\(\int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx\) [172]

Optimal result

Integrand size = 22, antiderivative size = 340 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}-\frac {(b c+a d) \sqrt {a+b x} \sqrt {c+d x}}{40 a c x^4}-\frac {\left (16 a b c d-7 (b c+a d)^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{240 a^2 c^2 x^3}-\frac {(b c+a d) \left (35 b^2 c^2-46 a b c d+35 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{960 a^3 c^3 x^2}+\frac {\left (105 b^4 c^4-40 a b^3 c^3 d-34 a^2 b^2 c^2 d^2-40 a^3 b c d^3+105 a^4 d^4\right ) \sqrt {a+b x} \sqrt {c+d x}}{1920 a^4 c^4 x}-\frac {(b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b c d+7 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{128 a^{9/2} c^{9/2}} \] Output:

-1/5*(b*x+a)^(1/2)*(d*x+c)^(1/2)/x^5-1/40*(a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^ 
(1/2)/a/c/x^4-1/240*(16*a*b*c*d-7*(a*d+b*c)^2)*(b*x+a)^(1/2)*(d*x+c)^(1/2) 
/a^2/c^2/x^3-1/960*(a*d+b*c)*(35*a^2*d^2-46*a*b*c*d+35*b^2*c^2)*(b*x+a)^(1 
/2)*(d*x+c)^(1/2)/a^3/c^3/x^2+1/1920*(105*a^4*d^4-40*a^3*b*c*d^3-34*a^2*b^ 
2*c^2*d^2-40*a*b^3*c^3*d+105*b^4*c^4)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^4/ 
x-1/128*(-a*d+b*c)^2*(a*d+b*c)*(7*a^2*d^2+2*a*b*c*d+7*b^2*c^2)*arctanh(c^( 
1/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))/a^(9/2)/c^(9/2)
 

Mathematica [A] (verified)

Time = 0.73 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (-105 b^4 c^4 x^4+10 a b^3 c^3 x^3 (7 c+4 d x)-2 a^2 b^2 c^2 x^2 \left (28 c^2+11 c d x-17 d^2 x^2\right )+2 a^3 b c x \left (24 c^3+8 c^2 d x-11 c d^2 x^2+20 d^3 x^3\right )+a^4 \left (384 c^4+48 c^3 d x-56 c^2 d^2 x^2+70 c d^3 x^3-105 d^4 x^4\right )\right )}{1920 a^4 c^4 x^5}-\frac {(b c-a d)^2 \left (7 b^3 c^3+9 a b^2 c^2 d+9 a^2 b c d^2+7 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )}{128 a^{9/2} c^{9/2}} \] Input:

Integrate[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]
 

Output:

-1/1920*(Sqrt[a + b*x]*Sqrt[c + d*x]*(-105*b^4*c^4*x^4 + 10*a*b^3*c^3*x^3* 
(7*c + 4*d*x) - 2*a^2*b^2*c^2*x^2*(28*c^2 + 11*c*d*x - 17*d^2*x^2) + 2*a^3 
*b*c*x*(24*c^3 + 8*c^2*d*x - 11*c*d^2*x^2 + 20*d^3*x^3) + a^4*(384*c^4 + 4 
8*c^3*d*x - 56*c^2*d^2*x^2 + 70*c*d^3*x^3 - 105*d^4*x^4)))/(a^4*c^4*x^5) - 
 ((b*c - a*d)^2*(7*b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + 7*a^3*d^3)*Ar 
cTanh[(Sqrt[a]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[a + b*x])])/(128*a^(9/2)*c^(9/ 
2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {108, 27, 168, 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.

\(\displaystyle \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx\)

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \int \frac {b c+a d+2 b d x}{x^5 \sqrt {a+b x} \sqrt {c+d x}}dx-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (-\frac {\int \frac {7 b^2 c^2-2 a b d c+7 a^2 d^2+6 b d (b c+a d) x}{2 x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (-\frac {\int \frac {7 b^2 c^2-2 a b d c+7 a^2 d^2+6 b d (b c+a d) x}{x^4 \sqrt {a+b x} \sqrt {c+d x}}dx}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {\int \frac {(b c+a d) \left (35 b^2 c^2-46 a b d c+35 a^2 d^2\right )+4 b d \left (7 b^2 c^2-2 a b d c+7 a^2 d^2\right ) x}{2 x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{3 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {\int \frac {(b c+a d) \left (35 b^2 c^2-46 a b d c+35 a^2 d^2\right )+4 b d \left (7 b^2 c^2-2 a b d c+7 a^2 d^2\right ) x}{x^3 \sqrt {a+b x} \sqrt {c+d x}}dx}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {-\frac {\int \frac {105 b^4 c^4-40 a b^3 d c^3-34 a^2 b^2 d^2 c^2-40 a^3 b d^3 c+105 a^4 d^4+2 b d (b c+a d) \left (35 b^2 c^2-46 a b d c+35 a^2 d^2\right ) x}{2 x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {-\frac {\int \frac {105 b^4 c^4-40 a b^3 d c^3-34 a^2 b^2 d^2 c^2-40 a^3 b d^3 c+105 a^4 d^4+2 b d (b c+a d) \left (35 b^2 c^2-46 a b d c+35 a^2 d^2\right ) x}{x^2 \sqrt {a+b x} \sqrt {c+d x}}dx}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {-\frac {-\frac {\int \frac {15 (b c-a d)^2 (b c+a d) \left (7 b^2 c^2+2 a b d c+7 a^2 d^2\right )}{2 x \sqrt {a+b x} \sqrt {c+d x}}dx}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {-\frac {-\frac {15 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{x \sqrt {a+b x} \sqrt {c+d x}}dx}{2 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {-\frac {-\frac {15 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) (b c-a d)^2 \int \frac {1}{\frac {c (a+b x)}{c+d x}-a}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{a c x}}{4 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{2 a c x^2}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{10} \left (-\frac {-\frac {-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c) \left (35 a^2 d^2-46 a b c d+35 b^2 c^2\right )}{2 a c x^2}-\frac {\frac {15 (b c-a d)^2 (a d+b c) \left (7 a^2 d^2+2 a b c d+7 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 a^4 d^4-40 a^3 b c d^3-34 a^2 b^2 c^2 d^2-40 a b^3 c^3 d+105 b^4 c^4\right )}{a c x}}{4 a c}}{6 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (\frac {7 b^2 c}{a}+\frac {7 a d^2}{c}-2 b d\right )}{3 x^3}}{8 a c}-\frac {\sqrt {a+b x} \sqrt {c+d x} (a d+b c)}{4 a c x^4}\right )-\frac {\sqrt {a+b x} \sqrt {c+d x}}{5 x^5}\)

Input:

Int[(Sqrt[a + b*x]*Sqrt[c + d*x])/x^6,x]
 

Output:

-1/5*(Sqrt[a + b*x]*Sqrt[c + d*x])/x^5 + (-1/4*((b*c + a*d)*Sqrt[a + b*x]* 
Sqrt[c + d*x])/(a*c*x^4) - (-1/3*(((7*b^2*c)/a - 2*b*d + (7*a*d^2)/c)*Sqrt 
[a + b*x]*Sqrt[c + d*x])/x^3 - (-1/2*((b*c + a*d)*(35*b^2*c^2 - 46*a*b*c*d 
 + 35*a^2*d^2)*Sqrt[a + b*x]*Sqrt[c + d*x])/(a*c*x^2) - (-(((105*b^4*c^4 - 
 40*a*b^3*c^3*d - 34*a^2*b^2*c^2*d^2 - 40*a^3*b*c*d^3 + 105*a^4*d^4)*Sqrt[ 
a + b*x]*Sqrt[c + d*x])/(a*c*x)) + (15*(b*c - a*d)^2*(b*c + a*d)*(7*b^2*c^ 
2 + 2*a*b*c*d + 7*a^2*d^2)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c 
 + d*x])])/(a^(3/2)*c^(3/2)))/(4*a*c))/(6*a*c))/(8*a*c))/10
 

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. \(812\) vs. \(2(296)=592\).

Time = 0.22 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.39

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \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^{5} d^{5} x^{5}-75 \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} b c \,d^{4} x^{5}-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} b^{2} c^{2} d^{3} x^{5}-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^{3} c^{3} d^{2} x^{5}-75 \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^{4} c^{4} d \,x^{5}+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 ) b^{5} c^{5} x^{5}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{4} d^{4} x^{4}+80 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{3} b c \,d^{3} x^{4}+68 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{2} b^{2} c^{2} d^{2} x^{4}+80 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a \,b^{3} c^{3} d \,x^{4}-210 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{4} c^{4} x^{4}+140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c \,d^{3} x^{3}-44 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{2} d^{2} x^{3}-44 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{3} d \,x^{3}+140 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a \,b^{3} c^{4} x^{3}-112 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c^{2} d^{2} x^{2}+32 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{3} d \,x^{2}-112 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{2} b^{2} c^{4} x^{2}+96 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{4} c^{3} d x +96 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {a c}\, a^{3} b \,c^{4} x +768 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a^{4} c^{4} \sqrt {a c}\right )}{3840 a^{4} c^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x^{5} \sqrt {a c}}\)	\(813\)

Input:

int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-1/3840*(b*x+a)^(1/2)*(d*x+c)^(1/2)/a^4/c^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^5*d^5*x^5-75*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*b*c*d^4*x^5-30*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*b^2*c^2*d^3*x^5-3 
0*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^3* 
c^3*d^2*x^5-75*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^4*c^4*d*x^5+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)*b^5*c^5*x^5-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^4*d^4 
*x^4+80*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a^3*b*c*d^3*x^4+68*(a*c)^(1/2) 
*((b*x+a)*(d*x+c))^(1/2)*a^2*b^2*c^2*d^2*x^4+80*(a*c)^(1/2)*((b*x+a)*(d*x+ 
c))^(1/2)*a*b^3*c^3*d*x^4-210*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b^4*c^4* 
x^4+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c*d^3*x^3-44*((b*x+a)*(d*x 
+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^2*d^2*x^3-44*((b*x+a)*(d*x+c))^(1/2)*(a*c)^ 
(1/2)*a^2*b^2*c^3*d*x^3+140*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a*b^3*c^4* 
x^3-112*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^2*d^2*x^2+32*((b*x+a)*(d 
*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^3*d*x^2-112*((b*x+a)*(d*x+c))^(1/2)*(a*c) 
^(1/2)*a^2*b^2*c^4*x^2+96*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^4*c^3*d*x+ 
96*((b*x+a)*(d*x+c))^(1/2)*(a*c)^(1/2)*a^3*b*c^4*x+768*((b*x+a)*(d*x+c))^( 
1/2)*a^4*c^4*(a*c)^(1/2))/((b*x+a)*(d*x+c))^(1/2)/x^5/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 2.94 (sec) , antiderivative size = 730, normalized size of antiderivative = 2.15 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=\left [\frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {a c} x^{5} \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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d - 34 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 11 \, a^{3} b^{2} c^{4} d - 11 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{7680 \, a^{5} c^{5} x^{5}}, \frac {15 \, {\left (7 \, b^{5} c^{5} - 5 \, a b^{4} c^{4} d - 2 \, a^{2} b^{3} c^{3} d^{2} - 2 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} + 7 \, a^{5} d^{5}\right )} \sqrt {-a c} x^{5} \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 (384 \, a^{5} c^{5} - {\left (105 \, a b^{4} c^{5} - 40 \, a^{2} b^{3} c^{4} d - 34 \, a^{3} b^{2} c^{3} d^{2} - 40 \, a^{4} b c^{2} d^{3} + 105 \, a^{5} c d^{4}\right )} x^{4} + 2 \, {\left (35 \, a^{2} b^{3} c^{5} - 11 \, a^{3} b^{2} c^{4} d - 11 \, a^{4} b c^{3} d^{2} + 35 \, a^{5} c^{2} d^{3}\right )} x^{3} - 8 \, {\left (7 \, a^{3} b^{2} c^{5} - 2 \, a^{4} b c^{4} d + 7 \, a^{5} c^{3} d^{2}\right )} x^{2} + 48 \, {\left (a^{4} b c^{5} + a^{5} c^{4} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3840 \, a^{5} c^{5} x^{5}}\right ] \] Input:

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x, algorithm="fricas")
 

Output:

[1/7680*(15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b^2*c^2 
*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*sqrt(a*c)*x^5*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*(384*a^5*c^5 - (105 
*a*b^4*c^5 - 40*a^2*b^3*c^4*d - 34*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 10 
5*a^5*c*d^4)*x^4 + 2*(35*a^2*b^3*c^5 - 11*a^3*b^2*c^4*d - 11*a^4*b*c^3*d^2 
 + 35*a^5*c^2*d^3)*x^3 - 8*(7*a^3*b^2*c^5 - 2*a^4*b*c^4*d + 7*a^5*c^3*d^2) 
*x^2 + 48*(a^4*b*c^5 + a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^5 
*x^5), 1/3840*(15*(7*b^5*c^5 - 5*a*b^4*c^4*d - 2*a^2*b^3*c^3*d^2 - 2*a^3*b 
^2*c^2*d^3 - 5*a^4*b*c*d^4 + 7*a^5*d^5)*sqrt(-a*c)*x^5*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*(384*a^5*c^5 - (105*a*b^4*c^5 - 40*a^2*b 
^3*c^4*d - 34*a^3*b^2*c^3*d^2 - 40*a^4*b*c^2*d^3 + 105*a^5*c*d^4)*x^4 + 2* 
(35*a^2*b^3*c^5 - 11*a^3*b^2*c^4*d - 11*a^4*b*c^3*d^2 + 35*a^5*c^2*d^3)*x^ 
3 - 8*(7*a^3*b^2*c^5 - 2*a^4*b*c^4*d + 7*a^5*c^3*d^2)*x^2 + 48*(a^4*b*c^5 
+ a^5*c^4*d)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(a^5*c^5*x^5)]
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=\int \frac {\sqrt {a + b x} \sqrt {c + d x}}{x^{6}}\, dx \] Input:

integrate((b*x+a)**(1/2)*(d*x+c)**(1/2)/x**6,x)
 

Output:

Integral(sqrt(a + b*x)*sqrt(c + d*x)/x**6, x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,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. 5923 vs. \(2 (296) = 592\).

Time = 9.39 (sec) , antiderivative size = 5923, normalized size of antiderivative = 17.42 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x, algorithm="giac")
 

Output:

-1/1920*b^5*(15*(7*sqrt(b*d)*b^5*c^5*abs(b) - 5*sqrt(b*d)*a*b^4*c^4*d*abs( 
b) - 2*sqrt(b*d)*a^2*b^3*c^3*d^2*abs(b) - 2*sqrt(b*d)*a^3*b^2*c^2*d^3*abs( 
b) - 5*sqrt(b*d)*a^4*b*c*d^4*abs(b) + 7*sqrt(b*d)*a^5*d^5*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^4*b^6*c^4) - 2*(105*sq 
rt(b*d)*b^23*c^14*abs(b) - 1090*sqrt(b*d)*a*b^22*c^13*d*abs(b) + 5091*sqrt 
(b*d)*a^2*b^21*c^12*d^2*abs(b) - 14100*sqrt(b*d)*a^3*b^20*c^11*d^3*abs(b) 
+ 25825*sqrt(b*d)*a^4*b^19*c^10*d^4*abs(b) - 33630*sqrt(b*d)*a^5*b^18*c^9* 
d^5*abs(b) + 34515*sqrt(b*d)*a^6*b^17*c^8*d^6*abs(b) - 33432*sqrt(b*d)*a^7 
*b^16*c^7*d^7*abs(b) + 34515*sqrt(b*d)*a^8*b^15*c^6*d^8*abs(b) - 33630*sqr 
t(b*d)*a^9*b^14*c^5*d^9*abs(b) + 25825*sqrt(b*d)*a^10*b^13*c^4*d^10*abs(b) 
 - 14100*sqrt(b*d)*a^11*b^12*c^3*d^11*abs(b) + 5091*sqrt(b*d)*a^12*b^11*c^ 
2*d^12*abs(b) - 1090*sqrt(b*d)*a^13*b^10*c*d^13*abs(b) + 105*sqrt(b*d)*a^1 
4*b^9*d^14*abs(b) - 945*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + 
(b*x + a)*b*d - a*b*d))^2*b^21*c^13*abs(b) + 6835*sqrt(b*d)*(sqrt(b*d)*sqr 
t(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2*a*b^20*c^12*d*abs(b) - 
 19950*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^19*c^11*d^2*abs(b) + 27650*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^18*c^10*d^3*abs(b) - 120 
75*sqrt(b*d)*(sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx=\text {Hanged} \] Input:

int(((a + b*x)^(1/2)*(c + d*x)^(1/2))/x^6,x)
 

Output:

\text{Hanged}
 

Reduce [B] (verification not implemented)

Time = 198.36 (sec) , antiderivative size = 1477, normalized size of antiderivative = 4.34 \[ \int \frac {\sqrt {a+b x} \sqrt {c+d x}}{x^6} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(1/2)*(d*x+c)^(1/2)/x^6,x)
 

Output:

( - 768*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**5 - 96*sqrt(c + d*x)*sqrt(a + 
b*x)*a**5*c**4*d*x + 112*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**3*d**2*x**2 - 
 140*sqrt(c + d*x)*sqrt(a + b*x)*a**5*c**2*d**3*x**3 + 210*sqrt(c + d*x)*s 
qrt(a + b*x)*a**5*c*d**4*x**4 - 96*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**5 
*x - 32*sqrt(c + d*x)*sqrt(a + b*x)*a**4*b*c**4*d*x**2 + 44*sqrt(c + d*x)* 
sqrt(a + b*x)*a**4*b*c**3*d**2*x**3 - 80*sqrt(c + d*x)*sqrt(a + b*x)*a**4* 
b*c**2*d**3*x**4 + 112*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**5*x**2 + 4 
4*sqrt(c + d*x)*sqrt(a + b*x)*a**3*b**2*c**4*d*x**3 - 68*sqrt(c + d*x)*sqr 
t(a + b*x)*a**3*b**2*c**3*d**2*x**4 - 140*sqrt(c + d*x)*sqrt(a + b*x)*a**2 
*b**3*c**5*x**3 - 80*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b**3*c**4*d*x**4 + 2 
10*sqrt(c + d*x)*sqrt(a + b*x)*a*b**4*c**5*x**4 + 105*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**5 - 75*sqrt(c)*sqrt(a)*log( - s 
qrt(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*b*c*d**4*x**5 - 30*sqrt(c)*sqrt(a)*log( - s 
qrt(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**2*c**2*d**3*x**5 - 30*sqrt(c)*sqrt(a)*lo 
g( - 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**5 - 75*sqrt(c)*sqrt 
(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + sqrt(d...