\(\int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx\) [313]

Optimal result

Integrand size = 22, antiderivative size = 142 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\frac {5 (b c-a d) (a+b x)^{3/2}}{3 c^2 (c+d x)^{3/2}}-\frac {(a+b x)^{5/2}}{c x (c+d x)^{3/2}}+\frac {5 a (b c-a d) \sqrt {a+b x}}{c^3 \sqrt {c+d x}}-\frac {5 a^{3/2} (b c-a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{7/2}} \] Output:

5/3*(-a*d+b*c)*(b*x+a)^(3/2)/c^2/(d*x+c)^(3/2)-(b*x+a)^(5/2)/c/x/(d*x+c)^( 
3/2)+5*a*(-a*d+b*c)*(b*x+a)^(1/2)/c^3/(d*x+c)^(1/2)-5*a^(3/2)*(-a*d+b*c)*a 
rctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(7/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.63 (sec) , antiderivative size = 1885, normalized size of antiderivative = 13.27 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

((Sqrt[c]*(-2*b^2*c^2*x^2 - 2*a*b*c*x*(7*c + 5*d*x) + a^2*(3*c^2 + 20*c*d* 
x + 15*d^2*x^2))*(16*a^3*d^2 - 4*a^2*d*(5*b*c - 7*b*d*x + 4*Sqrt[a - (b*c) 
/d]*d*Sqrt[a + b*x]) + b^2*(b*x*(5*c^2 - 10*c*d*x + d^2*x^2) - Sqrt[a - (b 
*c)/d]*Sqrt[a + b*x]*(c^2 - 10*c*d*x + 5*d^2*x^2)) + a*b*(4*Sqrt[a - (b*c) 
/d]*d*Sqrt[a + b*x]*(3*c - 5*d*x) + b*(5*c^2 - 30*c*d*x + 13*d^2*x^2))))/( 
x*(c + d*x)^(3/2)*(16*a^2*d^2*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]) + 4*a*b* 
d*(-3*c*Sqrt[a - (b*c)/d] + 5*Sqrt[a - (b*c)/d]*d*x + 5*c*Sqrt[a + b*x] - 
3*d*x*Sqrt[a + b*x]) + b^2*(c^2*(Sqrt[a - (b*c)/d] - 5*Sqrt[a + b*x]) + d^ 
2*x^2*(5*Sqrt[a - (b*c)/d] - Sqrt[a + b*x]) + c*d*x*(-10*Sqrt[a - (b*c)/d] 
 + 10*Sqrt[a + b*x])))) + (15*a^2*b*c*Sqrt[d]*ArcTan[(Sqrt[b*c - 2*a*d - ( 
2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqr 
t[a - (b*c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d 
]*Sqrt[b*c - a*d]] - (15*a^3*d^(3/2)*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt 
[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b* 
c)/d] - Sqrt[a + b*x]))])/Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b* 
c - a*d]] + ((15*I)*a^(3/2)*b^2*c^2*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[ 
a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c 
)/d] - Sqrt[a + b*x]))])/(Sqrt[b*c - a*d]*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a] 
*Sqrt[d]*Sqrt[b*c - a*d]]) - ((30*I)*a^(5/2)*b*c*d*ArcTan[(Sqrt[b*c - 2*a* 
d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt...
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {105, 105, 105, 104, 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.

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-((a + b*x)^(5/2)/(c*x*(c + d*x)^(3/2))) + (5*(b*c - a*d)*((2*(a + b*x)^(3 
/2))/(3*c*(c + d*x)^(3/2)) + (a*((2*Sqrt[a + b*x])/(c*Sqrt[c + d*x]) - (2* 
Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/c^(3/2)) 
)/c))/(2*c)
 

Defintions of rubi rules used

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 
1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f)))   Int[(a 
+ b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, 
e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] 
 ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(501\) vs. \(2(116)=232\).

Time = 0.24 (sec) , antiderivative size = 502, normalized size of antiderivative = 3.54

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} d^{3} x^{3}-15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b c \,d^{2} x^{3}+30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} c \,d^{2} x^{2}-30 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{2} d \,x^{2}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} c^{2} d x -15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b \,c^{3} x -30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} d^{2} x^{2}+20 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b c d \,x^{2}+4 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, b^{2} c^{2} x^{2}-40 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} c d x +28 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a b \,c^{2} x -6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} c^{2} \sqrt {a c}\right )}{6 c^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {a c}\, \left (x d +c \right )^{\frac {3}{2}}}\)	\(502\)

Input:

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

Output:

1/6*(b*x+a)^(1/2)*(15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2 
)+2*a*c)/x)*a^3*d^3*x^3-15*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c)) 
^(1/2)+2*a*c)/x)*a^2*b*c*d^2*x^3+30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a) 
*(d*x+c))^(1/2)+2*a*c)/x)*a^3*c*d^2*x^2-30*ln((a*d*x+b*c*x+2*(a*c)^(1/2)*( 
(b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c^2*d*x^2+15*ln((a*d*x+b*c*x+2*(a*c 
)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^3*c^2*d*x-15*ln((a*d*x+b*c*x+2 
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a^2*b*c^3*x-30*((b*x+a)*(d* 
x+c))^(1/2)*(a*c)^(1/2)*a^2*d^2*x^2+20*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2) 
*a*b*c*d*x^2+4*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^2*c^2*x^2-40*((b*x+a) 
*(d*x+c))^(1/2)*(a*c)^(1/2)*a^2*c*d*x+28*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/ 
2)*a*b*c^2*x-6*((b*x+a)*(d*x+c))^(1/2)*a^2*c^2*(a*c)^(1/2))/c^3/((b*x+a)*( 
d*x+c))^(1/2)/x/(a*c)^(1/2)/(d*x+c)^(3/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (116) = 232\).

Time = 0.61 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.57 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (3 \, a^{2} c^{2} - {\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \, {\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}, \frac {15 \, {\left ({\left (a b c d^{2} - a^{2} d^{3}\right )} x^{3} + 2 \, {\left (a b c^{2} d - a^{2} c d^{2}\right )} x^{2} + {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - 2 \, {\left (3 \, a^{2} c^{2} - {\left (2 \, b^{2} c^{2} + 10 \, a b c d - 15 \, a^{2} d^{2}\right )} x^{2} - 2 \, {\left (7 \, a b c^{2} - 10 \, a^{2} c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (c^{3} d^{2} x^{3} + 2 \, c^{4} d x^{2} + c^{5} x\right )}}\right ] \] Input:

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

Output:

[-1/12*(15*((a*b*c*d^2 - a^2*d^3)*x^3 + 2*(a*b*c^2*d - a^2*c*d^2)*x^2 + (a 
*b*c^3 - a^2*c^2*d)*x)*sqrt(a/c)*log((8*a^2*c^2 + (b^2*c^2 + 6*a*b*c*d + a 
^2*d^2)*x^2 + 4*(2*a*c^2 + (b*c^2 + a*c*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)* 
sqrt(a/c) + 8*(a*b*c^2 + a^2*c*d)*x)/x^2) + 4*(3*a^2*c^2 - (2*b^2*c^2 + 10 
*a*b*c*d - 15*a^2*d^2)*x^2 - 2*(7*a*b*c^2 - 10*a^2*c*d)*x)*sqrt(b*x + a)*s 
qrt(d*x + c))/(c^3*d^2*x^3 + 2*c^4*d*x^2 + c^5*x), 1/6*(15*((a*b*c*d^2 - a 
^2*d^3)*x^3 + 2*(a*b*c^2*d - a^2*c*d^2)*x^2 + (a*b*c^3 - a^2*c^2*d)*x)*sqr 
t(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqr 
t(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - 2*(3*a^2*c^2 - (2*b^2*c 
^2 + 10*a*b*c*d - 15*a^2*d^2)*x^2 - 2*(7*a*b*c^2 - 10*a^2*c*d)*x)*sqrt(b*x 
 + a)*sqrt(d*x + c))/(c^3*d^2*x^3 + 2*c^4*d*x^2 + c^5*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (116) = 232\).

Time = 0.85 (sec) , antiderivative size = 633, normalized size of antiderivative = 4.46 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{6} d {\left | b \right |} + 4 \, a b^{5} c^{5} d^{2} {\left | b \right |} - 11 \, a^{2} b^{4} c^{4} d^{3} {\left | b \right |} + 6 \, a^{3} b^{3} c^{3} d^{4} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}} + \frac {6 \, {\left (a b^{6} c^{6} d {\left | b \right |} - 3 \, a^{2} b^{5} c^{5} d^{2} {\left | b \right |} + 3 \, a^{3} b^{4} c^{4} d^{3} {\left | b \right |} - a^{4} b^{3} c^{3} d^{4} {\left | b \right |}\right )}}{b^{3} c^{7} d - a b^{2} c^{6} d^{2}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {5 \, {\left (\sqrt {b d} a^{2} b^{3} c - \sqrt {b d} a^{3} b^{2} d\right )} \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} b c^{3} {\left | b \right |}} - \frac {2 \, {\left (\sqrt {b d} a^{2} b^{5} c^{2} - 2 \, \sqrt {b d} a^{3} b^{4} c d + \sqrt {b d} a^{4} b^{3} d^{2} - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{3} c - \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{2} d\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} c^{3} {\left | b \right |}} \] Input:

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

Output:

2/3*sqrt(b*x + a)*((b^6*c^6*d*abs(b) + 4*a*b^5*c^5*d^2*abs(b) - 11*a^2*b^4 
*c^4*d^3*abs(b) + 6*a^3*b^3*c^3*d^4*abs(b))*(b*x + a)/(b^3*c^7*d - a*b^2*c 
^6*d^2) + 6*(a*b^6*c^6*d*abs(b) - 3*a^2*b^5*c^5*d^2*abs(b) + 3*a^3*b^4*c^4 
*d^3*abs(b) - a^4*b^3*c^3*d^4*abs(b))/(b^3*c^7*d - a*b^2*c^6*d^2))/(b^2*c 
+ (b*x + a)*b*d - a*b*d)^(3/2) - 5*(sqrt(b*d)*a^2*b^3*c - sqrt(b*d)*a^3*b^ 
2*d)*arctan(-1/2*(b^2*c + a*b*d - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d))^2)/(sqrt(-a*b*c*d)*b))/(sqrt(-a*b*c*d)*b*c^3*abs(b 
)) - 2*(sqrt(b*d)*a^2*b^5*c^2 - 2*sqrt(b*d)*a^3*b^4*c*d + sqrt(b*d)*a^4*b^ 
3*d^2 - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - 
a*b*d))^2*a^2*b^3*c - sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d))^2*a^3*b^2*d)/((b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2 - 
 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^2*c 
 - 2*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b 
*d + (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^4)*c^ 
3*abs(b))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.69 (sec) , antiderivative size = 1895, normalized size of antiderivative = 13.35 \[ \int \frac {(a+b x)^{5/2}}{x^2 (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 30*sqrt(c + d*x)*sqrt(a + b*x)*a**3*c**3*d**3 - 200*sqrt(c + d*x)*sqrt 
(a + b*x)*a**3*c**2*d**4*x - 150*sqrt(c + d*x)*sqrt(a + b*x)*a**3*c*d**5*x 
**2 + 6*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b*c**4*d**2 + 180*sqrt(c + d*x)*s 
qrt(a + b*x)*a**2*b*c**3*d**3*x + 130*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b*c 
**2*d**4*x**2 - 28*sqrt(c + d*x)*sqrt(a + b*x)*a*b**2*c**4*d**2*x - 4*sqrt 
(c + d*x)*sqrt(a + b*x)*b**3*c**4*d**2*x**2 - 75*sqrt(c)*sqrt(a)*log( - sq 
rt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) 
+ sqrt(b)*sqrt(c + d*x))*a**3*c**2*d**4*x - 150*sqrt(c)*sqrt(a)*log( - sqr 
t(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + 
 sqrt(b)*sqrt(c + d*x))*a**3*c*d**5*x**2 - 75*sqrt(c)*sqrt(a)*log( - sqrt( 
2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + s 
qrt(b)*sqrt(c + d*x))*a**3*d**6*x**3 + 90*sqrt(c)*sqrt(a)*log( - sqrt(2*sq 
rt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt( 
b)*sqrt(c + d*x))*a**2*b*c**3*d**3*x + 180*sqrt(c)*sqrt(a)*log( - sqrt(2*s 
qrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt 
(b)*sqrt(c + d*x))*a**2*b*c**2*d**4*x**2 + 90*sqrt(c)*sqrt(a)*log( - sqrt( 
2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + s 
qrt(b)*sqrt(c + d*x))*a**2*b*c*d**5*x**3 - 15*sqrt(c)*sqrt(a)*log( - sqrt( 
2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + s 
qrt(b)*sqrt(c + d*x))*a*b**2*c**4*d**2*x - 30*sqrt(c)*sqrt(a)*log( - sq...