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

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

[Out]

(b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(a + b/x)^(3/2)) + (b*(b*c - 2*a*d)*(5*b^2*c^
2 - a*b*c*d + a^2*d^2))/(a^3*c^2*(b*c - a*d)^3*Sqrt[a + b/x]) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*(a + b/x)
^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)) - (d^(7/2)*(9*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x]
)/Sqrt[b*c - a*d]])/(c^3*(b*c - a*d)^(7/2)) - ((5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^3)

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Rubi [A]  time = 0.447713, antiderivative size = 287, 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} \[ \frac{b (b c-2 a d) \left (a^2 d^2-a b c d+5 b^2 c^2\right )}{a^3 c^2 \sqrt{a+\frac{b}{x}} (b c-a d)^3}+\frac{b \left (6 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{3 a^2 c^2 \left (a+\frac{b}{x}\right )^{3/2} (b c-a d)^2}-\frac{(4 a d+5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{7/2} c^3}-\frac{d^{7/2} (9 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{c^3 (b c-a d)^{7/2}}+\frac{d (b c-2 a d)}{a c^2 \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right ) (b c-a d)}+\frac{x}{a c \left (a+\frac{b}{x}\right )^{3/2} \left (c+\frac{d}{x}\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(5*b^2*c^2 - 6*a*b*c*d + 6*a^2*d^2))/(3*a^2*c^2*(b*c - a*d)^2*(a + b/x)^(3/2)) + (b*(b*c - 2*a*d)*(5*b^2*c^
2 - a*b*c*d + a^2*d^2))/(a^3*c^2*(b*c - a*d)^3*Sqrt[a + b/x]) + (d*(b*c - 2*a*d))/(a*c^2*(b*c - a*d)*(a + b/x)
^(3/2)*(c + d/x)) + x/(a*c*(a + b/x)^(3/2)*(c + d/x)) - (d^(7/2)*(9*b*c - 4*a*d)*ArcTan[(Sqrt[d]*Sqrt[a + b/x]
)/Sqrt[b*c - a*d]])/(c^3*(b*c - a*d)^(7/2)) - ((5*b*c + 4*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(a^(7/2)*c^3)

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

Mathematica [C]  time = 0.13204, size = 178, normalized size = 0.62 \[ \frac{x \left ((a d-b c) \left (3 a c x (a d (c x+2 d)-b c (c x+d))-(c x+d) \left (-4 a^2 d^2-a b c d+5 b^2 c^2\right ) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{b}{a x}+1\right )\right )+a^2 d^2 (c x+d) (9 b c-4 a d) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{d \left (a+\frac{b}{x}\right )}{a d-b c}\right )\right )}{3 a^2 c^3 \sqrt{a+\frac{b}{x}} (a x+b) (c x+d) (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.017, size = 4644, normalized size = 16.2 \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)^(5/2)/(c+d/x)^2,x)

[Out]

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

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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{5}{2}}{\left (c + \frac{d}{x}\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 11.7287, size = 7776, normalized size = 27.09 \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)^(5/2)/(c+d/x)^2,x, algorithm="fricas")

[Out]

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

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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)**(5/2)/(c+d/x)**2,x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 1.23121, size = 779, normalized size = 2.71 \begin{align*} -\frac{1}{3} \, b{\left (\frac{3 \,{\left (9 \, b c d^{4} - 4 \, a d^{5}\right )} \arctan \left (\frac{d \sqrt{\frac{a x + b}{x}}}{\sqrt{b c d - a d^{2}}}\right )}{{\left (b^{4} c^{6} - 3 \, a b^{3} c^{5} d + 3 \, a^{2} b^{2} c^{4} d^{2} - a^{3} b c^{3} d^{3}\right )} \sqrt{b c d - a d^{2}}} - \frac{2 \,{\left (a b^{3} c - a^{2} b^{2} d + \frac{6 \,{\left (a x + b\right )} b^{3} c}{x} - \frac{12 \,{\left (a x + b\right )} a b^{2} d}{x}\right )} x}{{\left (a^{3} b^{3} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{5} b c d^{2} - a^{6} d^{3}\right )}{\left (a x + b\right )} \sqrt{\frac{a x + b}{x}}} + \frac{3 \,{\left (b^{4} c^{4} \sqrt{\frac{a x + b}{x}} - 4 \, a b^{3} c^{3} d \sqrt{\frac{a x + b}{x}} + 6 \, a^{2} b^{2} c^{2} d^{2} \sqrt{\frac{a x + b}{x}} - 4 \, a^{3} b c d^{3} \sqrt{\frac{a x + b}{x}} + 2 \, a^{4} d^{4} \sqrt{\frac{a x + b}{x}} + \frac{{\left (a x + b\right )} b^{3} c^{3} d \sqrt{\frac{a x + b}{x}}}{x} - \frac{3 \,{\left (a x + b\right )} a b^{2} c^{2} d^{2} \sqrt{\frac{a x + b}{x}}}{x} + \frac{3 \,{\left (a x + b\right )} a^{2} b c d^{3} \sqrt{\frac{a x + b}{x}}}{x} - \frac{2 \,{\left (a x + b\right )} a^{3} d^{4} \sqrt{\frac{a x + b}{x}}}{x}\right )}}{{\left (a^{3} b^{3} c^{5} - 3 \, a^{4} b^{2} c^{4} d + 3 \, a^{5} b c^{3} d^{2} - a^{6} c^{2} d^{3}\right )}{\left (a b c - a^{2} d - \frac{{\left (a x + b\right )} b c}{x} + \frac{2 \,{\left (a x + b\right )} a d}{x} - \frac{{\left (a x + b\right )}^{2} d}{x^{2}}\right )}} - \frac{3 \,{\left (5 \, b c + 4 \, a d\right )} \arctan \left (\frac{\sqrt{\frac{a x + b}{x}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3} b c^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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