\(\int \frac {(a+\frac {b}{x})^{5/2}}{(c+\frac {d}{x})^3} \, dx\) [24]

Optimal result

Integrand size = 21, antiderivative size = 237 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {(b c-3 a d) (b c-a d) \sqrt {a+\frac {b}{x}}}{2 c^2 d \left (c+\frac {d}{x}\right )^2}-\frac {\left (b^2 c^2+7 a b c d-12 a^2 d^2\right ) \sqrt {a+\frac {b}{x}}}{4 c^3 d \left (c+\frac {d}{x}\right )}+\frac {a \left (a+\frac {b}{x}\right )^{3/2} x}{c \left (c+\frac {d}{x}\right )^2}-\frac {\sqrt {b c-a d} \left (b^2 c^2+8 a b c d-24 a^2 d^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{4 c^4 d^{3/2}}+\frac {a^{3/2} (5 b c-6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{c^4} \] Output:

1/2*(-3*a*d+b*c)*(-a*d+b*c)*(a+b/x)^(1/2)/c^2/d/(c+d/x)^2-1/4*(-12*a^2*d^2 
+7*a*b*c*d+b^2*c^2)*(a+b/x)^(1/2)/c^3/d/(c+d/x)+a*(a+b/x)^(3/2)*x/c/(c+d/x 
)^2-1/4*(-a*d+b*c)^(1/2)*(-24*a^2*d^2+8*a*b*c*d+b^2*c^2)*arctan(d^(1/2)*(a 
+b/x)^(1/2)/(-a*d+b*c)^(1/2))/c^4/d^(3/2)+a^(3/2)*(-6*a*d+5*b*c)*arctanh(( 
a+b/x)^(1/2)/a^(1/2))/c^4
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\frac {\frac {c \sqrt {a+\frac {b}{x}} x \left (b^2 c^2 (-d+c x)-a b c d (7 d+11 c x)+2 a^2 d \left (6 d^2+9 c d x+2 c^2 x^2\right )\right )}{d (d+c x)^2}-\frac {\left (b^3 c^3+7 a b^2 c^2 d-32 a^2 b c d^2+24 a^3 d^3\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{d^{3/2} \sqrt {b c-a d}}-4 a^{3/2} (-5 b c+6 a d) \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right )}{4 c^4} \] Input:

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

Output:

((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) - ((b^3*c^3 + 7*a*b^2*c^2 
*d - 32*a^2*b*c*d^2 + 24*a^3*d^3)*ArcTan[(Sqrt[d]*Sqrt[a + b/x])/Sqrt[b*c 
- a*d]])/(d^(3/2)*Sqrt[b*c - a*d]) - 4*a^(3/2)*(-5*b*c + 6*a*d)*ArcTanh[Sq 
rt[a + b/x]/Sqrt[a]])/(4*c^4)
 

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {899, 109, 27, 166, 25, 168, 27, 174, 73, 218, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 899

\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^{5/2} x^2}{\left (c+\frac {d}{x}\right )^3}d\frac {1}{x}\)

\(\Big \downarrow \) 109

\(\displaystyle \frac {\int -\frac {\sqrt {a+\frac {b}{x}} \left (a (5 b c-6 a d)+\frac {b (2 b c-3 a d)}{x}\right ) x}{2 \left (c+\frac {d}{x}\right )^3}d\frac {1}{x}}{c}+\frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\int \frac {\sqrt {a+\frac {b}{x}} \left (a (5 b c-6 a d)+\frac {b (2 b c-3 a d)}{x}\right ) x}{\left (c+\frac {d}{x}\right )^3}d\frac {1}{x}}{2 c}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}-\frac {\int -\frac {\left (2 d (5 b c-6 a d) a^2+\frac {b \left (b^2 c^2+6 a b d c-9 a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c d}}{2 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\int \frac {\left (2 d (5 b c-6 a d) a^2+\frac {b \left (b^2 c^2+6 a b d c-9 a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )^2}d\frac {1}{x}}{2 c d}+\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}}{2 c}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{c \left (c+\frac {d}{x}\right )}-\frac {\int -\frac {\left (4 d (5 b c-6 a d) (b c-a d) a^2+\frac {b (b c-a d) \left (b^2 c^2+7 a b d c-12 a^2 d^2\right )}{x}\right ) x}{2 \sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c (b c-a d)}}{2 c d}+\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}}{2 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\int \frac {\left (4 d (5 b c-6 a d) (b c-a d) a^2+\frac {b (b c-a d) \left (b^2 c^2+7 a b d c-12 a^2 d^2\right )}{x}\right ) x}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{2 c (b c-a d)}+\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{c \left (c+\frac {d}{x}\right )}}{2 c d}+\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}}{2 c}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\frac {(b c-a d)^2 \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+\frac {b}{x}} \left (c+\frac {d}{x}\right )}d\frac {1}{x}}{c}+\frac {4 a^2 d (5 b c-6 a d) (b c-a d) \int \frac {x}{\sqrt {a+\frac {b}{x}}}d\frac {1}{x}}{c}}{2 c (b c-a d)}+\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{c \left (c+\frac {d}{x}\right )}}{2 c d}+\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}}{2 c}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\frac {2 (b c-a d)^2 \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \int \frac {1}{c-\frac {a d}{b}+\frac {d}{b x^2}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {8 a^2 d (5 b c-6 a d) (b c-a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}}{2 c (b c-a d)}+\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{c \left (c+\frac {d}{x}\right )}}{2 c d}+\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}}{2 c}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\frac {\frac {8 a^2 d (5 b c-6 a d) (b c-a d) \int \frac {1}{\frac {1}{b x^2}-\frac {a}{b}}d\sqrt {a+\frac {b}{x}}}{b c}+\frac {2 (b c-a d)^{3/2} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}}{2 c (b c-a d)}+\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{c \left (c+\frac {d}{x}\right )}}{2 c d}+\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}}{2 c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {a x \left (a+\frac {b}{x}\right )^{3/2}}{c \left (c+\frac {d}{x}\right )^2}-\frac {\frac {\sqrt {a+\frac {b}{x}} \left (-\frac {3 a^2 d}{c}+4 a b-\frac {b^2 c}{d}\right )}{\left (c+\frac {d}{x}\right )^2}+\frac {\frac {\sqrt {a+\frac {b}{x}} \left (-12 a^2 d^2+7 a b c d+b^2 c^2\right )}{c \left (c+\frac {d}{x}\right )}+\frac {\frac {2 (b c-a d)^{3/2} \left (-24 a^2 d^2+8 a b c d+b^2 c^2\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+\frac {b}{x}}}{\sqrt {b c-a d}}\right )}{c \sqrt {d}}-\frac {8 a^{3/2} d \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{x}}}{\sqrt {a}}\right ) (5 b c-6 a d) (b c-a d)}{c}}{2 c (b c-a d)}}{2 c d}}{2 c}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[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 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] - Simp[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] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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] && ILtQ[m, -1]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 899

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1035\) vs. \(2(209)=418\).

Time = 0.59 (sec) , antiderivative size = 1036, normalized size of antiderivative = 4.37

method	result	size
risch	\(\text {Expression too large to display}\)	\(1036\)
default	\(\text {Expression too large to display}\)	\(1640\)

Input:

int((a+b/x)^(5/2)/(c+1/x*d)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 1457, normalized size of antiderivative = 6.15 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="fricas")
 

Output:

[-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)*sqrt(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)*x*sqrt((a*x + b)/x)/(a*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 -...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{\frac {5}{2}}}{{\left (c + \frac {d}{x}\right )}^{3}} \,d x } \] Input:

integrate((a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 947 vs. \(2 (209) = 418\).

Time = 0.19 (sec) , antiderivative size = 947, normalized size of antiderivative = 4.00 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((a+b/x)^(5/2)/(c+d/x)^3,x, algorithm="giac")
 

Output:

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/sqr 
t(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(abs(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)*sgn(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...
 

Mupad [B] (verification not implemented)

Time = 2.94 (sec) , antiderivative size = 1476, normalized size of antiderivative = 6.23 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan((b^9*(a + b/x)^(1/2)*(a^3)^(1/2)*5i)/(8*((5*a^2*b^9)/8 + (8*a^3*b^8* 
d)/c - (159*a^4*b^7*d^2)/(8*c^2) + (45*a^5*b^6*d^3)/(4*c^3))) + (a*b^8*(a 
+ b/x)^(1/2)*(a^3)^(1/2)*8i)/(8*a^3*b^8 + (5*a^2*b^9*c)/(8*d) - (159*a^4*b 
^7*d)/(8*c) + (45*a^5*b^6*d^2)/(4*c^2)) - (a^2*b^7*d*(a + b/x)^(1/2)*(a^3) 
^(1/2)*159i)/(8*(8*a^3*b^8*c - (159*a^4*b^7*d)/8 + (5*a^2*b^9*c^2)/(8*d) + 
 (45*a^5*b^6*d^2)/(4*c))) + (a^3*b^6*d^2*(a + b/x)^(1/2)*(a^3)^(1/2)*45i)/ 
(4*(8*a^3*b^8*c^2 + (45*a^5*b^6*d^2)/4 + (5*a^2*b^9*c^3)/(8*d) - (159*a^4* 
b^7*c*d)/8)))*(6*a*d - 5*b*c)*(a^3)^(1/2)*1i)/c^4 - (((a + b/x)^(3/2)*(b^4 
*c^3 - 24*a^3*b*d^3 + 32*a^2*b^2*c*d^2 - 9*a*b^3*c^2*d))/(4*c^3*d) - (b*(a 
 + b/x)^(5/2)*(b^2*c^2 - 12*a^2*d^2 + 7*a*b*c*d))/(4*c^3) + (b*(a + b/x)^( 
1/2)*(12*a^4*d^3 - a*b^3*c^3 + 14*a^2*b^2*c^2*d - 25*a^3*b*c*d^2))/(4*c^3* 
d))/((a + b/x)^2*(3*a*d^2 - 2*b*c*d) - (a + b/x)*(3*a^2*d^2 + b^2*c^2 - 4* 
a*b*c*d) - d^2*(a + b/x)^3 + a^3*d^2 + a*b^2*c^2 - 2*a^2*b*c*d) + (log(- ( 
5*a^2*b^9*c^6 + 1728*a^8*b^3*d^6 + 64*a^3*b^8*c^5*d - 4752*a^7*b^4*c*d^5 - 
 59*a^4*b^7*c^4*d^2 - 1450*a^5*b^6*c^3*d^3 + 4464*a^6*b^5*c^2*d^4)/(16*c^9 
*d) - ((((a + b/x)^(1/2)*(b^8*c^6 + 1152*a^6*b^2*d^6 - 2496*a^5*b^3*c*d^5 
- 15*a^2*b^6*c^4*d^2 - 400*a^3*b^5*c^3*d^3 + 1760*a^4*b^4*c^2*d^4 + 14*a*b 
^7*c^5*d))/(8*c^6*d) - (((16*a*b^5*c^10*d^2 - 208*a^2*b^4*c^9*d^3 + 192*a^ 
3*b^3*c^8*d^4)/(16*c^9*d) - ((64*b^3*c^9*d^3 - 128*a*b^2*c^8*d^4)*(a + b/x 
)^(1/2)*(d^3*(a*d - b*c))^(1/2)*((b^2*c^2)/8 - 3*a^2*d^2 + a*b*c*d))/(8...
 

Reduce [B] (verification not implemented)

Time = 2.12 (sec) , antiderivative size = 3036, normalized size of antiderivative = 12.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^{5/2}}{\left (c+\frac {d}{x}\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(96*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqr 
t(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a**3*c**2*d 
**3*x**2 + 192*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2* 
sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))* 
a**3*c*d**4*x + 96*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqr 
t(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt( 
a))*a**3*d**5 - 80*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqr 
t(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt( 
a))*a**2*b*c**3*d**2*x**2 - 160*sqrt(d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a 
*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + sqrt(x)* 
sqrt(c)*sqrt(a))*a**2*b*c**2*d**3*x - 80*sqrt(d)*sqrt(a*d - b*c)*log(sqrt( 
c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b*c) + 
 sqrt(x)*sqrt(c)*sqrt(a))*a**2*b*c*d**4 + 12*sqrt(d)*sqrt(a*d - b*c)*log(s 
qrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2*a*d + b* 
c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b**2*c**4*d*x**2 + 24*sqrt(d)*sqrt(a*d - b 
*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2 
*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b**2*c**3*d**2*x + 12*sqrt(d)*sqr 
t(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - 
 b*c) - 2*a*d + b*c) + sqrt(x)*sqrt(c)*sqrt(a))*a*b**2*c**2*d**3 + 2*sqrt( 
d)*sqrt(a*d - b*c)*log(sqrt(c)*sqrt(a*x + b) - sqrt(2*sqrt(d)*sqrt(a)*s...