\(\int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx\) [385]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {2 a^3}{b^3 (b c-a d) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 \left (b^3 c^3+a^3 d^3\right ) \sqrt {a+b x}}{b^3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b^2 d^2}-\frac {3 (b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{5/2} d^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.01 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {3 a^3 d^2 (c+d x)+b^3 c^2 x (3 c+d x)+a b^2 c \left (3 c^2-c d x-2 d^2 x^2\right )+a^2 b d \left (-2 c^2-c d x+d^2 x^2\right )}{b^2 d^2 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}-\frac {3 (b c+a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {a+b x}}\right )}{b^{5/2} d^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {109, 27, 160, 66, 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.

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

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 160

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(2*a*x^2)/(b*(b*c - a*d)*Sqrt[a + b*x]*Sqrt[c + d*x]) - (-((Sqrt[a + b*x]* 
(c*(3*b^2*c^2 - 2*a*b*c*d + 3*a^2*d^2) + d*(b*c - 3*a*d)*(b*c - a*d)*x))/( 
b*d^2*(b*c - a*d)*Sqrt[c + d*x])) - (3*(a^2/b - (b*c^2)/d^2)*ArcTanh[(Sqrt 
[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(Sqrt[b]*Sqrt[d]))/(b*(b*c - 
a*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 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 160

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g 
+ e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d*(b*c - a*d)*(m + 1))), x] + Simp[(a*d*f*h*m + b*(d* 
(f*g + e*h) - c*f*h*(m + 2)))/(b^2*d)   Int[(a + b*x)^(m + 1)*(c + d*x)^n, 
x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m + n + 2, 0] && 
NeQ[m, -1] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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. \(905\) vs. \(2(138)=276\).

Time = 0.28 (sec) , antiderivative size = 906, normalized size of antiderivative = 5.59

method	result	size
default	\(-\frac {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 ) a^{3} b \,d^{4} x^{2}-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 ) a^{2} b^{2} c \,d^{3} x^{2}-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 ) a \,b^{3} c^{2} d^{2} x^{2}+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 ) b^{4} c^{3} d \,x^{2}+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 ) a^{4} d^{4} x -6 \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^{2} c^{2} d^{2} 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 ) b^{4} c^{4} x -2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b \,d^{3} x^{2}+4 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{2} c \,d^{2} x^{2}-2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, b^{3} c^{2} d \,x^{2}+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 ) a^{4} c \,d^{3}-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 ) a^{3} b \,c^{2} d^{2}-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 ) a^{2} b^{2} c^{3} d +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 ) a \,b^{3} c^{4}-6 a^{3} d^{3} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b c \,d^{2} x +2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a \,b^{2} c^{2} d x -6 b^{3} c^{3} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-6 a^{3} c \,d^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+4 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} b \,c^{2} d -6 a \,b^{2} c^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}}{2 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \left (a d -b c \right )^{2} \sqrt {d b}\, \sqrt {b x +a}\, \sqrt {x d +c}\, b^{2} d^{2}}\)	\(906\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (138) = 276\).

Time = 0.20 (sec) , antiderivative size = 910, normalized size of antiderivative = 5.62 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/4*(3*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3 + (b^4*c^3* 
d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + (b^4*c^4 - 2*a^2*b^2* 
c^2*d^2 + a^4*d^4)*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*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*(3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b* 
c*d^3 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*b^4*c^3*d - a 
*b^3*c^2*d^2 - a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c) 
)/(a*b^5*c^3*d^3 - 2*a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + (b^6*c^2*d^4 - 2*a* 
b^5*c*d^5 + a^2*b^4*d^6)*x^2 + (b^6*c^3*d^3 - a*b^5*c^2*d^4 - a^2*b^4*c*d^ 
5 + a^3*b^3*d^6)*x), 1/2*(3*(a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a 
^4*c*d^3 + (b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x^2 + ( 
b^4*c^4 - 2*a^2*b^2*c^2*d^2 + a^4*d^4)*x)*sqrt(-b*d)*arctan(1/2*(2*b*d*x + 
 b*c + a*d)*sqrt(-b*d)*sqrt(b*x + a)*sqrt(d*x + c)/(b^2*d^2*x^2 + a*b*c*d 
+ (b^2*c*d + a*b*d^2)*x)) + 2*(3*a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + 3*a^3*b 
*c*d^3 + (b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*b^4*c^3*d - 
a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + 3*a^3*b*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c 
))/(a*b^5*c^3*d^3 - 2*a^2*b^4*c^2*d^4 + a^3*b^3*c*d^5 + (b^6*c^2*d^4 - 2*a 
*b^5*c*d^5 + a^2*b^4*d^6)*x^2 + (b^6*c^3*d^3 - a*b^5*c^2*d^4 - a^2*b^4*c*d 
^5 + a^3*b^3*d^6)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3/(b*x+a)^(3/2)/(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. 340 vs. \(2 (138) = 276\).

Time = 0.22 (sec) , antiderivative size = 340, normalized size of antiderivative = 2.10 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx=\frac {4 \, a^{3} d}{{\left (\sqrt {b d} b c {\left | b \right |} - \sqrt {b d} a d {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}} + \frac {\sqrt {b x + a} {\left (\frac {{\left (b^{6} c^{2} d^{2} - 2 \, a b^{5} c d^{3} + a^{2} b^{4} d^{4}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{6} c d^{4} {\left | b \right |} + a^{2} b^{5} d^{5} {\left | b \right |}} + \frac {3 \, b^{7} c^{3} d - 3 \, a b^{6} c^{2} d^{2} + 3 \, a^{2} b^{5} c d^{3} - a^{3} b^{4} d^{4}}{b^{7} c^{2} d^{3} {\left | b \right |} - 2 \, a b^{6} c d^{4} {\left | b \right |} + a^{2} b^{5} d^{5} {\left | b \right |}}\right )}}{\sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {3 \, {\left (b c + a d\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{2 \, \sqrt {b d} b d^{2} {\left | b \right |}} \] Input:

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

Output:

4*a^3*d/((sqrt(b*d)*b*c*abs(b) - sqrt(b*d)*a*d*abs(b))*(b^2*c - a*b*d - (s 
qrt(b*d)*sqrt(b*x + a) - sqrt(b^2*c + (b*x + a)*b*d - a*b*d))^2)) + sqrt(b 
*x + a)*((b^6*c^2*d^2 - 2*a*b^5*c*d^3 + a^2*b^4*d^4)*(b*x + a)/(b^7*c^2*d^ 
3*abs(b) - 2*a*b^6*c*d^4*abs(b) + a^2*b^5*d^5*abs(b)) + (3*b^7*c^3*d - 3*a 
*b^6*c^2*d^2 + 3*a^2*b^5*c*d^3 - a^3*b^4*d^4)/(b^7*c^2*d^3*abs(b) - 2*a*b^ 
6*c*d^4*abs(b) + a^2*b^5*d^5*abs(b)))/sqrt(b^2*c + (b*x + a)*b*d - a*b*d) 
+ 3/2*(b*c + a*d)*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*d^2*abs(b))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 741, normalized size of antiderivative = 4.57 \[ \int \frac {x^3}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*s 
qrt(c + d*x))/sqrt(a*d - b*c))*a**3*c*d**3 - 3*sqrt(d)*sqrt(b)*sqrt(a + b* 
x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a* 
*3*d**4*x + 3*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + s 
qrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b*c**2*d**2 + 3*sqrt(d)*sqrt(b 
)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a 
*d - b*c))*a**2*b*c*d**3*x + 3*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)* 
sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**2*c**3*d + 3* 
sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c 
+ d*x))/sqrt(a*d - b*c))*a*b**2*c**2*d**2*x - 3*sqrt(d)*sqrt(b)*sqrt(a + b 
*x)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b 
**3*c**4 - 3*sqrt(d)*sqrt(b)*sqrt(a + b*x)*log((sqrt(d)*sqrt(a + b*x) + sq 
rt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*b**3*c**3*d*x + 2*sqrt(d)*sqrt(b)*sq 
rt(a + b*x)*a**3*c*d**3 + 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)*a**3*d**4*x + 2* 
sqrt(d)*sqrt(b)*sqrt(a + b*x)*b**3*c**4 + 2*sqrt(d)*sqrt(b)*sqrt(a + b*x)* 
b**3*c**3*d*x + 3*sqrt(c + d*x)*a**3*b*c*d**3 + 3*sqrt(c + d*x)*a**3*b*d** 
4*x - 2*sqrt(c + d*x)*a**2*b**2*c**2*d**2 - sqrt(c + d*x)*a**2*b**2*c*d**3 
*x + sqrt(c + d*x)*a**2*b**2*d**4*x**2 + 3*sqrt(c + d*x)*a*b**3*c**3*d - s 
qrt(c + d*x)*a*b**3*c**2*d**2*x - 2*sqrt(c + d*x)*a*b**3*c*d**3*x**2 + 3*s 
qrt(c + d*x)*b**4*c**3*d*x + sqrt(c + d*x)*b**4*c**2*d**2*x**2)/(sqrt(a...