\(\int \frac {x^3 \sqrt {a x+b x^2}}{(c+d x)^3} \, dx\) [45]

Optimal result

Integrand size = 24, antiderivative size = 311 \[ \int \frac {x^3 \sqrt {a x+b x^2}}{(c+d x)^3} \, dx=-\frac {\left (24 b^2 c^2-24 a b c d+a^2 d^2\right ) \sqrt {a x+b x^2}}{4 b d^4 (b c-a d)}+\frac {(12 b c-11 a d) x \sqrt {a x+b x^2}}{4 d^3 (b c-a d)}-\frac {x^3 \sqrt {a x+b x^2}}{2 d (c+d x)^2}-\frac {(8 b c-7 a d) x^2 \sqrt {a x+b x^2}}{4 d^2 (b c-a d) (c+d x)}+\frac {\left (48 b^2 c^2-12 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 b^{3/2} d^5}-\frac {c^{3/2} \left (48 b^2 c^2-84 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{4 d^5 (b c-a d)^{3/2}} \] Output:

-1/4*(a^2*d^2-24*a*b*c*d+24*b^2*c^2)*(b*x^2+a*x)^(1/2)/b/d^4/(-a*d+b*c)+1/ 
4*(-11*a*d+12*b*c)*x*(b*x^2+a*x)^(1/2)/d^3/(-a*d+b*c)-1/2*x^3*(b*x^2+a*x)^ 
(1/2)/d/(d*x+c)^2-1/4*(-7*a*d+8*b*c)*x^2*(b*x^2+a*x)^(1/2)/d^2/(-a*d+b*c)/ 
(d*x+c)+1/4*(-a^2*d^2-12*a*b*c*d+48*b^2*c^2)*arctanh(b^(1/2)*x/(b*x^2+a*x) 
^(1/2))/b^(3/2)/d^5-1/4*c^(3/2)*(35*a^2*d^2-84*a*b*c*d+48*b^2*c^2)*arctanh 
((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2))/d^5/(-a*d+b*c)^(3/2)
 

Mathematica [A] (verified)

Time = 11.23 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \sqrt {a x+b x^2}}{(c+d x)^3} \, dx=\frac {\sqrt {x (a+b x)} \left (-\frac {\sqrt {b} d \sqrt {x} \left (-a^2 d^2 (c+d x)^2+a b d \left (24 c^3+37 c^2 d x+9 c d^2 x^2-2 d^3 x^3\right )-2 b^2 c \left (12 c^3+18 c^2 d x+4 c d^2 x^2-d^3 x^3\right )\right )}{(-b c+a d) (c+d x)^2}+\frac {\left (48 b^2 c^2-12 a b c d-a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {1+\frac {b x}{a}}}-\frac {b^{3/2} c^{3/2} \left (48 b^2 c^2-84 a b c d+35 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )}{(b c-a d)^{3/2} \sqrt {a+b x}}\right )}{4 b^{3/2} d^5 \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(a + b*x)]*(-((Sqrt[b]*d*Sqrt[x]*(-(a^2*d^2*(c + d*x)^2) + a*b*d*( 
24*c^3 + 37*c^2*d*x + 9*c*d^2*x^2 - 2*d^3*x^3) - 2*b^2*c*(12*c^3 + 18*c^2* 
d*x + 4*c*d^2*x^2 - d^3*x^3)))/((-(b*c) + a*d)*(c + d*x)^2)) + ((48*b^2*c^ 
2 - 12*a*b*c*d - a^2*d^2)*ArcSinh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[a]*Sqr 
t[1 + (b*x)/a]) - (b^(3/2)*c^(3/2)*(48*b^2*c^2 - 84*a*b*c*d + 35*a^2*d^2)* 
ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/((b*c - a*d)^( 
3/2)*Sqrt[a + b*x])))/(4*b^(3/2)*d^5*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {1261, 108, 27, 166, 27, 171, 27, 171, 25, 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.

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

\(\Big \downarrow \) 1261

\(\displaystyle \frac {\sqrt {a x+b x^2} \int \frac {x^{7/2} \sqrt {a+b x}}{(c+d x)^3}dx}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\int \frac {x^{5/2} (7 a+8 b x)}{2 \sqrt {a+b x} (c+d x)^2}dx}{2 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\int \frac {x^{5/2} (7 a+8 b x)}{\sqrt {a+b x} (c+d x)^2}dx}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {-\frac {\int -\frac {x^{3/2} (5 a (8 b c-7 a d)+4 b (12 b c-11 a d) x)}{2 \sqrt {a+b x} (c+d x)}dx}{d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\int \frac {x^{3/2} (5 a (8 b c-7 a d)+4 b (12 b c-11 a d) x)}{\sqrt {a+b x} (c+d x)}dx}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {\int -\frac {2 b \sqrt {x} \left (3 a c (12 b c-11 a d)+2 \left (24 b^2 c^2-24 a b d c+a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)}dx}{2 b d}+\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {\int \frac {\sqrt {x} \left (3 a c (12 b c-11 a d)+2 \left (24 b^2 c^2-24 a b d c+a^2 d^2\right ) x\right )}{\sqrt {a+b x} (c+d x)}dx}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {\frac {\int -\frac {a c \left (24 b^2 c^2-24 a b d c+a^2 d^2\right )+(b c-a d) \left (48 b^2 c^2-12 a b d c-a^2 d^2\right ) x}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {-\frac {\int \frac {a c \left (24 b^2 c^2-24 a b d c+a^2 d^2\right )+(b c-a d) \left (48 b^2 c^2-12 a b d c-a^2 d^2\right ) x}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 175

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {-\frac {\frac {(b c-a d) \left (-a^2 d^2-12 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x}}dx}{d}-\frac {b c^2 \left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 65

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {-\frac {\frac {2 (b c-a d) \left (-a^2 d^2-12 a b c d+48 b^2 c^2\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {b c^2 \left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {a+b x} (c+d x)}dx}{d}}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {-\frac {\frac {2 (b c-a d) \left (-a^2 d^2-12 a b c d+48 b^2 c^2\right ) \int \frac {1}{1-\frac {b x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}-\frac {2 b c^2 \left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) (b c-a d) \left (-a^2 d^2-12 a b c d+48 b^2 c^2\right )}{\sqrt {b} d}-\frac {2 b c^2 \left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \int \frac {1}{c-\frac {(b c-a d) x}{a+b x}}d\frac {\sqrt {x}}{\sqrt {a+b x}}}{d}}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {a x+b x^2} \left (\frac {\frac {\frac {2 x^{3/2} \sqrt {a+b x} (12 b c-11 a d)}{d}-\frac {-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) (b c-a d) \left (-a^2 d^2-12 a b c d+48 b^2 c^2\right )}{\sqrt {b} d}-\frac {2 b c^{3/2} \left (35 a^2 d^2-84 a b c d+48 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {x} \sqrt {b c-a d}}{\sqrt {c} \sqrt {a+b x}}\right )}{d \sqrt {b c-a d}}}{b d}-2 \sqrt {x} \sqrt {a+b x} \left (-\frac {a^2 d}{b}+24 a c-\frac {24 b c^2}{d}\right )}{d}}{2 d (b c-a d)}-\frac {x^{5/2} \sqrt {a+b x} (8 b c-7 a d)}{d (c+d x) (b c-a d)}}{4 d}-\frac {x^{7/2} \sqrt {a+b x}}{2 d (c+d x)^2}\right )}{\sqrt {x} \sqrt {a+b x}}\)

Input:

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

Output:

(Sqrt[a*x + b*x^2]*(-1/2*(x^(7/2)*Sqrt[a + b*x])/(d*(c + d*x)^2) + (-(((8* 
b*c - 7*a*d)*x^(5/2)*Sqrt[a + b*x])/(d*(b*c - a*d)*(c + d*x))) + ((2*(12*b 
*c - 11*a*d)*x^(3/2)*Sqrt[a + b*x])/d - (-2*(24*a*c - (24*b*c^2)/d - (a^2* 
d)/b)*Sqrt[x]*Sqrt[a + b*x] - ((2*(b*c - a*d)*(48*b^2*c^2 - 12*a*b*c*d - a 
^2*d^2)*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/(Sqrt[b]*d) - (2*b*c^(3/ 
2)*(48*b^2*c^2 - 84*a*b*c*d + 35*a^2*d^2)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x] 
)/(Sqrt[c]*Sqrt[a + b*x])])/(d*Sqrt[b*c - a*d]))/(b*d))/d)/(2*d*(b*c - a*d 
)))/(4*d)))/(Sqrt[x]*Sqrt[a + b*x])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 166

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

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 = 0.75 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(-\frac {3 \left (4 \left (d x +c \right )^{2} b^{\frac {5}{2}} x \left (b^{2} c^{2}-\frac {7}{4} a b c d +\frac {35}{48} a^{2} d^{2}\right ) a \,c^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+\left (\frac {a b x \left (d x +c \right )^{2} \left (a d -b c \right ) \left (a^{2} d^{2}+12 a b c d -48 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{12}+b^{\frac {3}{2}} d \left (\left (-b^{2} c^{4}+\frac {11}{12} a b \,c^{3} d \right ) \left (x \left (b x +a \right )\right )^{\frac {3}{2}}-\frac {\sqrt {x \left (b x +a \right )}\, x \left (a d -b c \right ) \left (-12 \left (-b x +a \right ) b \,c^{3}+d a \left (-35 b x +a \right ) c^{2}+2 a \,d^{2} x \left (-4 b x +a \right ) c +a \,d^{3} x^{2} \left (2 b x +a \right )\right )}{12}\right )\right ) \sqrt {c \left (a d -b c \right )}\right )}{\sqrt {c \left (a d -b c \right )}\, a \,b^{\frac {5}{2}} d^{5} x \left (d x +c \right )^{2} \left (a d -b c \right )}\)	\(290\)
risch	\(\frac {\left (2 b d x +a d -12 b c \right ) x \left (b x +a \right )}{4 b \,d^{4} \sqrt {x \left (b x +a \right )}}-\frac {\frac {\left (a^{2} d^{2}+12 a b c d -48 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 {16 b \,c^{2} \left (3 a d -5 b c \right ) \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}}}}+\frac {8 b \,c^{3} \left (4 a d -5 b c \right ) \left (\frac {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}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}-\frac {8 b \,c^{4} \left (a d -b c \right ) \left (\frac {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}}}}{2 c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 \left (a d -2 b c \right ) d \left (\frac {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}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{4 c \left (a d -b c \right )}-\frac {b \,d^{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 )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{4}}}{8 d^{4} b}\)	\(971\)
default	\(\text {Expression too large to display}\)	\(1927\)

Input:

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

Output:

-3*(4*(d*x+c)^2*b^(5/2)*x*(b^2*c^2-7/4*a*b*c*d+35/48*a^2*d^2)*a*c^2*arctan 
((x*(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2))+(1/12*a*b*x*(d*x+c)^2*(a*d-b*c 
)*(a^2*d^2+12*a*b*c*d-48*b^2*c^2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))+b^( 
3/2)*d*((-b^2*c^4+11/12*a*b*c^3*d)*(x*(b*x+a))^(3/2)-1/12*(x*(b*x+a))^(1/2 
)*x*(a*d-b*c)*(-12*(-b*x+a)*b*c^3+d*a*(-35*b*x+a)*c^2+2*a*d^2*x*(-4*b*x+a) 
*c+a*d^3*x^2*(2*b*x+a))))*(c*(a*d-b*c))^(1/2))/(c*(a*d-b*c))^(1/2)/a/b^(5/ 
2)/d^5/x/(d*x+c)^2/(a*d-b*c)
 

Fricas [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 2216, normalized size of antiderivative = 7.13 \[ \int \frac {x^3 \sqrt {a x+b x^2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*((48*b^3*c^5 - 60*a*b^2*c^4*d + 11*a^2*b*c^3*d^2 + a^3*c^2*d^3 + (48 
*b^3*c^3*d^2 - 60*a*b^2*c^2*d^3 + 11*a^2*b*c*d^4 + a^3*d^5)*x^2 + 2*(48*b^ 
3*c^4*d - 60*a*b^2*c^3*d^2 + 11*a^2*b*c^2*d^3 + a^3*c*d^4)*x)*sqrt(b)*log( 
2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + (48*b^4*c^5 - 84*a*b^3*c^4*d + 
35*a^2*b^2*c^3*d^2 + (48*b^4*c^3*d^2 - 84*a*b^3*c^2*d^3 + 35*a^2*b^2*c*d^4 
)*x^2 + 2*(48*b^4*c^4*d - 84*a*b^3*c^3*d^2 + 35*a^2*b^2*c^2*d^3)*x)*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)) + 2*(24*b^3*c^4*d - 24*a*b^2*c^3*d^2 + a^ 
2*b*c^2*d^3 - 2*(b^3*c*d^4 - a*b^2*d^5)*x^3 + (8*b^3*c^2*d^3 - 9*a*b^2*c*d 
^4 + a^2*b*d^5)*x^2 + (36*b^3*c^3*d^2 - 37*a*b^2*c^2*d^3 + 2*a^2*b*c*d^4)* 
x)*sqrt(b*x^2 + a*x))/(b^3*c^3*d^5 - a*b^2*c^2*d^6 + (b^3*c*d^7 - a*b^2*d^ 
8)*x^2 + 2*(b^3*c^2*d^6 - a*b^2*c*d^7)*x), -1/8*(2*(48*b^4*c^5 - 84*a*b^3* 
c^4*d + 35*a^2*b^2*c^3*d^2 + (48*b^4*c^3*d^2 - 84*a*b^3*c^2*d^3 + 35*a^2*b 
^2*c*d^4)*x^2 + 2*(48*b^4*c^4*d - 84*a*b^3*c^3*d^2 + 35*a^2*b^2*c^2*d^3)*x 
)*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)) + (48*b^3*c^5 - 60*a*b^2*c^4*d + 11*a^2*b*c^3*d^2 + 
 a^3*c^2*d^3 + (48*b^3*c^3*d^2 - 60*a*b^2*c^2*d^3 + 11*a^2*b*c*d^4 + a^3*d 
^5)*x^2 + 2*(48*b^3*c^4*d - 60*a*b^2*c^3*d^2 + 11*a^2*b*c^2*d^3 + a^3*c*d^ 
4)*x)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 2*(24*b^3*c^4 
*d - 24*a*b^2*c^3*d^2 + a^2*b*c^2*d^3 - 2*(b^3*c*d^4 - a*b^2*d^5)*x^3 +...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x^3*(b*x^2+a*x)^(1/2)/(d*x+c)^3,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>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 573 vs. \(2 (275) = 550\).

Time = 0.16 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.84 \[ \int \frac {x^3 \sqrt {a x+b x^2}}{(c+d x)^3} \, dx=\frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (\frac {2 \, x}{d^{3}} - \frac {12 \, b c d^{8} - a d^{9}}{b d^{12}}\right )} - \frac {{\left (48 \, b^{2} c^{4} - 84 \, a b c^{3} d + 35 \, a^{2} c^{2} d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} + a c d}}\right )}{4 \, {\left (b c d^{5} - a d^{6}\right )} \sqrt {-b c^{2} + a c d}} - \frac {32 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} b^{2} c^{4} d - 44 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a b c^{3} d^{2} + 13 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} c^{2} d^{3} + 56 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b^{\frac {5}{2}} c^{5} - 60 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a b^{\frac {3}{2}} c^{4} d + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} \sqrt {b} c^{3} d^{2} + 56 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a b^{2} c^{5} - 64 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{2} b c^{4} d + 11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} c^{3} d^{2} + 14 \, a^{2} b^{\frac {3}{2}} c^{5} - 13 \, a^{3} \sqrt {b} c^{4} d}{4 \, {\left (b c d^{5} - a d^{6}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} c + a c\right )}^{2}} - \frac {{\left (48 \, b^{2} c^{2} - 12 \, a b c d - a^{2} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{8 \, b^{\frac {3}{2}} d^{5}} \] Input:

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

Output:

1/4*sqrt(b*x^2 + a*x)*(2*x/d^3 - (12*b*c*d^8 - a*d^9)/(b*d^12)) - 1/4*(48* 
b^2*c^4 - 84*a*b*c^3*d + 35*a^2*c^2*d^2)*arctan(-((sqrt(b)*x - sqrt(b*x^2 
+ a*x))*d + sqrt(b)*c)/sqrt(-b*c^2 + a*c*d))/((b*c*d^5 - a*d^6)*sqrt(-b*c^ 
2 + a*c*d)) - 1/4*(32*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*b^2*c^4*d - 44*(sq 
rt(b)*x - sqrt(b*x^2 + a*x))^3*a*b*c^3*d^2 + 13*(sqrt(b)*x - sqrt(b*x^2 + 
a*x))^3*a^2*c^2*d^3 + 56*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*b^(5/2)*c^5 - 6 
0*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a*b^(3/2)*c^4*d + 7*(sqrt(b)*x - sqrt( 
b*x^2 + a*x))^2*a^2*sqrt(b)*c^3*d^2 + 56*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a 
*b^2*c^5 - 64*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^2*b*c^4*d + 11*(sqrt(b)*x 
- sqrt(b*x^2 + a*x))*a^3*c^3*d^2 + 14*a^2*b^(3/2)*c^5 - 13*a^3*sqrt(b)*c^4 
*d)/((b*c*d^5 - a*d^6)*((sqrt(b)*x - sqrt(b*x^2 + a*x))^2*d + 2*(sqrt(b)*x 
 - sqrt(b*x^2 + a*x))*sqrt(b)*c + a*c)^2) - 1/8*(48*b^2*c^2 - 12*a*b*c*d - 
 a^2*d^2)*log(abs(2*(sqrt(b)*x - sqrt(b*x^2 + a*x))*sqrt(b) + a))/(b^(3/2) 
*d^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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