\(\int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx\) [267]

Optimal result

Integrand size = 22, antiderivative size = 197 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {b (b c+11 a d) \sqrt {a+b x} \sqrt {c+d x}}{4 d}+\frac {3}{2} b (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}-\frac {a^{3/2} (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 d^{3/2}} \] Output:

1/4*b*(11*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/d+3/2*b*(b*x+a)^(3/2)*(d*x+ 
c)^(1/2)-(b*x+a)^(5/2)*(d*x+c)^(1/2)/x-a^(3/2)*(a*d+5*b*c)*arctanh(c^(1/2) 
*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/c^(1/2)-1/4*b^(1/2)*(-15*a^2*d^2-10* 
a*b*c*d+b^2*c^2)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/d^(3 
/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-4 a^2 d+9 a b d x+b^2 x (c+2 d x)\right )}{d x}-\frac {4 a^{3/2} (5 b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {c}}-\frac {\sqrt {b} \left (b^2 c^2-10 a b c d-15 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{d^{3/2}}\right ) \] Input:

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

Output:

((Sqrt[a + b*x]*Sqrt[c + d*x]*(-4*a^2*d + 9*a*b*d*x + b^2*x*(c + 2*d*x)))/ 
(d*x) - (4*a^(3/2)*(5*b*c + a*d)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]* 
Sqrt[a + b*x])])/Sqrt[c] - (Sqrt[b]*(b^2*c^2 - 10*a*b*c*d - 15*a^2*d^2)*Ar 
cTanh[(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[a + b*x])])/d^(3/2))/4
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.06, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {108, 27, 171, 27, 171, 27, 175, 66, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {d \sqrt {a+b x} (2 a (5 b c+a d)+b (b c+11 a d) x)}{x \sqrt {c+d x}}dx}{2 d}+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {\sqrt {a+b x} (2 a (5 b c+a d)+b (b c+11 a d) x)}{x \sqrt {c+d x}}dx+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {4 a^2 d (5 b c+a d)-b \left (b^2 c^2-10 a b d c-15 a^2 d^2\right ) x}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{d}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{d}\right )+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\int \frac {4 a^2 d (5 b c+a d)-b \left (b^2 c^2-10 a b d c-15 a^2 d^2\right ) x}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{d}\right )+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {4 a^2 d (a d+5 b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-b \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 d}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{d}\right )+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {4 a^2 d (a d+5 b c) \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx-2 b \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 d}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{d}\right )+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {8 a^2 d (a d+5 b c) \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}-2 b \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{2 d}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{d}\right )+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {-\frac {8 a^{3/2} d (a d+5 b c) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}}-\frac {2 \sqrt {b} \left (-15 a^2 d^2-10 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}}{2 d}+\frac {b \sqrt {a+b x} \sqrt {c+d x} (11 a d+b c)}{d}\right )+3 b (a+b x)^{3/2} \sqrt {c+d x}\right )-\frac {(a+b x)^{5/2} \sqrt {c+d x}}{x}\)

Input:

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

Output:

-(((a + b*x)^(5/2)*Sqrt[c + d*x])/x) + (3*b*(a + b*x)^(3/2)*Sqrt[c + d*x] 
+ ((b*(b*c + 11*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x])/d + ((-8*a^(3/2)*d*(5*b* 
c + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] 
 - (2*Sqrt[b]*(b^2*c^2 - 10*a*b*c*d - 15*a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a 
+ b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[d])/(2*d))/2)/2
 

Defintions of rubi rules used

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 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

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 108

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/(b*(m + 1))) 
, x] - Simp[1/(b*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* 
x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c 
, d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 
*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

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

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. \(433\) vs. \(2(155)=310\).

Time = 0.23 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.20

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a^{2} b \,d^{2} x \sqrt {a c}+10 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) a \,b^{2} c d x \sqrt {a c}-\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{3} c^{2} x \sqrt {a c}-4 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} d^{2} x \sqrt {d b}-20 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b c d x \sqrt {d b}+4 b^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}+18 a b d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}+2 b^{2} c x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}-8 a^{2} d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\right )}{8 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, \sqrt {d b}\, d x}\)	\(434\)

Input:

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

Output:

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(15*ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1 
/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a^2*b*d^2*x*(a*c)^(1/2)+10*ln(1/2*(2 
*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d+b*c)/(d*b)^(1/2))*a*b^2*c 
*d*x*(a*c)^(1/2)-ln(1/2*(2*b*d*x+2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+a*d 
+b*c)/(d*b)^(1/2))*b^3*c^2*x*(a*c)^(1/2)-4*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^2*x*(d*b)^(1/2)-20*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*x*(d*b)^(1/2)+4* 
b^2*d*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)+18*a*b*d*x*((b*x 
+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)+2*b^2*c*x*((b*x+a)*(d*x+c))^(1/ 
2)*(d*b)^(1/2)*(a*c)^(1/2)-8*a^2*d*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a* 
c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/(a*c)^(1/2)/(d*b)^(1/2)/d/x
 

Fricas [A] (verification not implemented)

Time = 1.33 (sec) , antiderivative size = 1074, normalized size of antiderivative = 5.45 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/16*((b^2*c^2 - 10*a*b*c*d - 15*a^2*d^2)*x*sqrt(b/d)*log(8*b^2*d^2*x^2 
+ b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + 
 a)*sqrt(d*x + c)*sqrt(b/d) + 8*(b^2*c*d + a*b*d^2)*x) - 4*(5*a*b*c*d + a^ 
2*d^2)*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*(2*b^2*d*x^2 - 4*a^2*d + (b^2*c + 9*a*b*d) 
*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d*x), 1/8*((b^2*c^2 - 10*a*b*c*d - 15*a^ 
2*d^2)*x*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d* 
x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) + 2*(5*a*b*c*d 
+ a^2*d^2)*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) + 2*(2*b^2*d*x^2 - 4*a^2*d + (b^2*c + 9*a* 
b*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d*x), 1/16*(8*(5*a*b*c*d + a^2*d^2)* 
x*sqrt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c 
)*sqrt(-a/c)/(a*b*d*x^2 + a^2*c + (a*b*c + a^2*d)*x)) - (b^2*c^2 - 10*a*b* 
c*d - 15*a^2*d^2)*x*sqrt(b/d)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^ 
2*d^2 + 4*(2*b*d^2*x + b*c*d + a*d^2)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b/d 
) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d*x^2 - 4*a^2*d + (b^2*c + 9*a*b*d 
)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(d*x), 1/8*(4*(5*a*b*c*d + a^2*d^2)*x*sq 
rt(-a/c)*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(b*x + a)*sqrt(d*x + c)...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(5/2)*(d*x+c)^(1/2)/x^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. 556 vs. \(2 (155) = 310\).

Time = 0.37 (sec) , antiderivative size = 556, normalized size of antiderivative = 2.82 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {1}{8} \, {\left (2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} {\left | b \right |}}{b^{2}} + \frac {b^{4} c d {\left | b \right |} + 7 \, a b^{3} d^{2} {\left | b \right |}}{b^{5} d^{2}}\right )} + \frac {8 \, {\left (5 \, a^{2} b c d {\left | b \right |} + a^{3} d^{2} {\left | b \right |}\right )} \arctan \left (\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} \sqrt {b d} b} - \frac {16 \, {\left (a^{2} b^{3} c^{2} d {\left | b \right |} - 2 \, a^{3} b^{2} c d^{2} {\left | b \right |} + a^{4} b d^{3} {\left | b \right |} - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b c d {\left | b \right |} - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} d^{2} {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} \sqrt {b d}} + \frac {{\left (b^{2} c^{2} {\left | b \right |} - 10 \, a b c d {\left | b \right |} - 15 \, a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d} b d}\right )} b \] Input:

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

Output:

1/8*(2*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*sqrt(b*x + a)*(2*(b*x + a)*abs( 
b)/b^2 + (b^4*c*d*abs(b) + 7*a*b^3*d^2*abs(b))/(b^5*d^2)) + 8*(5*a^2*b*c*d 
*abs(b) + a^3*d^2*abs(b))*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)*sqrt(b*d)*b) - 16*(a^2*b^3*c^2*d*abs(b) - 2*a^3*b^2*c*d^2*abs(b) 
+ a^4*b*d^3*abs(b) - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d 
 - a*b*d))^2*a^2*b*c*d*abs(b) - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b 
*x + a)*b*d - a*b*d))^2*a^3*d^2*abs(b))/((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)*sqrt(b*d)) + (b^2*c^2*abs(b) - 10*a*b*c*d*abs(b) - 15*a^2*d^2*abs(b))*l 
og((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)/(sqr 
t(b*d)*b*d))*b
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 8.46 (sec) , antiderivative size = 539, normalized size of antiderivative = 2.74 \[ \int \frac {(a+b x)^{5/2} \sqrt {c+d x}}{x^2} \, dx=\frac {-4 \sqrt {d x +c}\, \sqrt {b x +a}\, a^{2} c \,d^{2}+9 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c \,d^{2} x +\sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c^{2} d x +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c \,d^{2} x^{2}+2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a^{2} d^{3} x +10 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b c \,d^{2} x +2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a^{2} d^{3} x +10 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b c \,d^{2} x -2 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a^{2} d^{3} x -10 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a b c \,d^{2} x +15 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} c \,d^{2} x +10 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b \,c^{2} d x -\sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{3} x}{4 c \,d^{2} x} \] Input:

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

Output:

( - 4*sqrt(c + d*x)*sqrt(a + b*x)*a**2*c*d**2 + 9*sqrt(c + d*x)*sqrt(a + b 
*x)*a*b*c*d**2*x + sqrt(c + d*x)*sqrt(a + b*x)*b**2*c**2*d*x + 2*sqrt(c + 
d*x)*sqrt(a + b*x)*b**2*c*d**2*x**2 + 2*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**2*d**3*x + 10*sqrt(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqr 
t(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
 + d*x))*a*b*c*d**2*x + 2*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 
**2*d**3*x + 10*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*b*c*d**2* 
x - 2*sqrt(c)*sqrt(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 
2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*a**2*d**3*x - 10*sqrt(c)*sqrt 
(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)* 
sqrt(b)*sqrt(a) + 2*b*d*x)*a*b*c*d**2*x + 15*sqrt(d)*sqrt(b)*log((sqrt(d)* 
sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*c*d**2*x + 10 
*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt( 
a*d - b*c))*a*b*c**2*d*x - sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sq 
rt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**2*c**3*x)/(4*c*d**2*x)