\(\int \frac {x^3}{(c+d x) \sqrt {a x+b x^2}} \, dx\) [143]

Optimal result

Integrand size = 24, antiderivative size = 168 \[ \int \frac {x^3}{(c+d x) \sqrt {a x+b x^2}} \, dx=-\frac {(4 b c+3 a d) \sqrt {a x+b x^2}}{4 b^2 d^2}+\frac {x \sqrt {a x+b x^2}}{2 b d}+\frac {\left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{5/2} d^3}-\frac {2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^3 \sqrt {b c-a d}} \] Output:

-1/4*(3*a*d+4*b*c)*(b*x^2+a*x)^(1/2)/b^2/d^2+1/2*x*(b*x^2+a*x)^(1/2)/b/d+1 
/4*(3*a^2*d^2+4*a*b*c*d+8*b^2*c^2)*arctanh(b^(1/2)*x/(b*x^2+a*x)^(1/2))/b^ 
(5/2)/d^3-2*c^(5/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/ 
d^3/(-a*d+b*c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.64 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.79 \[ \int \frac {x^3}{(c+d x) \sqrt {a x+b x^2}} \, dx=-\frac {\sqrt {x} \left (\sqrt {b} \left (d \sqrt {b c-a d} \sqrt {x} (a+b x) (4 b c+3 a d-2 b d x)+8 b c^{3/2} \left (-i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {a+b x} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )+8 b c^{3/2} \left (i \sqrt {a} \sqrt {d}+\sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {a+b x} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )\right )+2 \sqrt {b c-a d} \left (8 b^2 c^2+4 a b c d+3 a^2 d^2\right ) \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )\right )}{4 b^{5/2} d^3 \sqrt {b c-a d} \sqrt {x (a+b x)}} \] Input:

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

Output:

-1/4*(Sqrt[x]*(Sqrt[b]*(d*Sqrt[b*c - a*d]*Sqrt[x]*(a + b*x)*(4*b*c + 3*a*d 
 - 2*b*d*x) + 8*b*c^(3/2)*((-I)*Sqrt[a]*Sqrt[d] + Sqrt[b*c - a*d])*Sqrt[-( 
b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[a + b*x]*ArcTan 
[(Sqrt[-(b*c) + 2*a*d - (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]*Sqrt[x])/(S 
qrt[c]*(Sqrt[a] - Sqrt[a + b*x]))] + 8*b*c^(3/2)*(I*Sqrt[a]*Sqrt[d] + Sqrt 
[b*c - a*d])*Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c - a*d]]* 
Sqrt[a + b*x]*ArcTan[(Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b*c 
 - a*d]]*Sqrt[x])/(Sqrt[c]*(Sqrt[a] - Sqrt[a + b*x]))]) + 2*Sqrt[b*c - a*d 
]*(8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x]*ArcTanh[(Sqrt[b]*Sqrt[ 
x])/(Sqrt[a] - Sqrt[a + b*x])]))/(b^(5/2)*d^3*Sqrt[b*c - a*d]*Sqrt[x*(a + 
b*x)])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.29, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1261, 113, 27, 171, 27, 175, 65, 104, 219, 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.

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

\(\Big \downarrow \) 1261

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

\(\Big \downarrow \) 113

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 175

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

\(\Big \downarrow \) 65

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

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b d}-\frac {\frac {\sqrt {x} \sqrt {a+b x} (3 a d+4 b c)}{b d}-\frac {\frac {2 \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {16 b^2 c^3 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{2 b d}}{4 b d}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b d}-\frac {\frac {\sqrt {x} \sqrt {a+b x} (3 a d+4 b c)}{b d}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right )}{\sqrt {b} d}-\frac {16 b^2 c^3 \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{2 b d}}{4 b d}\right )}{\sqrt {a x+b x^2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {x} \sqrt {a+b x} \left (\frac {x^{3/2} \sqrt {a+b x}}{2 b d}-\frac {\frac {\sqrt {x} \sqrt {a+b x} (3 a d+4 b c)}{b d}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \left (3 a^2 d^2+4 a b c d+8 b^2 c^2\right )}{\sqrt {b} d}-\frac {16 b^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{d \sqrt {b c-a d}}}{2 b d}}{4 b d}\right )}{\sqrt {a x+b x^2}}\)

Input:

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

Output:

(Sqrt[x]*Sqrt[a + b*x]*((x^(3/2)*Sqrt[a + b*x])/(2*b*d) - (((4*b*c + 3*a*d 
)*Sqrt[x]*Sqrt[a + b*x])/(b*d) - ((2*(8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*A 
rcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(Sqrt[b]*d) - (16*b^2*c^(5/2)*Arc 
Tanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/(d*Sqrt[b*c - a*d 
]))/(2*b*d))/(4*b*d)))/Sqrt[a*x + b*x^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 65

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2   Sub 
st[Int[1/(b - d*x^2), x], x, Sqrt[b*x]/Sqrt[c + d*x]], x] /; FreeQ[{b, c, d 
}, x] &&  !GtQ[c, 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 113

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

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

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]
 

rule 1261

Int[((e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((b_.)*(x_) + (c_.)*(x_)^2) 
^(p_), x_Symbol] :> Simp[(e*x)^m*((b*x + c*x^2)^p/(x^(m + p)*(b + c*x)^p)) 
  Int[x^(m + p)*(f + g*x)^n*(b + c*x)^p, x], x] /; FreeQ[{b, c, e, f, g, m, 
 n}, x] &&  !IGtQ[n, 0]
 

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.74

method	result	size
pseudoelliptic	\(-\frac {\frac {d \sqrt {x \left (b x +a \right )}\, \left (-2 b d x +3 a d +4 b c \right )}{4 b^{2}}-\frac {\left (3 a^{2} d^{2}+4 a b c d +8 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{4 b^{\frac {5}{2}}}-\frac {2 c^{3} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{d^{3}}\)	\(125\)
risch	\(-\frac {\left (-2 b d x +3 a d +4 b c \right ) x \left (b x +a \right )}{4 b^{2} d^{2} \sqrt {x \left (b x +a \right )}}+\frac {\frac {\left (3 a^{2} d^{2}+4 a b c d +8 b^{2} c^{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {8 c^{3} b^{2} \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{8 b^{2} d^{2}}\)	\(239\)
default	\(\frac {c^{2} \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d^{3} \sqrt {b}}+\frac {\frac {x \sqrt {b \,x^{2}+a x}}{2 b}-\frac {3 a \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )}{4 b}}{d}-\frac {c \left (\frac {\sqrt {b \,x^{2}+a x}}{b}-\frac {a \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{2 b^{\frac {3}{2}}}\right )}{d^{2}}+\frac {c^{3} \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{4} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\)	\(295\)

Input:

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

Output:

-1/d^3*(1/4*d*(x*(b*x+a))^(1/2)*(-2*b*d*x+3*a*d+4*b*c)/b^2-1/4*(3*a^2*d^2+ 
4*a*b*c*d+8*b^2*c^2)/b^(5/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))-2*c^3/(c 
*(a*d-b*c))^(1/2)*arctan((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 696, normalized size of antiderivative = 4.14 \[ \int \frac {x^3}{(c+d x) \sqrt {a x+b x^2}} \, dx=\left [\frac {8 \, b^{3} c^{2} \sqrt {\frac {c}{b c - a d}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {\frac {c}{b c - a d}}}{d x + c}\right ) + {\left (8 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + 2 \, {\left (2 \, b^{2} d^{2} x - 4 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a x}}{8 \, b^{3} d^{3}}, -\frac {16 \, b^{3} c^{2} \sqrt {-\frac {c}{b c - a d}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {-\frac {c}{b c - a d}}}{b c x + a c}\right ) - {\left (8 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - 2 \, {\left (2 \, b^{2} d^{2} x - 4 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a x}}{8 \, b^{3} d^{3}}, \frac {4 \, b^{3} c^{2} \sqrt {\frac {c}{b c - a d}} \log \left (\frac {a c + {\left (2 \, b c - a d\right )} x - 2 \, \sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {\frac {c}{b c - a d}}}{d x + c}\right ) - {\left (8 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) + {\left (2 \, b^{2} d^{2} x - 4 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a x}}{4 \, b^{3} d^{3}}, -\frac {8 \, b^{3} c^{2} \sqrt {-\frac {c}{b c - a d}} \arctan \left (-\frac {\sqrt {b x^{2} + a x} {\left (b c - a d\right )} \sqrt {-\frac {c}{b c - a d}}}{b c x + a c}\right ) + {\left (8 \, b^{2} c^{2} + 4 \, a b c d + 3 \, a^{2} d^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-b}}{b x + a}\right ) - {\left (2 \, b^{2} d^{2} x - 4 \, b^{2} c d - 3 \, a b d^{2}\right )} \sqrt {b x^{2} + a x}}{4 \, b^{3} d^{3}}\right ] \] Input:

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

Output:

[1/8*(8*b^3*c^2*sqrt(c/(b*c - a*d))*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b* 
x^2 + a*x)*(b*c - a*d)*sqrt(c/(b*c - a*d)))/(d*x + c)) + (8*b^2*c^2 + 4*a* 
b*c*d + 3*a^2*d^2)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 
2*(2*b^2*d^2*x - 4*b^2*c*d - 3*a*b*d^2)*sqrt(b*x^2 + a*x))/(b^3*d^3), -1/8 
*(16*b^3*c^2*sqrt(-c/(b*c - a*d))*arctan(-sqrt(b*x^2 + a*x)*(b*c - a*d)*sq 
rt(-c/(b*c - a*d))/(b*c*x + a*c)) - (8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*sq 
rt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 2*(2*b^2*d^2*x - 4*b^ 
2*c*d - 3*a*b*d^2)*sqrt(b*x^2 + a*x))/(b^3*d^3), 1/4*(4*b^3*c^2*sqrt(c/(b* 
c - a*d))*log((a*c + (2*b*c - a*d)*x - 2*sqrt(b*x^2 + a*x)*(b*c - a*d)*sqr 
t(c/(b*c - a*d)))/(d*x + c)) - (8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*sqrt(-b 
)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + (2*b^2*d^2*x - 4*b^2*c*d 
- 3*a*b*d^2)*sqrt(b*x^2 + a*x))/(b^3*d^3), -1/4*(8*b^3*c^2*sqrt(-c/(b*c - 
a*d))*arctan(-sqrt(b*x^2 + a*x)*(b*c - a*d)*sqrt(-c/(b*c - a*d))/(b*c*x + 
a*c)) + (8*b^2*c^2 + 4*a*b*c*d + 3*a^2*d^2)*sqrt(-b)*arctan(sqrt(b*x^2 + a 
*x)*sqrt(-b)/(b*x + a)) - (2*b^2*d^2*x - 4*b^2*c*d - 3*a*b*d^2)*sqrt(b*x^2 
 + a*x))/(b^3*d^3)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3/(d*x+c)/(b*x^2+a*x)^(1/2),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-(2*b*c)/d^2)^2>0)', see `as 
sume?` for
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.99 \[ \int \frac {x^3}{(c+d x) \sqrt {a x+b x^2}} \, dx=\frac {8 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}+8 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}-3 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,d^{3}-\sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c \,d^{2}+2 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} d^{3} x +4 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{2} d -2 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c \,d^{2} x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{3}+\sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b c \,d^{2}+4 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,b^{2} c^{2} d -8 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{3} c^{3}}{4 b^{3} d^{3} \left (a d -b c \right )} \] Input:

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

Output:

(8*sqrt(c)*sqrt(a*d - b*c)*atan((sqrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - 
 sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*b**3*c**2 + 8*sqrt(c)*sqrt(a* 
d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*s 
qrt(b))/(sqrt(c)*sqrt(b)))*b**3*c**2 - 3*sqrt(x)*sqrt(a + b*x)*a**2*b*d**3 
 - sqrt(x)*sqrt(a + b*x)*a*b**2*c*d**2 + 2*sqrt(x)*sqrt(a + b*x)*a*b**2*d* 
*3*x + 4*sqrt(x)*sqrt(a + b*x)*b**3*c**2*d - 2*sqrt(x)*sqrt(a + b*x)*b**3* 
c*d**2*x + 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**3*d 
**3 + sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*b*c*d**2 
 + 4*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b**2*c**2*d 
- 8*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*b**3*c**3)/(4*b 
**3*d**3*(a*d - b*c))