\(\int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx\) [367]

Optimal result

Integrand size = 22, antiderivative size = 327 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {3 \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} \sqrt {c+d x}}{64 b^5 d^2}+\frac {\left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \sqrt {a+b x} (c+d x)^{3/2}}{32 b^4 d^2 (b c-a d)}+\frac {2 a^3 (c+d x)^{5/2}}{b^3 (b c-a d) \sqrt {a+b x}}-\frac {(b c+7 a d) \sqrt {a+b x} (c+d x)^{5/2}}{8 b^3 d^2}+\frac {(a+b x)^{3/2} (c+d x)^{5/2}}{4 b^3 d}+\frac {3 (b c-a d) \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{64 b^{11/2} d^{5/2}} \] Output:

3/64*(-105*a^3*d^3+35*a^2*b*c*d^2+5*a*b^2*c^2*d+b^3*c^3)*(b*x+a)^(1/2)*(d* 
x+c)^(1/2)/b^5/d^2+1/32*(-105*a^3*d^3+35*a^2*b*c*d^2+5*a*b^2*c^2*d+b^3*c^3 
)*(b*x+a)^(1/2)*(d*x+c)^(3/2)/b^4/d^2/(-a*d+b*c)+2*a^3*(d*x+c)^(5/2)/b^3/( 
-a*d+b*c)/(b*x+a)^(1/2)-1/8*(7*a*d+b*c)*(b*x+a)^(1/2)*(d*x+c)^(5/2)/b^3/d^ 
2+1/4*(b*x+a)^(3/2)*(d*x+c)^(5/2)/b^3/d+3/64*(-a*d+b*c)*(-105*a^3*d^3+35*a 
^2*b*c*d^2+5*a*b^2*c^2*d+b^3*c^3)*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d 
*x+c)^(1/2))/b^(11/2)/d^(5/2)
 

Mathematica [A] (verified)

Time = 10.47 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.80 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {\sqrt {c+d x} \left (-\frac {\sqrt {d} \left (315 a^4 d^3+105 a^3 b d^2 (-3 c+d x)+a^2 b^2 d \left (13 c^2-119 c d x-42 d^2 x^2\right )-b^4 x \left (-3 c^3+2 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )+a b^3 \left (3 c^3+11 c^2 d x+44 c d^2 x^2+24 d^3 x^3\right )\right )}{\sqrt {a+b x}}+\frac {3 \sqrt {b c-a d} \left (b^3 c^3+5 a b^2 c^2 d+35 a^2 b c d^2-105 a^3 d^3\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{64 b^5 d^{5/2}} \] Input:

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

Output:

(Sqrt[c + d*x]*(-((Sqrt[d]*(315*a^4*d^3 + 105*a^3*b*d^2*(-3*c + d*x) + a^2 
*b^2*d*(13*c^2 - 119*c*d*x - 42*d^2*x^2) - b^4*x*(-3*c^3 + 2*c^2*d*x + 24* 
c*d^2*x^2 + 16*d^3*x^3) + a*b^3*(3*c^3 + 11*c^2*d*x + 44*c*d^2*x^2 + 24*d^ 
3*x^3)))/Sqrt[a + b*x]) + (3*Sqrt[b*c - a*d]*(b^3*c^3 + 5*a*b^2*c^2*d + 35 
*a^2*b*c*d^2 - 105*a^3*d^3)*ArcSinh[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[b*c - a*d 
]])/Sqrt[(b*(c + d*x))/(b*c - a*d)]))/(64*b^5*d^(5/2))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.77, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {108, 27, 170, 27, 164, 60, 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.

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

\(\Big \downarrow \) 108

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 170

\(\displaystyle \frac {3 \left (\frac {\int -\frac {d x \sqrt {c+d x} (12 a c-(b c-21 a d) x)}{2 \sqrt {a+b x}}dx}{4 b d}+\frac {3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b}\right )}{b}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 \left (\frac {3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b}-\frac {\int \frac {x \sqrt {c+d x} (12 a c-(b c-21 a d) x)}{\sqrt {a+b x}}dx}{8 b}\right )}{b}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {3 \left (\frac {3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{12 b^2 d^2}-\frac {\left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}}dx}{8 b^2 d^2}}{8 b}\right )}{b}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {3 \left (\frac {3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{12 b^2 d^2}-\frac {\left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {(b c-a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{8 b^2 d^2}}{8 b}\right )}{b}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 66

\(\displaystyle \frac {3 \left (\frac {3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{12 b^2 d^2}-\frac {\left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {(b c-a d) \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {\sqrt {a+b x}}{\sqrt {c+d x}}}{b}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{8 b^2 d^2}}{8 b}\right )}{b}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {3 \left (\frac {3 x^2 \sqrt {a+b x} (c+d x)^{3/2}}{4 b}-\frac {\frac {\sqrt {a+b x} (c+d x)^{3/2} \left (-105 a^2 d^2-4 b d x (b c-21 a d)+14 a b c d+3 b^2 c^2\right )}{12 b^2 d^2}-\frac {\left (-105 a^3 d^3+35 a^2 b c d^2+5 a b^2 c^2 d+b^3 c^3\right ) \left (\frac {(b c-a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} \sqrt {d}}+\frac {\sqrt {a+b x} \sqrt {c+d x}}{b}\right )}{8 b^2 d^2}}{8 b}\right )}{b}-\frac {2 x^3 (c+d x)^{3/2}}{b \sqrt {a+b x}}\)

Input:

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

Output:

(-2*x^3*(c + d*x)^(3/2))/(b*Sqrt[a + b*x]) + (3*((3*x^2*Sqrt[a + b*x]*(c + 
 d*x)^(3/2))/(4*b) - ((Sqrt[a + b*x]*(c + d*x)^(3/2)*(3*b^2*c^2 + 14*a*b*c 
*d - 105*a^2*d^2 - 4*b*d*(b*c - 21*a*d)*x))/(12*b^2*d^2) - ((b^3*c^3 + 5*a 
*b^2*c^2*d + 35*a^2*b*c*d^2 - 105*a^3*d^3)*((Sqrt[a + b*x]*Sqrt[c + d*x])/ 
b + ((b*c - a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*x])/(Sqrt[b]*Sqrt[c + d*x])]) 
/(b^(3/2)*Sqrt[d])))/(8*b^2*d^2))/(8*b)))/b
 

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 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, 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 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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

rule 170

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] 
 && IntegerQ[m]
 

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. \(960\) vs. \(2(285)=570\).

Time = 0.25 (sec) , antiderivative size = 961, normalized size of antiderivative = 2.94

method	result	size
default	\(\frac {\sqrt {x d +c}\, \left (32 b^{4} d^{3} x^{4} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-48 a \,b^{3} d^{3} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+48 b^{4} c \,d^{2} x^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+315 \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} b \,d^{4} x -420 \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^{2} c \,d^{3} x +90 \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^{3} c^{2} d^{2} x +12 \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^{4} c^{3} 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 ) b^{5} c^{4} x +84 a^{2} b^{2} d^{3} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-88 a \,b^{3} c \,d^{2} x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+4 b^{4} c^{2} d \,x^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+315 \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^{5} d^{4}-420 \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} b c \,d^{3}+90 \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^{2} c^{2} d^{2}+12 \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^{3} 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^{4} c^{4}-210 a^{3} b \,d^{3} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+238 a^{2} b^{2} c \,d^{2} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-22 a \,b^{3} c^{2} d x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-6 b^{4} c^{3} x \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-630 a^{4} d^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}+630 a^{3} b c \,d^{2} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-26 a^{2} b^{2} c^{2} d \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}-6 a \,b^{3} c^{3} \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\right )}{128 \sqrt {\left (b x +a \right ) \left (x d +c \right )}\, \sqrt {d b}\, \sqrt {b x +a}\, b^{5} d^{2}}\)	\(961\)

Input:

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

Output:

1/128*(d*x+c)^(1/2)*(32*b^4*d^3*x^4*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-48 
*a*b^3*d^3*x^3*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+48*b^4*c*d^2*x^3*((b*x+ 
a)*(d*x+c))^(1/2)*(d*b)^(1/2)+315*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*b*d^4*x-420*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^2*c*d^3*x+90*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^2*b^3*c^2*d^2*x+12*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^4*c^3*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))*b^5*c^4*x+84*a^2*b^2*d^3*x^2*((b 
*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-88*a*b^3*c*d^2*x^2*((b*x+a)*(d*x+c))^(1/2 
)*(d*b)^(1/2)+4*b^4*c^2*d*x^2*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+315*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^ 
5*d^4-420*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*b*c*d^3+90*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^2*c^2*d^2+12*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^3*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^4* 
c^4-210*a^3*b*d^3*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)+238*a^2*b^2*c*d^2* 
x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-22*a*b^3*c^2*d*x*((b*x+a)*(d*x+c))^( 
1/2)*(d*b)^(1/2)-6*b^4*c^3*x*((b*x+a)*(d*x+c))^(1/2)*(d*b)^(1/2)-630*a^...
 

Fricas [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 788, normalized size of antiderivative = 2.41 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/256*(3*(a*b^4*c^4 + 4*a^2*b^3*c^3*d + 30*a^3*b^2*c^2*d^2 - 140*a^4*b*c* 
d^3 + 105*a^5*d^4 + (b^5*c^4 + 4*a*b^4*c^3*d + 30*a^2*b^3*c^2*d^2 - 140*a^ 
3*b^2*c*d^3 + 105*a^4*b*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*(16*b^5*d^4*x^4 - 3*a*b^4*c^3*d - 1 
3*a^2*b^3*c^2*d^2 + 315*a^3*b^2*c*d^3 - 315*a^4*b*d^4 + 24*(b^5*c*d^3 - a* 
b^4*d^4)*x^3 + 2*(b^5*c^2*d^2 - 22*a*b^4*c*d^3 + 21*a^2*b^3*d^4)*x^2 - (3* 
b^5*c^3*d + 11*a*b^4*c^2*d^2 - 119*a^2*b^3*c*d^3 + 105*a^3*b^2*d^4)*x)*sqr 
t(b*x + a)*sqrt(d*x + c))/(b^7*d^3*x + a*b^6*d^3), -1/128*(3*(a*b^4*c^4 + 
4*a^2*b^3*c^3*d + 30*a^3*b^2*c^2*d^2 - 140*a^4*b*c*d^3 + 105*a^5*d^4 + (b^ 
5*c^4 + 4*a*b^4*c^3*d + 30*a^2*b^3*c^2*d^2 - 140*a^3*b^2*c*d^3 + 105*a^4*b 
*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*(16 
*b^5*d^4*x^4 - 3*a*b^4*c^3*d - 13*a^2*b^3*c^2*d^2 + 315*a^3*b^2*c*d^3 - 31 
5*a^4*b*d^4 + 24*(b^5*c*d^3 - a*b^4*d^4)*x^3 + 2*(b^5*c^2*d^2 - 22*a*b^4*c 
*d^3 + 21*a^2*b^3*d^4)*x^2 - (3*b^5*c^3*d + 11*a*b^4*c^2*d^2 - 119*a^2*b^3 
*c*d^3 + 105*a^3*b^2*d^4)*x)*sqrt(b*x + a)*sqrt(d*x + c))/(b^7*d^3*x + a*b 
^6*d^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(d*x+c)^(3/2)/(b*x+a)^(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 [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.26 \[ \int \frac {x^3 (c+d x)^{3/2}}{(a+b x)^{3/2}} \, dx=\frac {1}{64} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} {\left (2 \, {\left (b x + a\right )} {\left (4 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b x + a\right )} d {\left | b \right |}}{b^{7}} + \frac {3 \, b^{28} c d^{6} {\left | b \right |} - 11 \, a b^{27} d^{7} {\left | b \right |}}{b^{34} d^{6}}\right )} + \frac {b^{29} c^{2} d^{5} {\left | b \right |} - 58 \, a b^{28} c d^{6} {\left | b \right |} + 105 \, a^{2} b^{27} d^{7} {\left | b \right |}}{b^{34} d^{6}}\right )} - \frac {3 \, b^{30} c^{3} d^{4} {\left | b \right |} + 15 \, a b^{29} c^{2} d^{5} {\left | b \right |} - 279 \, a^{2} b^{28} c d^{6} {\left | b \right |} + 325 \, a^{3} b^{27} d^{7} {\left | b \right |}}{b^{34} d^{6}}\right )} \sqrt {b x + a} + \frac {4 \, {\left (a^{3} b^{2} c^{2} d {\left | b \right |} - 2 \, a^{4} b c d^{2} {\left | b \right |} + a^{5} d^{3} {\left | b \right |}\right )}}{{\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )} \sqrt {b d} b^{5}} - \frac {3 \, {\left (b^{4} c^{4} {\left | b \right |} + 4 \, a b^{3} c^{3} d {\left | b \right |} + 30 \, a^{2} b^{2} c^{2} d^{2} {\left | b \right |} - 140 \, a^{3} b c d^{3} {\left | b \right |} + 105 \, a^{4} d^{4} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{128 \, \sqrt {b d} b^{6} d^{2}} \] Input:

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

Output:

1/64*sqrt(b^2*c + (b*x + a)*b*d - a*b*d)*(2*(b*x + a)*(4*(b*x + a)*(2*(b*x 
 + a)*d*abs(b)/b^7 + (3*b^28*c*d^6*abs(b) - 11*a*b^27*d^7*abs(b))/(b^34*d^ 
6)) + (b^29*c^2*d^5*abs(b) - 58*a*b^28*c*d^6*abs(b) + 105*a^2*b^27*d^7*abs 
(b))/(b^34*d^6)) - (3*b^30*c^3*d^4*abs(b) + 15*a*b^29*c^2*d^5*abs(b) - 279 
*a^2*b^28*c*d^6*abs(b) + 325*a^3*b^27*d^7*abs(b))/(b^34*d^6))*sqrt(b*x + a 
) + 4*(a^3*b^2*c^2*d*abs(b) - 2*a^4*b*c*d^2*abs(b) + a^5*d^3*abs(b))/((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(b*d)*b^5) - 3/128*(b^4*c^4*abs(b) + 4*a*b^3*c^3*d*abs(b) + 30*a 
^2*b^2*c^2*d^2*abs(b) - 140*a^3*b*c*d^3*abs(b) + 105*a^4*d^4*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^6*d^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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