\(\int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx\) [171]

Optimal result

Integrand size = 22, antiderivative size = 251 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{24 a c x^3}-\frac {\left (12 a b c d-5 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{96 a^2 c^2 x^2}-\frac {(b c+a d) \left (15 b^2 c^2-22 a b c d+15 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{192 a^3 c^3 x}+\frac {(b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{64 a^{7/2} c^{7/2}} \] Output:

-1/4*(b*x+a)^(1/2)*(d*x+c)^(1/2)/x^4-1/24*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^ 
(1/2)/a/c/x^3-1/96*(12*a*b*c*d-5*(a*d+b*c)^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/ 
a^2/c^2/x^2-1/192*(a*d+b*c)*(15*a^2*d^2-22*a*b*c*d+15*b^2*c^2)*(b*x+a)^(1/ 
2)*(d*x+c)^(1/2)/a^3/c^3/x+1/64*(-a*d+b*c)^2*(5*a^2*d^2+6*a*b*c*d+5*b^2*c^ 
2)*arctanh(c^(1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(7/2)/c^(7/2)
 

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (15 b^3 c^3 x^3-a b^2 c^2 x^2 (10 c+7 d x)+a^2 b c x \left (8 c^2+4 c d x-7 d^2 x^2\right )+a^3 \left (48 c^3+8 c^2 d x-10 c d^2 x^2+15 d^3 x^3\right )\right )}{192 a^3 c^3 x^4}+\frac {(b c-a d)^2 \left (5 b^2 c^2+6 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{64 a^{7/2} c^{7/2}} \] Input:

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

Output:

-1/192*(Sqrt[a + b*x]*Sqrt[c + d*x]*(15*b^3*c^3*x^3 - a*b^2*c^2*x^2*(10*c 
+ 7*d*x) + a^2*b*c*x*(8*c^2 + 4*c*d*x - 7*d^2*x^2) + a^3*(48*c^3 + 8*c^2*d 
*x - 10*c*d^2*x^2 + 15*d^3*x^3)))/(a^3*c^3*x^4) + ((b*c - a*d)^2*(5*b^2*c^ 
2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a 
 + b*x])])/(64*a^(7/2)*c^(7/2))
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {108, 27, 168, 27, 168, 27, 168, 27, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{4} \int \frac {b c+a d+2 b d x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \int \frac {b c+a d+2 b d x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {\int \frac {(b c+a d) \left (15 b^2 c^2-22 a b d c+15 a^2 d^2\right )+2 b d \left (5 b^2 c^2-2 a b d c+5 a^2 d^2\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}+\frac {5 a d^2}{c}-2 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {\int \frac {(b c+a d) \left (15 b^2 c^2-22 a b d c+15 a^2 d^2\right )+2 b d \left (5 b^2 c^2-2 a b d c+5 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}+\frac {5 a d^2}{c}-2 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {-\frac {\int \frac {3 (b c-a d)^2 \left (5 b^2 c^2+6 a b d c+5 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}+\frac {5 a d^2}{c}-2 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {-\frac {3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}+\frac {5 a d^2}{c}-2 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {-\frac {3 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}+\frac {5 a d^2}{c}-2 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{8} \left (-\frac {-\frac {\frac {3 (b c-a d)^2 \left (5 a^2 d^2+6 a b c d+5 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (15 a^2 d^2-22 a b c d+15 b^2 c^2\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {5 b^2 c}{a}+\frac {5 a d^2}{c}-2 b d\right )}{2 x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{3 a c x^3}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{4 x^4}\)

Input:

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

Output:

-1/4*(Sqrt[a + b*x]*Sqrt[c + d*x])/x^4 + (-1/3*((b*c + a*d)*Sqrt[a + b*x]* 
Sqrt[c + d*x])/(a*c*x^3) - (-1/2*(((5*b^2*c)/a - 2*b*d + (5*a*d^2)/c)*Sqrt 
[a + b*x]*Sqrt[c + d*x])/x^2 - (-(((b*c + a*d)*(15*b^2*c^2 - 22*a*b*c*d + 
15*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (3*(b*c - a*d)^2*(5*b^ 
2*c^2 + 6*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sq 
rt[c + d*x])])/(a^(3/2)*c^(3/2)))/(4*a*c))/(6*a*c))/8
 

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 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 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 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. \(592\) vs. \(2(213)=426\).

Time = 0.22 (sec) , antiderivative size = 593, normalized size of antiderivative = 2.36

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{4} d^{4} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{3} b c \,d^{3} x^{4}-6 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} b^{2} c^{2} d^{2} x^{4}-12 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a \,b^{3} c^{3} d \,x^{4}+15 \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) b^{4} c^{4} x^{4}-30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} d^{3} x^{3}+14 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b c \,d^{2} x^{3}+14 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a \,b^{2} c^{2} d \,x^{3}-30 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, b^{3} c^{3} x^{3}+20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} c \,d^{2} x^{2}-8 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b \,c^{2} d \,x^{2}+20 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a \,b^{2} c^{3} x^{2}-16 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} c^{2} d x -16 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b \,c^{3} x -96 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} c^{3} \sqrt {a c}\right )}{384 a^{3} c^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{4} \sqrt {a c}}\)	\(593\)

Input:

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

Output:

1/384*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^3/c^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^4*d^4*x^4-12*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*b*c*d^3*x^4-6*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^2*c^2*d^2*x^4-12*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^3*c^3*d* 
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^4 
*c^4*x^4-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*d^3*x^3+14*((b*x+a)*(d 
*x+c))^(1/2)*(a*c)^(1/2)*a^2*b*c*d^2*x^3+14*((b*x+a)*(d*x+c))^(1/2)*(a*c)^ 
(1/2)*a*b^2*c^2*d*x^3-30*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*b^3*c^3*x^3+2 
0*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c*d^2*x^2-8*(a*c)^(1/2)*((b*x+a) 
*(d*x+c))^(1/2)*a^2*b*c^2*d*x^2+20*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b 
^2*c^3*x^2-16*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*c^2*d*x-16*(a*c)^(1/ 
2)*((b*x+a)*(d*x+c))^(1/2)*a^2*b*c^3*x-96*((b*x+a)*(d*x+c))^(1/2)*a^3*c^3* 
(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^4/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 1.30 (sec) , antiderivative size = 568, normalized size of antiderivative = 2.26 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx=\left [\frac {3 \, {\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt {a c} x^{4} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \, {\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {a c} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) - 4 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d - 7 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{768 \, a^{4} c^{4} x^{4}}, -\frac {3 \, {\left (5 \, b^{4} c^{4} - 4 \, a b^{3} c^{3} d - 2 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + 5 \, a^{4} d^{4}\right )} \sqrt {-a c} x^{4} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {-a c} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (a b c d x^{2} + a^{2} c^{2} + {\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) + 2 \, {\left (48 \, a^{4} c^{4} + {\left (15 \, a b^{3} c^{4} - 7 \, a^{2} b^{2} c^{3} d - 7 \, a^{3} b c^{2} d^{2} + 15 \, a^{4} c d^{3}\right )} x^{3} - 2 \, {\left (5 \, a^{2} b^{2} c^{4} - 2 \, a^{3} b c^{3} d + 5 \, a^{4} c^{2} d^{2}\right )} x^{2} + 8 \, {\left (a^{3} b c^{4} + a^{4} c^{3} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{384 \, a^{4} c^{4} x^{4}}\right ] \] Input:

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

Output:

[1/768*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + 
 5*a^4*d^4)*sqrt(a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 - 7*a^2*b^2*c 
^3*d - 7*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 2*a^3*b*c^ 
3*d + 5*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt 
(d*x + c))/(a^4*c^4*x^4), -1/384*(3*(5*b^4*c^4 - 4*a*b^3*c^3*d - 2*a^2*b^2 
*c^2*d^2 - 4*a^3*b*c*d^3 + 5*a^4*d^4)*sqrt(-a*c)*x^4*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*(48*a^4*c^4 + (15*a*b^3*c^4 - 7*a^2*b^2*c^ 
3*d - 7*a^3*b*c^2*d^2 + 15*a^4*c*d^3)*x^3 - 2*(5*a^2*b^2*c^4 - 2*a^3*b*c^3 
*d + 5*a^4*c^2*d^2)*x^2 + 8*(a^3*b*c^4 + a^4*c^3*d)*x)*sqrt(b*x + a)*sqrt( 
d*x + c))/(a^4*c^4*x^4)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx=\int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{5}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5,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. 3834 vs. \(2 (213) = 426\).

Time = 1.33 (sec) , antiderivative size = 3834, normalized size of antiderivative = 15.27 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx=\text {Too large to display} \] Input:

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

Output:

1/192*(3*(5*sqrt(b*d)*b^5*c^4*abs(b) - 4*sqrt(b*d)*a*b^4*c^3*d*abs(b) - 2* 
sqrt(b*d)*a^2*b^3*c^2*d^2*abs(b) - 4*sqrt(b*d)*a^3*b^2*c*d^3*abs(b) + 5*sq 
rt(b*d)*a^4*b*d^4*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)*a^3*b*c^3) - 2*(15*sqrt(b*d)*b^19*c^11*abs(b) - 127*sqrt(b*d)*a* 
b^18*c^10*d*abs(b) + 469*sqrt(b*d)*a^2*b^17*c^9*d^2*abs(b) - 965*sqrt(b*d) 
*a^3*b^16*c^8*d^3*abs(b) + 1126*sqrt(b*d)*a^4*b^15*c^7*d^4*abs(b) - 518*sq 
rt(b*d)*a^5*b^14*c^6*d^5*abs(b) - 518*sqrt(b*d)*a^6*b^13*c^5*d^6*abs(b) + 
1126*sqrt(b*d)*a^7*b^12*c^4*d^7*abs(b) - 965*sqrt(b*d)*a^8*b^11*c^3*d^8*ab 
s(b) + 469*sqrt(b*d)*a^9*b^10*c^2*d^9*abs(b) - 127*sqrt(b*d)*a^10*b^9*c*d^ 
10*abs(b) + 15*sqrt(b*d)*a^11*b^8*d^11*abs(b) - 105*sqrt(b*d)*(sqrt(b*d)*s 
qrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*b^17*c^10*abs(b) + 5 
54*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^16*c^9*d*abs(b) - 1013*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^15*c^8*d^2*abs(b) + 248*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^14 
*c^7*d^3*abs(b) + 1886*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^13*c^6*d^4*abs(b) - 3140*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^12*c^5*d^5 
*abs(b) + 1886*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x +...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 2.64 (sec) , antiderivative size = 1458, normalized size of antiderivative = 5.81 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^5} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 192*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**4*d - 32*sqrt(c + d*x)*sqrt(a 
+ b*x)*a**5*c**3*d**2*x + 40*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**2*d**3*x* 
*2 - 60*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c*d**4*x**3 - 192*sqrt(c + d*x)*s 
qrt(a + b*x)*a**4*b*c**5 - 64*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**4*d*x 
+ 24*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**3*d**2*x**2 - 32*sqrt(c + d*x)* 
sqrt(a + b*x)*a**4*b*c**2*d**3*x**3 - 32*sqrt(c + d*x)*sqrt(a + b*x)*a**3* 
b**2*c**5*x + 24*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**4*d*x**2 + 56*sq 
rt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**3*d**2*x**3 + 40*sqrt(c + d*x)*sqrt 
(a + b*x)*a**2*b**3*c**5*x**2 - 32*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c 
**4*d*x**3 - 60*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**5*x**3 - 30*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**5*d**5*x**4 - 6*sqrt(c)*sqrt( 
a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sq 
rt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**4*b*c*d**4*x**4 + 36*sqrt(c)*sqrt( 
a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d)*sq 
rt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a**3*b**2*c**2*d**3*x**4 + 36*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*b**3*c**3*d**2*x**4 - 6*sq 
rt(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**4*c**4*d*x**4 - 30...