\(\int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx\) [357]

Optimal result

Integrand size = 22, antiderivative size = 161 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c^3 \sqrt {a+b x}}{3 d^3 (b c-a d) (c+d x)^{3/2}}+\frac {2 c^2 (7 b c-9 a d) \sqrt {a+b x}}{3 d^3 (b c-a d)^2 \sqrt {c+d x}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b d^3}-\frac {(5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \] Output:

-2/3*c^3*(b*x+a)^(1/2)/d^3/(-a*d+b*c)/(d*x+c)^(3/2)+2/3*c^2*(-9*a*d+7*b*c) 
*(b*x+a)^(1/2)/d^3/(-a*d+b*c)^2/(d*x+c)^(1/2)+(b*x+a)^(1/2)*(d*x+c)^(1/2)/ 
b/d^3-(a*d+5*b*c)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))/b^( 
3/2)/d^(7/2)
 

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.97 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (3 a^2 d^2 (c+d x)^2-2 a b c d \left (11 c^2+15 c d x+3 d^2 x^2\right )+b^2 c^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 b d^3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {(5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \] Input:

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

Output:

(Sqrt[a + b*x]*(3*a^2*d^2*(c + d*x)^2 - 2*a*b*c*d*(11*c^2 + 15*c*d*x + 3*d 
^2*x^2) + b^2*c^2*(15*c^2 + 20*c*d*x + 3*d^2*x^2)))/(3*b*d^3*(b*c - a*d)^2 
*(c + d*x)^(3/2)) - ((5*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b 
]*Sqrt[c + d*x])])/(b^(3/2)*d^(7/2))
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.22, 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.

\(\displaystyle \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx\)

\(\Big \downarrow \) 109

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 160

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

\(\Big \downarrow \) 66

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

(-2*c*x^2*Sqrt[a + b*x])/(3*d*(b*c - a*d)*(c + d*x)^(3/2)) + ((Sqrt[a + b* 
x]*(c*(15*b^2*c^2 - 22*a*b*c*d + 3*a^2*d^2) + d*(5*b*c - 3*a*d)*(b*c - a*d 
)*x))/(b*d^2*(b*c - a*d)*Sqrt[c + d*x]) - (3*(b*c - a*d)*(5*b*c + a*d)*Arc 
Tanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])])/(b^(3/2)*d^(5/2)))/ 
(3*d*(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. \(927\) vs. \(2(133)=266\).

Time = 0.27 (sec) , antiderivative size = 928, normalized size of antiderivative = 5.76

method	result	size
default	\(-\frac {\sqrt {b x +a}\, \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 ) a^{3} d^{5} x^{2}+9 \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 c \,d^{4} x^{2}-27 \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^{2} c^{2} d^{3} x^{2}+15 \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^{3} c^{3} d^{2} x^{2}+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^{3} c \,d^{4} 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 ) a^{2} b \,c^{2} d^{3} x -54 \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^{2} c^{3} d^{2} x +30 \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^{3} c^{4} d x -6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} d^{4} x^{2}+12 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a b c \,d^{3} x^{2}-6 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{2} 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 ) a^{3} c^{2} d^{3}+9 \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 \,c^{3} d^{2}-27 \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^{2} c^{4} d +15 \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^{3} c^{5}-12 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} c \,d^{3} x +60 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a b \,c^{2} d^{2} x -40 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{2} c^{3} d x -6 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, a^{2} c^{2} d^{2}+44 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, a b \,c^{3} d -30 \sqrt {d b}\, \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, b^{2} c^{4}\right )}{6 \sqrt {d b}\, b \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, d^{3} \left (x d +c \right )^{\frac {3}{2}}}\)	\(928\)

Input:

int(x^3/(b*x+a)^(1/2)/(d*x+c)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6*(b*x+a)^(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*d^5*x^2+9*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*c*d^4*x^2-27*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^2*c^2*d^3 
*x^2+15*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^3*c^3*d^2*x^2+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^3*c*d^4*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^2*b*c^2*d^3*x-54*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^2* 
c^3*d^2*x+30*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^3*c^4*d*x-6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*d^4*x 
^2+12*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c*d^3*x^2-6*(d*b)^(1/2)*((b* 
x+a)*(d*x+c))^(1/2)*b^2*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))*a^3*c^2*d^3+9*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*c^3*d^2-27*l 
n(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^2*c^4*d+15*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^3*c^5-12*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*a^2*c*d^3 
*x+60*(d*b)^(1/2)*((b*x+a)*(d*x+c))^(1/2)*a*b*c^2*d^2*x-40*(d*b)^(1/2)*((b 
*x+a)*(d*x+c))^(1/2)*b^2*c^3*d*x-6*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (133) = 266\).

Time = 0.31 (sec) , antiderivative size = 896, normalized size of antiderivative = 5.57 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(x^3/(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="fricas")
 

Output:

[1/12*(3*(5*b^3*c^5 - 9*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (5*b 
^3*c^3*d^2 - 9*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(5*b^3*c^4 
*d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + a^3*c*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*(15*b^3*c^4* 
d - 22*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + 3*(b^3*c^2*d^3 - 2*a*b^2*c*d^4 + 
a^2*b*d^5)*x^2 + 2*(10*b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 3*a^2*b*c*d^4)*x)* 
sqrt(b*x + a)*sqrt(d*x + c))/(b^4*c^4*d^4 - 2*a*b^3*c^3*d^5 + a^2*b^2*c^2* 
d^6 + (b^4*c^2*d^6 - 2*a*b^3*c*d^7 + a^2*b^2*d^8)*x^2 + 2*(b^4*c^3*d^5 - 2 
*a*b^3*c^2*d^6 + a^2*b^2*c*d^7)*x), 1/6*(3*(5*b^3*c^5 - 9*a*b^2*c^4*d + 3* 
a^2*b*c^3*d^2 + a^3*c^2*d^3 + (5*b^3*c^3*d^2 - 9*a*b^2*c^2*d^3 + 3*a^2*b*c 
*d^4 + a^3*d^5)*x^2 + 2*(5*b^3*c^4*d - 9*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + 
 a^3*c*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*(15*b^3*c^4*d - 22*a*b^2*c^3*d^2 + 3*a^2*b*c^2*d^3 + 3*(b^3*c^2*d^3 - 2 
*a*b^2*c*d^4 + a^2*b*d^5)*x^2 + 2*(10*b^3*c^3*d^2 - 15*a*b^2*c^2*d^3 + 3*a 
^2*b*c*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^4*c^4*d^4 - 2*a*b^3*c^3*d^5 
 + a^2*b^2*c^2*d^6 + (b^4*c^2*d^6 - 2*a*b^3*c*d^7 + a^2*b^2*d^8)*x^2 + 2*( 
b^4*c^3*d^5 - 2*a*b^3*c^2*d^6 + a^2*b^2*c*d^7)*x)]
 

Sympy [F]

\[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^{3}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**3/(b*x+a)**(1/2)/(d*x+c)**(5/2),x)
 

Output:

Integral(x**3/(sqrt(a + b*x)*(c + d*x)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^3/(b*x+a)^(1/2)/(d*x+c)^(5/2),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. 373 vs. \(2 (133) = 266\).

Time = 0.19 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.32 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c^{2} d^{4} {\left | b \right |} - 2 \, a b^{5} c d^{5} {\left | b \right |} + a^{2} b^{4} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}} + \frac {2 \, {\left (10 \, b^{7} c^{3} d^{3} {\left | b \right |} - 18 \, a b^{6} c^{2} d^{4} {\left | b \right |} + 9 \, a^{2} b^{5} c d^{5} {\left | b \right |} - 3 \, a^{3} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{4} d^{2} {\left | b \right |} - 14 \, a b^{7} c^{3} d^{3} {\left | b \right |} + 12 \, a^{2} b^{6} c^{2} d^{4} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{5} {\left | b \right |} + a^{4} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b c {\left | b \right |} + a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}} \] Input:

integrate(x^3/(b*x+a)^(1/2)/(d*x+c)^(5/2),x, algorithm="giac")
 

Output:

1/3*((b*x + a)*(3*(b^6*c^2*d^4*abs(b) - 2*a*b^5*c*d^5*abs(b) + a^2*b^4*d^6 
*abs(b))*(b*x + a)/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7) + 2*(10*b^7 
*c^3*d^3*abs(b) - 18*a*b^6*c^2*d^4*abs(b) + 9*a^2*b^5*c*d^5*abs(b) - 3*a^3 
*b^4*d^6*abs(b))/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^2*b^5*d^7)) + 3*(5*b^8*c 
^4*d^2*abs(b) - 14*a*b^7*c^3*d^3*abs(b) + 12*a^2*b^6*c^2*d^4*abs(b) - 4*a^ 
3*b^5*c*d^5*abs(b) + a^4*b^4*d^6*abs(b))/(b^7*c^2*d^5 - 2*a*b^6*c*d^6 + a^ 
2*b^5*d^7))*sqrt(b*x + a)/(b^2*c + (b*x + a)*b*d - a*b*d)^(3/2) + (5*b*c*a 
bs(b) + a*d*abs(b))*log(abs(-sqrt(b*d)*sqrt(b*x + a) + sqrt(b^2*c + (b*x + 
 a)*b*d - a*b*d)))/(sqrt(b*d)*b^2*d^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^3}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \] Input:

int(x^3/((a + b*x)^(1/2)*(c + d*x)^(5/2)),x)
 

Output:

int(x^3/((a + b*x)^(1/2)*(c + d*x)^(5/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 1076, normalized size of antiderivative = 6.68 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx =\text {Too large to display} \] Input:

int(x^3/(b*x+a)^(1/2)/(d*x+c)^(5/2),x)
 

Output:

(6*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b*c**2*d**3 + 12*sqrt(c + d*x)*sqrt(a 
+ b*x)*a**2*b*c*d**4*x + 6*sqrt(c + d*x)*sqrt(a + b*x)*a**2*b*d**5*x**2 - 
44*sqrt(c + d*x)*sqrt(a + b*x)*a*b**2*c**3*d**2 - 60*sqrt(c + d*x)*sqrt(a 
+ b*x)*a*b**2*c**2*d**3*x - 12*sqrt(c + d*x)*sqrt(a + b*x)*a*b**2*c*d**4*x 
**2 + 30*sqrt(c + d*x)*sqrt(a + b*x)*b**3*c**4*d + 40*sqrt(c + d*x)*sqrt(a 
 + b*x)*b**3*c**3*d**2*x + 6*sqrt(c + d*x)*sqrt(a + b*x)*b**3*c**2*d**3*x* 
*2 - 6*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x)) 
/sqrt(a*d - b*c))*a**3*c**2*d**3 - 12*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a 
+ b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**3*c*d**4*x - 6*sqrt(d) 
*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b* 
c))*a**3*d**5*x**2 - 18*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt( 
b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b*c**3*d**2 - 36*sqrt(d)*sqrt(b)*l 
og((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a**2*b 
*c**2*d**3*x - 18*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqr 
t(c + d*x))/sqrt(a*d - b*c))*a**2*b*c*d**4*x**2 + 54*sqrt(d)*sqrt(b)*log(( 
sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**2*c** 
4*d + 108*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + d* 
x))/sqrt(a*d - b*c))*a*b**2*c**3*d**2*x + 54*sqrt(d)*sqrt(b)*log((sqrt(d)* 
sqrt(a + b*x) + sqrt(b)*sqrt(c + d*x))/sqrt(a*d - b*c))*a*b**2*c**2*d**3*x 
**2 - 30*sqrt(d)*sqrt(b)*log((sqrt(d)*sqrt(a + b*x) + sqrt(b)*sqrt(c + ...