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

Optimal. Leaf size=237 \[ -\frac{\sqrt{a+\frac{b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{4 c^3 d \left (c+\frac{d}{x}\right )}-\frac{\sqrt{b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 d^{3/2}}+\frac{a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}+\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left (c+\frac{d}{x}\right )^2}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c \left (c+\frac{d}{x}\right )^2} \]

[Out]

((b*c - 3*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(2*c^2*d*(c + d/x)^2) - ((b^2*c^2 + 7*a*b*c*d - 12*a^2*d^2)*Sqrt[a +
b/x])/(4*c^3*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)/(c*(c + d/x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 2
4*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*d^(3/2)) + (a^(3/2)*(5*b*c - 6*a*d)*ArcTanh
[Sqrt[a + b/x]/Sqrt[a]])/c^4

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Rubi [A]  time = 0.372679, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {375, 98, 149, 151, 156, 63, 208, 205} \[ -\frac{\sqrt{a+\frac{b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{4 c^3 d \left (c+\frac{d}{x}\right )}-\frac{\sqrt{b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 d^{3/2}}+\frac{a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}+\frac{\sqrt{a+\frac{b}{x}} (b c-3 a d) (b c-a d)}{2 c^2 d \left (c+\frac{d}{x}\right )^2}+\frac{a x \left (a+\frac{b}{x}\right )^{3/2}}{c \left (c+\frac{d}{x}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^(5/2)/(c + d/x)^3,x]

[Out]

((b*c - 3*a*d)*(b*c - a*d)*Sqrt[a + b/x])/(2*c^2*d*(c + d/x)^2) - ((b^2*c^2 + 7*a*b*c*d - 12*a^2*d^2)*Sqrt[a +
b/x])/(4*c^3*d*(c + d/x)) + (a*(a + b/x)^(3/2)*x)/(c*(c + d/x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 2
4*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a*d]])/(4*c^4*d^(3/2)) + (a^(3/2)*(5*b*c - 6*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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 )^{5/2}}{\left (c+\frac{d}{x}\right )^3} \, dx &=-\operatorname{Subst}\left (\int \frac{(a+b x)^{5/2}}{x^2 (c+d x)^3} \, dx,x,\frac{1}{x}\right )\\ &=\frac{a \left (a+\frac{b}{x}\right )^{3/2} x}{c \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+b x} \left (-\frac{1}{2} a (5 b c-6 a d)-\frac{1}{2} b (2 b c-3 a d) x\right )}{x (c+d x)^3} \, dx,x,\frac{1}{x}\right )}{c}\\ &=\frac{(b c-3 a d) (b c-a d) \sqrt{a+\frac{b}{x}}}{2 c^2 d \left (c+\frac{d}{x}\right )^2}+\frac{a \left (a+\frac{b}{x}\right )^{3/2} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{a^2 d (5 b c-6 a d)+\frac{1}{2} b \left (b^2 c^2+6 a b c d-9 a^2 d^2\right ) x}{x \sqrt{a+b x} (c+d x)^2} \, dx,x,\frac{1}{x}\right )}{2 c^2 d}\\ &=\frac{(b c-3 a d) (b c-a d) \sqrt{a+\frac{b}{x}}}{2 c^2 d \left (c+\frac{d}{x}\right )^2}-\frac{\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt{a+\frac{b}{x}}}{4 c^3 d \left (c+\frac{d}{x}\right )}+\frac{a \left (a+\frac{b}{x}\right )^{3/2} x}{c \left (c+\frac{d}{x}\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{-a^2 d (5 b c-6 a d) (b c-a d)-\frac{1}{4} b (b c-a d) \left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) x}{x \sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{2 c^3 d (b c-a d)}\\ &=\frac{(b c-3 a d) (b c-a d) \sqrt{a+\frac{b}{x}}}{2 c^2 d \left (c+\frac{d}{x}\right )^2}-\frac{\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt{a+\frac{b}{x}}}{4 c^3 d \left (c+\frac{d}{x}\right )}+\frac{a \left (a+\frac{b}{x}\right )^{3/2} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{\left (a^2 (5 b c-6 a d)\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 c^4}-\frac{\left ((b c-a d) \left (b^2 c^2+8 a b c d-24 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x} (c+d x)} \, dx,x,\frac{1}{x}\right )}{8 c^4 d}\\ &=\frac{(b c-3 a d) (b c-a d) \sqrt{a+\frac{b}{x}}}{2 c^2 d \left (c+\frac{d}{x}\right )^2}-\frac{\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt{a+\frac{b}{x}}}{4 c^3 d \left (c+\frac{d}{x}\right )}+\frac{a \left (a+\frac{b}{x}\right )^{3/2} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{\left (a^2 (5 b c-6 a d)\right ) \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 ((b c-a d) \left (b^2 c^2+8 a b c d-24 a^2 d^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c-\frac{a d}{b}+\frac{d x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{4 b c^4 d}\\ &=\frac{(b c-3 a d) (b c-a d) \sqrt{a+\frac{b}{x}}}{2 c^2 d \left (c+\frac{d}{x}\right )^2}-\frac{\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt{a+\frac{b}{x}}}{4 c^3 d \left (c+\frac{d}{x}\right )}+\frac{a \left (a+\frac{b}{x}\right )^{3/2} x}{c \left (c+\frac{d}{x}\right )^2}-\frac{\sqrt{b c-a d} \left (b^2 c^2+8 a b c d-24 a^2 d^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{4 c^4 d^{3/2}}+\frac{a^{3/2} (5 b c-6 a d) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{c^4}\\ \end{align*}

Mathematica [A]  time = 0.546294, size = 191, normalized size = 0.81 \[ \frac{\frac{c x \sqrt{a+\frac{b}{x}} \left (2 a^2 d \left (2 c^2 x^2+9 c d x+6 d^2\right )-a b c d (11 c x+7 d)+b^2 c^2 (c x-d)\right )}{d (c x+d)^2}-\frac{\sqrt{b c-a d} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b c-a d}}\right )}{d^{3/2}}-4 a^{3/2} (6 a d-5 b c) \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{4 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^(5/2)/(c + d/x)^3,x]

[Out]

((c*Sqrt[a + b/x]*x*(b^2*c^2*(-d + c*x) - a*b*c*d*(7*d + 11*c*x) + 2*a^2*d*(6*d^2 + 9*c*d*x + 2*c^2*x^2)))/(d*
(d + c*x)^2) - (Sqrt[b*c - a*d]*(b^2*c^2 + 8*a*b*c*d - 24*a^2*d^2)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c - a
*d]])/d^(3/2) - 4*a^(3/2)*(-5*b*c + 6*a*d)*ArcTanh[Sqrt[a + b/x]/Sqrt[a]])/(4*c^4)

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Maple [B]  time = 0.011, size = 1638, normalized size = 6.9 \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)^(5/2)/(c+d/x)^3,x)

[Out]

-1/8*(30*a^(5/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(1/2)*x*b*c^4*d^2+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-40*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^3-20*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^
2-20*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+14*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+2*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+2*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-32*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+7*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^3-64*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+14*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+2*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b
)*x)^(1/2)*x^2*b^2*c^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
^2*b^3*c^5*d+2*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^2-6*a^(3/2)*((a*d-b*c)*d/c^2)^(1/2)*((a*x+b)*x)^(3/2)*b*c^5*d+4*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)*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*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^
4-32*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+7*a^(5/2)*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))*b^2*c^2*d^4+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^5-2*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)/c^5/((a*d-b*c)*d/c^2)^(1/2)/a^(3/2)/(c*x+d)^2/d^2/((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{5}{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)^(5/2)/(c+d/x)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.89564, size = 3040, normalized size = 12.83 \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)^(5/2)/(c+d/x)^3,x, algorithm="fricas")

[Out]

[-1/8*(4*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^2*d^2)*x^2 + 2*(5*a*b*c^2*d^2 - 6*a^2*c*d^3)*x)*sqr
t(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^2*d^4 + (b^2*c^4 + 8*a
*b*c^3*d - 24*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt(-(b*c - a*d)/d)*log((2*d
*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) - 2*(4*a^2*c^3*d*x^3 + (b^2*c^4
- 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(c^6*d
*x^2 + 2*c^5*d^2*x + c^4*d^3), 1/4*((b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^2*d^4 + (b^2*c^4 + 8*a*b*c^3*d - 24*a^2*
c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt((b*c - a*d)/d)*arctan(-d*sqrt((b*c - a*d)/
d)*sqrt((a*x + b)/x)/(b*c - a*d)) - 2*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^2*d^2)*x^2 + 2*(5*a*b*
c^2*d^2 - 6*a^2*c*d^3)*x)*sqrt(a)*log(2*a*x - 2*sqrt(a)*x*sqrt((a*x + b)/x) + b) + (4*a^2*c^3*d*x^3 + (b^2*c^4
- 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x + b)/x))/(c^6*
d*x^2 + 2*c^5*d^2*x + c^4*d^3), -1/8*(8*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^2*d^2)*x^2 + 2*(5*a*
b*c^2*d^2 - 6*a^2*c*d^3)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^
2*d^4 + (b^2*c^4 + 8*a*b*c^3*d - 24*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt(-(
b*c - a*d)/d)*log((2*d*x*sqrt(-(b*c - a*d)/d)*sqrt((a*x + b)/x) + b*d - (b*c - 2*a*d)*x)/(c*x + d)) - 2*(4*a^2
*c^3*d*x^3 + (b^2*c^4 - 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqr
t((a*x + b)/x))/(c^6*d*x^2 + 2*c^5*d^2*x + c^4*d^3), 1/4*((b^2*c^2*d^2 + 8*a*b*c*d^3 - 24*a^2*d^4 + (b^2*c^4 +
8*a*b*c^3*d - 24*a^2*c^2*d^2)*x^2 + 2*(b^2*c^3*d + 8*a*b*c^2*d^2 - 24*a^2*c*d^3)*x)*sqrt((b*c - a*d)/d)*arcta
n(-d*sqrt((b*c - a*d)/d)*sqrt((a*x + b)/x)/(b*c - a*d)) - 4*(5*a*b*c*d^3 - 6*a^2*d^4 + (5*a*b*c^3*d - 6*a^2*c^
2*d^2)*x^2 + 2*(5*a*b*c^2*d^2 - 6*a^2*c*d^3)*x)*sqrt(-a)*arctan(sqrt(-a)*sqrt((a*x + b)/x)/a) + (4*a^2*c^3*d*x
^3 + (b^2*c^4 - 11*a*b*c^3*d + 18*a^2*c^2*d^2)*x^2 - (b^2*c^3*d + 7*a*b*c^2*d^2 - 12*a^2*c*d^3)*x)*sqrt((a*x +
b)/x))/(c^6*d*x^2 + 2*c^5*d^2*x + c^4*d^3)]

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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)**(5/2)/(c+d/x)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.37821, size = 1276, normalized size = 5.38 \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)^(5/2)/(c+d/x)^3,x, algorithm="giac")

[Out]

sqrt(a*x^2 + b*x)*a^2*sgn(x)/c^3 - 1/2*(5*a^2*b*c*sgn(x) - 6*a^3*d*sgn(x))*log(abs(2*(sqrt(a)*x - sqrt(a*x^2 +
b*x))*sqrt(a) + b))/(sqrt(a)*c^4) + 1/4*(b^3*c^3*sgn(x) + 7*a*b^2*c^2*d*sgn(x) - 32*a^2*b*c*d^2*sgn(x) + 24*a
^3*d^3*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*d) + 1/4*(sqrt(a)*b^3*c^3*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 7*a^(3/2)*b^2*c^2*d*arctan(sqrt(a)*d/
sqrt(b*c*d - a*d^2)) - 32*a^(5/2)*b*c*d^2*arctan(sqrt(a)*d/sqrt(b*c*d - a*d^2)) + 24*a^(7/2)*d^3*arctan(sqrt(a
)*d/sqrt(b*c*d - a*d^2)) + 10*sqrt(b*c*d - a*d^2)*a^2*b*c*d*log(abs(b)) - 12*sqrt(b*c*d - a*d^2)*a^3*d^2*log(a
bs(b)) - sqrt(b*c*d - a*d^2)*a*b^2*c^2 + 11*sqrt(b*c*d - a*d^2)*a^2*b*c*d - 10*sqrt(b*c*d - a*d^2)*a^3*d^2)*sg
n(x)/(sqrt(b*c*d - a*d^2)*sqrt(a)*c^4*d) - 1/4*((sqrt(a)*x - sqrt(a*x^2 + b*x))^3*sqrt(a)*b^3*c^4*sgn(x) - 17*
(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(3/2)*b^2*c^3*d*sgn(x) + 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(5/2)*b*c^
2*d^2*sgn(x) - 24*(sqrt(a)*x - sqrt(a*x^2 + b*x))^3*a^(7/2)*c*d^3*sgn(x) - 5*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2
*a*b^3*c^3*d*sgn(x) - 3*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^2*b^2*c^2*d^2*sgn(x) + 48*(sqrt(a)*x - sqrt(a*x^2
+ b*x))^2*a^3*b*c*d^3*sgn(x) - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))^2*a^4*d^4*sgn(x) - (sqrt(a)*x - sqrt(a*x^2 +
b*x))*sqrt(a)*b^4*c^3*d*sgn(x) - 11*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(3/2)*b^3*c^2*d^2*sgn(x) + 52*(sqrt(a)*
x - sqrt(a*x^2 + b*x))*a^(5/2)*b^2*c*d^3*sgn(x) - 40*(sqrt(a)*x - sqrt(a*x^2 + b*x))*a^(7/2)*b*d^4*sgn(x) - a*
b^4*c^2*d^2*sgn(x) + 11*a^2*b^3*c*d^3*sgn(x) - 10*a^3*b^2*d^4*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*d)