\(\int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx\) [253]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{c^{3/2}}+\frac {2 b^{3/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{d^{3/2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.00 (sec) , antiderivative size = 430, normalized size of antiderivative = 3.61 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 (b c-a d) \sqrt {a+b x}}{c d \sqrt {c+d x}}+\frac {2 i a^{3/2} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b c^{5/2}}-\frac {2 a^{3/2} \left (\sqrt {a} \sqrt {d}+i \sqrt {b c-a d}\right ) \sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {b c-2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {c+d x}}{\sqrt {c} \sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{b c^{5/2}}-\frac {4 b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \left (\sqrt {a-\frac {b c}{d}}-\sqrt {a+b x}\right )}\right )}{d^{3/2}} \] Input:

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

Output:

(-2*(b*c - a*d)*Sqrt[a + b*x])/(c*d*Sqrt[c + d*x]) + ((2*I)*a^(3/2)*(I*Sqr 
t[a]*Sqrt[d] + Sqrt[b*c - a*d])*Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*S 
qrt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c 
- a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x] 
))])/(b*c^(5/2)) - (2*a^(3/2)*(Sqrt[a]*Sqrt[d] + I*Sqrt[b*c - a*d])*Sqrt[b 
*c - 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*ArcTan[(Sqrt[b*c - 2*a 
*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[c + d*x])/(Sqrt[c]*Sqrt[d 
]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/(b*c^(5/2)) - (4*b^(3/2)*ArcTanh[ 
(Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*(Sqrt[a - (b*c)/d] - Sqrt[a + b*x]))])/d^ 
(3/2)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 104

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*(b*c - a*d)*Sqrt[a + b*x])/(c*d*Sqrt[c + d*x]) + ((-2*a^(3/2)*d*ArcTan 
h[(Sqrt[c]*Sqrt[a + b*x])/(Sqrt[a]*Sqrt[c + d*x])])/Sqrt[c] + (2*b^(3/2)*c 
*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/Sqrt[d])/(c*d)
 

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 109

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(b*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f 
*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) 
 Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) 
+ c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) 
 + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, 
d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || 
IntegersQ[m, n + p] || IntegersQ[p, m + n])
 

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. \(305\) vs. \(2(91)=182\).

Time = 0.22 (sec) , antiderivative size = 306, normalized size of antiderivative = 2.57

method	result	size
default	\(\frac {\left (-\ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} d^{2} x \sqrt {d b}+\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) b^{2} c d x \sqrt {a c}-\sqrt {d b}\, \ln \left (\frac {a d x +b c x +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}+2 a c}{x}\right ) a^{2} c d +\ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+a d +b c}{2 \sqrt {d b}}\right ) \sqrt {a c}\, b^{2} c^{2}+2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, a d -2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, b c \right ) \sqrt {b x +a}}{\sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {a c}\, \sqrt {x d +c}\, c d}\)	\(306\)

Input:

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

Output:

(-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*d^2* 
x*(d*b)^(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^2*c*d*x*(a*c)^(1/2)-(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+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+2*(( 
b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*(a*c)^(1/2)*a*d-2*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)*(a*c)^(1/2)*b*c)*(b*x+a)^(1/2)/((b*x+a)*(d*x+c))^(1/2)/(d*b) 
^(1/2)/(a*c)^(1/2)/(d*x+c)^(1/2)/c/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 211 vs. \(2 (91) = 182\).

Time = 0.38 (sec) , antiderivative size = 956, normalized size of antiderivative = 8.03 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/2*((b*c*d*x + b*c^2)*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) + (a*d^2*x + a*c*d)*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*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/(c*d^2*x + c^2*d), -1/2*(2*(b*c 
*d*x + b*c^2)*sqrt(-b/d)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sq 
rt(d*x + c)*sqrt(-b/d)/(b^2*d*x^2 + a*b*c + (b^2*c + a*b*d)*x)) - (a*d^2*x 
 + a*c*d)*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*(b*c - a*d)*sqrt(b*x + a)*sqrt(d*x + c))/ 
(c*d^2*x + c^2*d), 1/2*(2*(a*d^2*x + a*c*d)*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*c*d*x + b*c^2)*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*(b*c - a*d)*sqrt(b* 
x + a)*sqrt(d*x + c))/(c*d^2*x + c^2*d), ((a*d^2*x + a*c*d)*sqrt(-a/c)*arc 
tan(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*c*d*x + b*c^2)*sqrt(-b/d)*arcta 
n(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-b/d)/(b^2...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x+a)^(3/2)/x/(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. 205 vs. \(2 (91) = 182\).

Time = 0.19 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=-\frac {2 \, \sqrt {b d} a^{2} b \arctan \left (-\frac {b^{2} c + a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}}{2 \, \sqrt {-a b c d} b}\right )}{\sqrt {-a b c d} c {\left | b \right |}} - \frac {\sqrt {b d} b^{2} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{d^{2} {\left | b \right |}} - \frac {2 \, {\left (b^{3} c {\left | b \right |} - a b^{2} d {\left | b \right |}\right )} \sqrt {b x + a}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} b^{2} c d} \] Input:

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

Output:

-2*sqrt(b*d)*a^2*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)*c*abs(b)) - sqrt(b*d)*b^2*log((sqrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + ( 
b*x + a)*b*d - a*b*d))^2)/(d^2*abs(b)) - 2*(b^3*c*abs(b) - a*b^2*d*abs(b)) 
*sqrt(b*x + a)/(sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*b^2*c*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 472, normalized size of antiderivative = 3.97 \[ \int \frac {(a+b x)^{3/2}}{x (c+d x)^{3/2}} \, dx=\frac {2 \sqrt {d x +c}\, \sqrt {b x +a}\, a c \,d^{2}-2 \sqrt {d x +c}\, \sqrt {b x +a}\, b \,c^{2} d +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a c \,d^{2}+\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,d^{3} x +\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a c \,d^{2}+\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+a d +b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}\right ) a \,d^{3} x -\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a c \,d^{2}-\sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {b}\, \sqrt {d x +c}\, \sqrt {b x +a}+2 \sqrt {d}\, \sqrt {c}\, \sqrt {b}\, \sqrt {a}+2 b d x \right ) a \,d^{3} x +2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b \,c^{3}+2 \sqrt {d}\, \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {d}\, \sqrt {b x +a}+\sqrt {b}\, \sqrt {d x +c}}{\sqrt {a d -b c}}\right ) b \,c^{2} d x +2 \sqrt {d}\, \sqrt {b}\, a \,c^{2} d +2 \sqrt {d}\, \sqrt {b}\, a c \,d^{2} x -2 \sqrt {d}\, \sqrt {b}\, b \,c^{3}-2 \sqrt {d}\, \sqrt {b}\, b \,c^{2} d x}{c^{2} d^{2} \left (d x +c \right )} \] Input:

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

Output:

(2*sqrt(c + d*x)*sqrt(a + b*x)*a*c*d**2 - 2*sqrt(c + d*x)*sqrt(a + b*x)*b* 
c**2*d + 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*c*d**2 + sqrt 
(c)*sqrt(a)*log( - sqrt(2*sqrt(d)*sqrt(c)*sqrt(b)*sqrt(a) + a*d + b*c) + s 
qrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))*a*d**3*x + sqrt(c)*sqrt(a)*l 
og(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*c*d**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)*s 
qrt(c + d*x))*a*d**3*x - 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*c*d**2 - 
 sqrt(c)*sqrt(a)*log(2*sqrt(d)*sqrt(b)*sqrt(c + d*x)*sqrt(a + b*x) + 2*sqr 
t(d)*sqrt(c)*sqrt(b)*sqrt(a) + 2*b*d*x)*a*d**3*x + 2*sqrt(d)*sqrt(b)*log(( 
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b*c**3 + 2 
*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt( 
a*d - b*c))*b*c**2*d*x + 2*sqrt(d)*sqrt(b)*a*c**2*d + 2*sqrt(d)*sqrt(b)*a* 
c*d**2*x - 2*sqrt(d)*sqrt(b)*b*c**3 - 2*sqrt(d)*sqrt(b)*b*c**2*d*x)/(c**2* 
d**2*(c + d*x))