d^3+24*a^(9/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*d^6-6*
a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*b^2*c^4*d^2+12*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(
1/2)*x^3*c^5*d-6*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x^3*b*c^6+24*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^
(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*x^2*c^3*d^3-12*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)
^(3/2)*x*c^5*d-48*a^(7/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b*
c^3*d^3+27*a^(5/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b^2*c^4*d
^2+48*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)*d/c^2)^(1/2)*x*c^2*d^4-36*a^(7/2)*(
(a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*c^3*d^3-96*a^(7/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*
c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x*b*c^2*d^4+54*a^(5/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x
+b*c*x-b*d)/(c*x+d))*x*b^2*c^3*d^3-6*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x^2*b^2*c^6-3*a^(3/2)*l
n((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*b^3*c^5*d-6*a^(3/2)*ln((2*((a
*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x*b^3*c^4*d^2+2*a^(3/2)*((a*d-b*c)*d/c^2)
^(1/2)*((a*x+b)*x)^(3/2)*b*c^5*d-12*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*b^2*c^5*d+24*a^(9/2)*l
n((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*x^2*c^2*d^4-8*a^(5/2)*((a*d-b*c)*
d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*c^4*d^2+24*ln(1/2*(2*((a*x+b)*x)^(1/2)*a^(1/2)+2*a*x+b)/a^(1/2))*a^4*((a*d-b*c)
*d/c^2)^(1/2)*c*d^5-24*a^(7/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c^2*d^4-48*a^(7/2)*ln((2*((a*d-b*c)*d
/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b*c*d^5+27*a^(5/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*(
(a*x+b)*x)^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b^2*c^2*d^4-3*a^(3/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)
^(1/2)*c-2*a*d*x+b*c*x-b*d)/(c*x+d))*b^3*c^3*d^3+48*a^(9/2)*ln((2*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*c-
2*a*d*x+b*c*x-b*d)/(c*x+d))*x*c*d^5+6*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*x*b*c^6)*x*((a*x+b)/x)
^(1/2)/d/c^5/((a*d-b*c)*d/c^2)^(1/2)/a^(3/2)/(c*x+d)^2/(a*d-b*c)/((a*x+b)*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((a+b/x)^(3/2)/(c+d/x)^3,x, algorithm="maxima")

[Out]

integrate((a + b/x)^(3/2)/(c + d/x)^3, x)

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Fricas [B]  time = 1.81602, size = 3646, normalized size = 17.44 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)/(c+d/x)^3,x, algorithm="fricas")

[Out]

[-1/8*(12*(b^2*c^2*d^3 - 3*a*b*c*d^4 + 2*a^2*d^5 + (b^2*c^4*d - 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2 + 2*(b^2*c^
3*d^2 - 3*a*b*c^2*d^3 + 2*a^2*c*d^4)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + 3*(b^2*c^2*d^
2 - 8*a*b*c*d^3 + 8*a^2*d^4 + (b^2*c^4 - 8*a*b*c^3*d + 8*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 8*a*b*c^2*d^2 + 8*a
^2*c*d^3)*x)*sqrt(-b*c*d + a*d^2)*log((b*d - (b*c - 2*a*d)*x + 2*sqrt(-b*c*d + a*d^2)*x*sqrt((a*x + b)/x))/(c*
x + d)) - 2*(4*(a*b*c^4*d - a^2*c^3*d^2)*x^3 - (5*b^2*c^4*d - 23*a*b*c^3*d^2 + 18*a^2*c^2*d^3)*x^2 - 3*(b^2*c^
3*d^2 - 5*a*b*c^2*d^3 + 4*a^2*c*d^4)*x)*sqrt((a*x + b)/x))/(b*c^5*d^3 - a*c^4*d^4 + (b*c^7*d - a*c^6*d^2)*x^2
+ 2*(b*c^6*d^2 - a*c^5*d^3)*x), -1/8*(24*(b^2*c^2*d^3 - 3*a*b*c*d^4 + 2*a^2*d^5 + (b^2*c^4*d - 3*a*b*c^3*d^2 +
2*a^2*c^2*d^3)*x^2 + 2*(b^2*c^3*d^2 - 3*a*b*c^2*d^3 + 2*a^2*c*d^4)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)
/x)/a) + 3*(b^2*c^2*d^2 - 8*a*b*c*d^3 + 8*a^2*d^4 + (b^2*c^4 - 8*a*b*c^3*d + 8*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d
- 8*a*b*c^2*d^2 + 8*a^2*c*d^3)*x)*sqrt(-b*c*d + a*d^2)*log((b*d - (b*c - 2*a*d)*x + 2*sqrt(-b*c*d + a*d^2)*x*
sqrt((a*x + b)/x))/(c*x + d)) - 2*(4*(a*b*c^4*d - a^2*c^3*d^2)*x^3 - (5*b^2*c^4*d - 23*a*b*c^3*d^2 + 18*a^2*c^
2*d^3)*x^2 - 3*(b^2*c^3*d^2 - 5*a*b*c^2*d^3 + 4*a^2*c*d^4)*x)*sqrt((a*x + b)/x))/(b*c^5*d^3 - a*c^4*d^4 + (b*c
^7*d - a*c^6*d^2)*x^2 + 2*(b*c^6*d^2 - a*c^5*d^3)*x), 1/4*(3*(b^2*c^2*d^2 - 8*a*b*c*d^3 + 8*a^2*d^4 + (b^2*c^4
- 8*a*b*c^3*d + 8*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 8*a*b*c^2*d^2 + 8*a^2*c*d^3)*x)*sqrt(b*c*d - a*d^2)*arcta
n(sqrt(b*c*d - a*d^2)*x*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 6*(b^2*c^2*d^3 - 3*a*b*c*d^4 + 2*a^2*d^5 + (b^2*c^4
*d - 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2 + 2*(b^2*c^3*d^2 - 3*a*b*c^2*d^3 + 2*a^2*c*d^4)*x)*sqrt(a)*log(2*a*x -
2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (4*(a*b*c^4*d - a^2*c^3*d^2)*x^3 - (5*b^2*c^4*d - 23*a*b*c^3*d^2 + 18*a^
2*c^2*d^3)*x^2 - 3*(b^2*c^3*d^2 - 5*a*b*c^2*d^3 + 4*a^2*c*d^4)*x)*sqrt((a*x + b)/x))/(b*c^5*d^3 - a*c^4*d^4 +
(b*c^7*d - a*c^6*d^2)*x^2 + 2*(b*c^6*d^2 - a*c^5*d^3)*x), 1/4*(3*(b^2*c^2*d^2 - 8*a*b*c*d^3 + 8*a^2*d^4 + (b^2
*c^4 - 8*a*b*c^3*d + 8*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d - 8*a*b*c^2*d^2 + 8*a^2*c*d^3)*x)*sqrt(b*c*d - a*d^2)*a
rctan(sqrt(b*c*d - a*d^2)*x*sqrt((a*x + b)/x)/(a*d*x + b*d)) - 12*(b^2*c^2*d^3 - 3*a*b*c*d^4 + 2*a^2*d^5 + (b^
2*c^4*d - 3*a*b*c^3*d^2 + 2*a^2*c^2*d^3)*x^2 + 2*(b^2*c^3*d^2 - 3*a*b*c^2*d^3 + 2*a^2*c*d^4)*x)*sqrt(-a)*arcta
n(sqrt(-a)*sqrt((a*x + b)/x)/a) + (4*(a*b*c^4*d - a^2*c^3*d^2)*x^3 - (5*b^2*c^4*d - 23*a*b*c^3*d^2 + 18*a^2*c^
2*d^3)*x^2 - 3*(b^2*c^3*d^2 - 5*a*b*c^2*d^3 + 4*a^2*c*d^4)*x)*sqrt((a*x + b)/x))/(b*c^5*d^3 - a*c^4*d^4 + (b*c
^7*d - a*c^6*d^2)*x^2 + 2*(b*c^6*d^2 - a*c^5*d^3)*x)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**(3/2)/(c+d/x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.31728, size = 981, normalized size = 4.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^(3/2)/(c+d/x)^3,x, algorithm="giac")

[Out]

sqrt(a*x^2 + b*x)*a*sgn(x)/c^3 + 3/4*(b^2*c^2*sgn(x) - 8*a*b*c*d*sgn(x) + 8*a^2*d^2*sgn(x))*arctan(-((sqrt(a)*
x - sqrt(a*x^2 + b*x))*c + sqrt(a)*d)/sqrt(b*c*d - a*d^2))/(sqrt(b*c*d - a*d^2)*c^4) - 3/2*(a*b*c*sgn(x) - 2*a
^2*d*sgn(x))*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a) + b))/(sqrt(a)*c^4) + 1/4*(3*sqrt(a)*b^2*c^2*ar
ctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) - 24*a^(3/2)*b*c*d*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(5/2)*d^2*
arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 6*sqrt(b*c*d - a*d^2)*a*b*c*log(abs(b)) - 12*sqrt(b*c*d - a*d^2)*a^2*d
*log(abs(b)) + 5*sqrt(b*c*d - a*d^2)*a*b*c - 10*sqrt(b*c*d - a*d^2)*a^2*d)*sgn(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)
*c^4) + 1/4*(5*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*sqrt(a)*b^2*c^3*sgn(x) - 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3
*a^(3/2)*b*c^2*d*sgn(x) + 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(5/2)*c*d^2*sgn(x) - (sqrt(a)*x - sqrt(a*x^2
+ b*x))^2*a*b^2*c^2*d*sgn(x) - 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^2*b*c*d^2*sgn(x) + 40*(sqrt(a)*x - sqrt(
a*x^2 + b*x))^2*a^3*d^3*sgn(x) + 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))*sqrt(a)*b^3*c^2*d*sgn(x) - 28*(sqrt(a)*x -
sqrt(a*x^2 + b*x))*a^(3/2)*b^2*c*d^2*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(5/2)*b*d^3*sgn(x) - 5*a*b^
3*c*d^2*sgn(x) + 10*a^2*b^2*d^3*sgn(x))/(((sqrt(a)*x - sqrt(a*x^2 + b*x))^2*c + 2*(sqrt(a)*x - sqrt(a*x^2 + b*
x))*sqrt(a)*d + b*d)^2*sqrt(a)*c^4)