\(\int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx\) [306]

Optimal result

Integrand size = 22, antiderivative size = 226 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=\frac {5 (b c-7 a d) (b c-a d)^2 \sqrt {a+b x}}{8 a c^4 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d) (a+b x)^{3/2}}{24 a c^3 x \sqrt {c+d x}}-\frac {(b c-7 a d) (a+b x)^{5/2}}{12 a c^2 x^2 \sqrt {c+d x}}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}-\frac {5 (b c-7 a d) (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{9/2}} \] Output:

5/8*(-7*a*d+b*c)*(-a*d+b*c)^2*(b*x+a)^(1/2)/a/c^4/(d*x+c)^(1/2)-5/24*(-7*a 
*d+b*c)*(-a*d+b*c)*(b*x+a)^(3/2)/a/c^3/x/(d*x+c)^(1/2)-1/12*(-7*a*d+b*c)*( 
b*x+a)^(5/2)/a/c^2/x^2/(d*x+c)^(1/2)-1/3*(b*x+a)^(7/2)/a/c/x^3/(d*x+c)^(1/ 
2)-5/8*(-7*a*d+b*c)*(-a*d+b*c)^2*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d* 
x+c)^(1/2))/a^(1/2)/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.76 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=-\frac {\sqrt {a+b x} \left (3 b^2 c^2 x^2 (11 c+27 d x)+2 a b c x \left (13 c^2-34 c d x-95 d^2 x^2\right )+a^2 \left (8 c^3-14 c^2 d x+35 c d^2 x^2+105 d^3 x^3\right )\right )}{24 c^4 x^3 \sqrt {c+d x}}+\frac {5 (b c-a d)^2 (-b c+7 a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{8 \sqrt {a} c^{9/2}} \] Input:

Integrate[(a + b*x)^(5/2)/(x^4*(c + d*x)^(3/2)),x]
 

Output:

-1/24*(Sqrt[a + b*x]*(3*b^2*c^2*x^2*(11*c + 27*d*x) + 2*a*b*c*x*(13*c^2 - 
34*c*d*x - 95*d^2*x^2) + a^2*(8*c^3 - 14*c^2*d*x + 35*c*d^2*x^2 + 105*d^3* 
x^3)))/(c^4*x^3*Sqrt[c + d*x]) + (5*(b*c - a*d)^2*(-(b*c) + 7*a*d)*ArcTanh 
[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/(8*Sqrt[a]*c^(9/2))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {107, 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.

\(\displaystyle \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 107

\(\displaystyle \frac {(b c-7 a d) \int \frac {(a+b x)^{5/2}}{x^3 (c+d x)^{3/2}}dx}{6 a c}-\frac {(a+b x)^{7/2}}{3 a c x^3 \sqrt {c+d x}}\)

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 105

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

Int[(a + b*x)^(5/2)/(x^4*(c + d*x)^(3/2)),x]
 

Output:

-1/3*(a + b*x)^(7/2)/(a*c*x^3*Sqrt[c + d*x]) + ((b*c - 7*a*d)*(-1/2*(a + b 
*x)^(5/2)/(c*x^2*Sqrt[c + d*x]) + (5*(b*c - a*d)*(-((a + b*x)^(3/2)/(c*x*S 
qrt[c + d*x])) + (3*(b*c - a*d)*((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)) 
)/(2*c)))/(4*c)))/(6*a*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 107

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> 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] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 
 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x 
] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[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. \(703\) vs. \(2(188)=376\).

Time = 0.24 (sec) , antiderivative size = 704, normalized size of antiderivative = 3.12

method	result	size
default	\(\frac {\sqrt {b x +a}\, \left (105 \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^{4} x^{4}-225 \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^{3} x^{4}+135 \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 \,b^{2} c^{2} d^{2} x^{4}-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 ) b^{3} c^{3} d \,x^{4}+105 \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^{3} x^{3}-225 \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^{2} x^{3}+135 \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 \,b^{2} c^{3} d \,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 ) b^{3} c^{4} x^{3}-210 a^{2} d^{3} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+380 a b c \,d^{2} x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}-162 b^{2} c^{2} d \,x^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}-70 a^{2} c \,d^{2} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+136 a b \,c^{2} d \,x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}-66 b^{2} c^{3} x^{2} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+28 a^{2} c^{2} d x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}-52 a b \,c^{3} x \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}-16 a^{2} c^{3} \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\right )}{48 c^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{3} \sqrt {a c}\, \sqrt {x d +c}}\)	\(704\)

Input:

int((b*x+a)^(5/2)/x^4/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/48*(b*x+a)^(1/2)*(105*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^4*x^4-225*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^3*x^4+135*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*b^2*c^2*d^2*x^4-15*ln((a*d*x+b*c*x+2*(a*c) 
^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*b^3*c^3*d*x^4+105*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^3*x^3-225*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^2*x^3 
+135*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*b^2 
*c^3*d*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)*b^3*c^4*x^3-210*a^2*d^3*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+380*a* 
b*c*d^2*x^3*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-162*b^2*c^2*d*x^3*(a*c)^(1 
/2)*((b*x+a)*(d*x+c))^(1/2)-70*a^2*c*d^2*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c)) 
^(1/2)+136*a*b*c^2*d*x^2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-66*b^2*c^3*x^ 
2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+28*a^2*c^2*d*x*(a*c)^(1/2)*((b*x+a)* 
(d*x+c))^(1/2)-52*a*b*c^3*x*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)-16*a^2*c^3 
*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/c^4/((b*x+a)*(d*x+c))^(1/2)/x^3/(a*c 
)^(1/2)/(d*x+c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.80 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{4} + {\left (b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 15 \, a^{2} b c^{2} d^{2} - 7 \, a^{3} c d^{3}\right )} x^{3}\right )} \sqrt {a c} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \, {\left (8 \, a^{3} c^{4} + {\left (81 \, a b^{2} c^{3} d - 190 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{3} + {\left (33 \, a b^{2} c^{4} - 68 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{4} - 7 \, a^{3} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (a c^{5} d x^{4} + a c^{6} x^{3}\right )}}, \frac {15 \, {\left ({\left (b^{3} c^{3} d - 9 \, a b^{2} c^{2} d^{2} + 15 \, a^{2} b c d^{3} - 7 \, a^{3} d^{4}\right )} x^{4} + {\left (b^{3} c^{4} - 9 \, a b^{2} c^{3} d + 15 \, a^{2} b c^{2} d^{2} - 7 \, a^{3} c d^{3}\right )} x^{3}\right )} \sqrt {-a c} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left (8 \, a^{3} c^{4} + {\left (81 \, a b^{2} c^{3} d - 190 \, a^{2} b c^{2} d^{2} + 105 \, a^{3} c d^{3}\right )} x^{3} + {\left (33 \, a b^{2} c^{4} - 68 \, a^{2} b c^{3} d + 35 \, a^{3} c^{2} d^{2}\right )} x^{2} + 2 \, {\left (13 \, a^{2} b c^{4} - 7 \, a^{3} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (a c^{5} d x^{4} + a c^{6} x^{3}\right )}}\right ] \] Input:

integrate((b*x+a)^(5/2)/x^4/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

[-1/96*(15*((b^3*c^3*d - 9*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - 7*a^3*d^4)*x^4 
 + (b^3*c^4 - 9*a*b^2*c^3*d + 15*a^2*b*c^2*d^2 - 7*a^3*c*d^3)*x^3)*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 + (b*c 
+ a*d)*x)*sqrt(a*c)*sqrt(b*x + a)*sqrt(d*x + c) + 8*(a*b*c^2 + a^2*c*d)*x) 
/x^2) + 4*(8*a^3*c^4 + (81*a*b^2*c^3*d - 190*a^2*b*c^2*d^2 + 105*a^3*c*d^3 
)*x^3 + (33*a*b^2*c^4 - 68*a^2*b*c^3*d + 35*a^3*c^2*d^2)*x^2 + 2*(13*a^2*b 
*c^4 - 7*a^3*c^3*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a*c^5*d*x^4 + a*c^6*x 
^3), 1/48*(15*((b^3*c^3*d - 9*a*b^2*c^2*d^2 + 15*a^2*b*c*d^3 - 7*a^3*d^4)* 
x^4 + (b^3*c^4 - 9*a*b^2*c^3*d + 15*a^2*b*c^2*d^2 - 7*a^3*c*d^3)*x^3)*sqrt 
(-a*c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(b*x + a)*sqrt(d* 
x + c)/(a*b*c*d*x^2 + a^2*c^2 + (a*b*c^2 + a^2*c*d)*x)) - 2*(8*a^3*c^4 + ( 
81*a*b^2*c^3*d - 190*a^2*b*c^2*d^2 + 105*a^3*c*d^3)*x^3 + (33*a*b^2*c^4 - 
68*a^2*b*c^3*d + 35*a^3*c^2*d^2)*x^2 + 2*(13*a^2*b*c^4 - 7*a^3*c^3*d)*x)*s 
qrt(b*x + a)*sqrt(d*x + c))/(a*c^5*d*x^4 + a*c^6*x^3)]
 

Sympy [F]

\[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=\int \frac {\left (a + b x\right )^{\frac {5}{2}}}{x^{4} \left (c + d x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((b*x+a)**(5/2)/x**4/(d*x+c)**(3/2),x)
 

Output:

Integral((a + b*x)**(5/2)/(x**4*(c + d*x)**(3/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(5/2)/x^4/(d*x+c)^(3/2),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. 2200 vs. \(2 (188) = 376\).

Time = 3.23 (sec) , antiderivative size = 2200, normalized size of antiderivative = 9.73 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(5/2)/x^4/(d*x+c)^(3/2),x, algorithm="giac")
 

Output:

-2*(b^4*c^2*d - 2*a*b^3*c*d^2 + a^2*b^2*d^3)*sqrt(b*x + a)/(sqrt(b^2*c + ( 
b*x + a)*b*d - a*b*d)*c^4*abs(b)) - 5/8*(sqrt(b*d)*b^5*c^3 - 9*sqrt(b*d)*a 
*b^4*c^2*d + 15*sqrt(b*d)*a^2*b^3*c*d^2 - 7*sqrt(b*d)*a^3*b^2*d^3)*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^4*abs(b)) - 1/12*(3 
3*sqrt(b*d)*b^15*c^8 - 292*sqrt(b*d)*a*b^14*c^7*d + 1116*sqrt(b*d)*a^2*b^1 
3*c^6*d^2 - 2412*sqrt(b*d)*a^3*b^12*c^5*d^3 + 3230*sqrt(b*d)*a^4*b^11*c^4* 
d^4 - 2748*sqrt(b*d)*a^5*b^10*c^3*d^5 + 1452*sqrt(b*d)*a^6*b^9*c^2*d^6 - 4 
36*sqrt(b*d)*a^7*b^8*c*d^7 + 57*sqrt(b*d)*a^8*b^7*d^8 - 165*sqrt(b*d)*(sqr 
t(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^13*c^7 + 1 
017*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b* 
d))^2*a*b^12*c^6*d - 2277*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^11*c^5*d^2 + 1953*sqrt(b*d)*(sqrt(b*d)*s 
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^3*b^10*c^4*d^3 + 3 
93*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d 
))^2*a^4*b^9*c^3*d^4 - 1917*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2* 
c + (b*x + a)*b*d - a*b*d))^2*a^5*b^8*c^2*d^5 + 1281*sqrt(b*d)*(sqrt(b*d)* 
sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a^6*b^7*c*d^6 - 285 
*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d)) 
^2*a^7*b^6*d^7 + 330*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx=\int \frac {{\left (a+b\,x\right )}^{5/2}}{x^4\,{\left (c+d\,x\right )}^{3/2}} \,d x \] Input:

int((a + b*x)^(5/2)/(x^4*(c + d*x)^(3/2)),x)
 

Output:

int((a + b*x)^(5/2)/(x^4*(c + d*x)^(3/2)), x)
 

Reduce [B] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 2164, normalized size of antiderivative = 9.58 \[ \int \frac {(a+b x)^{5/2}}{x^4 (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(5/2)/x^4/(d*x+c)^(3/2),x)
 

Output:

( - 112*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**4*d + 196*sqrt(c + d*x)*sqrt(a 
 + b*x)*a**4*c**3*d**2*x - 490*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c**2*d**3* 
x**2 - 1470*sqrt(c + d*x)*sqrt(a + b*x)*a**4*c*d**4*x**3 - 80*sqrt(c + d*x 
)*sqrt(a + b*x)*a**3*b*c**5 - 224*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*c**4* 
d*x + 602*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b*c**3*d**2*x**2 + 1610*sqrt(c 
+ d*x)*sqrt(a + b*x)*a**3*b*c**2*d**3*x**3 - 260*sqrt(c + d*x)*sqrt(a + b* 
x)*a**2*b**2*c**5*x + 218*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*c**4*d*x** 
2 + 766*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**2*c**3*d**2*x**3 - 330*sqrt(c 
+ d*x)*sqrt(a + b*x)*a*b**3*c**5*x**2 - 810*sqrt(c + d*x)*sqrt(a + b*x)*a* 
b**3*c**4*d*x**3 - 735*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) + sqrt(b)*sqrt(c + d*x))*a 
**4*c*d**4*x**3 - 735*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) + sqrt(b)*sqrt(c + d*x))*a* 
*4*d**5*x**4 + 1050*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) + sqrt(b)*sqrt(c + d*x))*a**3 
*b*c**2*d**3*x**3 + 1050*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqr 
t(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x)) 
*a**3*b*c*d**4*x**4 + 180*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sq 
rt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x) 
)*a**2*b**2*c**3*d**2*x**3 + 180*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*...