\(\int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx\) [227]

Optimal result

Integrand size = 22, antiderivative size = 191 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {3}{4} (3 b c+a d) \sqrt {a+b x} \sqrt {c+d x}+\frac {3}{2} d (a+b x)^{3/2} \sqrt {c+d x}-\frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x}-3 \sqrt {a} \sqrt {c} (b c+a d) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 \sqrt {b} \sqrt {d}} \] Output:

3/4*(a*d+3*b*c)*(b*x+a)^(1/2)*(d*x+c)^(1/2)+3/2*d*(b*x+a)^(3/2)*(d*x+c)^(1 
/2)-(b*x+a)^(3/2)*(d*x+c)^(3/2)/x-3*a^(1/2)*c^(1/2)*(a*d+b*c)*arctanh(c^(1 
/2)*(b*x+a)^(1/2)/a^(1/2)/(d*x+c)^(1/2))+3/4*(a^2*d^2+6*a*b*c*d+b^2*c^2)*a 
rctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^(1/2)/d^(1/2)
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=-3 \sqrt {a} \sqrt {c} (b c+a d) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c+d x}}{\sqrt {c} \sqrt {a+b x}}\right )+\frac {1}{4} \left (\frac {\sqrt {a+b x} \sqrt {c+d x} (b x (5 c+2 d x)+a (-4 c+5 d x))}{x}+\frac {3 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{\sqrt {b} \sqrt {d}}\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.04, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(((a + b*x)^(3/2)*(c + d*x)^(3/2))/x) + (3*(b*Sqrt[a + b*x]*(c + d*x)^(3/ 
2) + ((b*c + 3*a*d)*Sqrt[a + b*x]*Sqrt[c + d*x] + (-8*Sqrt[a]*Sqrt[c]*(b*c 
 + a*d)*ArcTanh[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])] + (2*(b^2 
*c^2 + 6*a*b*c*d + a^2*d^2)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[ 
c + d*x])])/(Sqrt[b]*Sqrt[d]))/2)/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. \(418\) vs. \(2(149)=298\).

Time = 0.22 (sec) , antiderivative size = 419, normalized size of antiderivative = 2.19

method	result	size
default	\(\frac {\sqrt {b x +a}\, \sqrt {x d +c}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) \sqrt {a c}\, a^{2} d^{2} x +18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) \sqrt {a c}\, a b c d x +3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) \sqrt {a c}\, b^{2} c^{2} x -12 \sqrt {d b}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} c d x -12 \sqrt {d b}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a b \,c^{2} x +4 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b d \,x^{2}+10 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a d x +10 \sqrt {d b}\, \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b c x -8 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a c \sqrt {d b}\, \sqrt {a c}\right )}{8 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, x \sqrt {d b}\, \sqrt {a c}}\)	\(419\)

Input:

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

Output:

1/8*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(3*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*c)^(1/2)*a^2*d^2*x+18*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*c)^(1/2 
)*a*b*c*d*x+3*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*c)^(1/2)*b^2*c^2*x-12*(d*b)^(1/2)*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*c*d*x-12*(d*b)^(1/2)*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*c^2*x+4*(d*b 
)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*b*d*x^2+10*(d*b)^(1/2)*(a*c)^( 
1/2)*((b*x+a)*(d*x+c))^(1/2)*a*d*x+10*(d*b)^(1/2)*(a*c)^(1/2)*((b*x+a)*(d* 
x+c))^(1/2)*b*c*x-8*((b*x+a)*(d*x+c))^(1/2)*a*c*(d*b)^(1/2)*(a*c)^(1/2))/( 
(b*x+a)*(d*x+c))^(1/2)/x/(d*b)^(1/2)/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 1073, normalized size of antiderivative = 5.62 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/16*(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)*x*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*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x 
+ a)*sqrt(d*x + c) + 8*(b^2*c*d + a*b*d^2)*x) + 12*(b^2*c*d + a*b*d^2)*sqr 
t(a*c)*x*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*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d + a*b*d^2)*x)*sqrt 
(b*x + a)*sqrt(d*x + c))/(b*d*x), -1/8*(3*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)* 
sqrt(-b*d)*x*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqr 
t(d*x + c)/(b^2*d^2*x^2 + a*b*c*d + (b^2*c*d + a*b*d^2)*x)) - 6*(b^2*c*d + 
 a*b*d^2)*sqrt(a*c)*x*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) - 2*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d + a*b 
*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), 1/16*(24*(b^2*c*d + a*b*d^2 
)*sqrt(-a*c)*x*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)) + 3*(b^2*c 
^2 + 6*a*b*c*d + a^2*d^2)*sqrt(b*d)*x*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*x + b*c + a*d)*sqrt(b*d)*sqrt(b*x + a)*sqrt(d*x + 
 c) + 8*(b^2*c*d + a*b*d^2)*x) + 4*(2*b^2*d^2*x^2 - 4*a*b*c*d + 5*(b^2*c*d 
 + a*b*d^2)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b*d*x), 1/8*(12*(b^2*c*d + a* 
b*d^2)*sqrt(-a*c)*x*arctan(1/2*(2*a*c + (b*c + a*d)*x)*sqrt(-a*c)*sqrt(...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(3/2)*(d*x+c)^(3/2)/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. 559 vs. \(2 (149) = 298\).

Time = 0.44 (sec) , antiderivative size = 559, normalized size of antiderivative = 2.93 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {1}{8} \, {\left (2 \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b^{3}} + \frac {5 \, b^{4} c d^{2} {\left | b \right |} + 3 \, a b^{3} d^{3} {\left | b \right |}}{b^{6} d^{2}}\right )} + \frac {24 \, {\left (a b c^{2} d {\left | b \right |} + a^{2} c d^{2} {\left | b \right |}\right )} \arctan \left (\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} \sqrt {b d} b} - \frac {16 \, {\left (a b^{3} c^{3} d {\left | b \right |} - 2 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + a^{3} b c d^{3} {\left | b \right |} - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b c^{2} d {\left | b \right |} - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} c d^{2} {\left | b \right |}\right )}}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2} - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{2} c - 2 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b d + {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4}\right )} \sqrt {b d}} - \frac {3 \, {\left (b^{2} c^{2} {\left | b \right |} + 6 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{\sqrt {b d} b^{2}}\right )} b \] Input:

integrate((b*x+a)^(3/2)*(d*x+c)^(3/2)/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)*d*ab 
s(b)/b^3 + (5*b^4*c*d^2*abs(b) + 3*a*b^3*d^3*abs(b))/(b^6*d^2)) + 24*(a*b* 
c^2*d*abs(b) + a^2*c*d^2*abs(b))*arctan(1/2*(b^2*c + a*b*d - (sqrt(b*d)*sq 
rt(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*b^3*c^3*d*abs(b) - 2*a^2*b^2*c^2*d^2* 
abs(b) + a^3*b*c*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*b*c^2*d*abs(b) - (sqrt(b*d)*sqrt(b*x + a) - sqrt(b 
^2*c + (b*x + a)*b*d - a*b*d))^2*a^2*c*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)) - 3*(b^2*c^2*abs(b) + 6*a*b*c*d*abs(b) + a^2*d^2* 
abs(b))*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d) 
)^2)/(sqrt(b*d)*b^2))*b
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 22.21 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.74 \[ \int \frac {(a+b x)^{3/2} (c+d x)^{3/2}}{x^2} \, dx=\frac {-4 \sqrt {d x +c}\, \sqrt {b x +a}\, a b c d +5 \sqrt {d x +c}\, \sqrt {b x +a}\, a b \,d^{2} x +5 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} c d x +2 \sqrt {d x +c}\, \sqrt {b x +a}\, b^{2} d^{2} x^{2}+6 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b \,d^{2} x +6 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c d x +6 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a b \,d^{2} x +6 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) b^{2} c d x -6 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a b \,d^{2} x -6 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) b^{2} c d x +3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a^{2} d^{2} x +18 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) a b c d x +3 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b^{2} c^{2} x}{4 b d x} \] Input:

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

Output:

( - 4*sqrt(c + d*x)*sqrt(a + b*x)*a*b*c*d + 5*sqrt(c + d*x)*sqrt(a + b*x)* 
a*b*d**2*x + 5*sqrt(c + d*x)*sqrt(a + b*x)*b**2*c*d*x + 2*sqrt(c + d*x)*sq 
rt(a + b*x)*b**2*d**2*x**2 + 6*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*d**2*x + 6*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))*b* 
*2*c*d*x + 6*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*d**2*x + 6 
*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))*b**2*c*d*x - 6*sqrt(c)*sqr 
t(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*d**2*x - 6*sqrt(c)*sqrt(a)*log(2*sqrt(d)*s 
qrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2 
*b*d*x)*b**2*c*d*x + 3*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b 
)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*d**2*x + 18*sqrt(d)*sqrt(b)*log((sq 
rt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b*c*d*x + 
3*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt 
(a*d - b*c))*b**2*c**2*x)/(4*b*d*x)