\(\int \frac {x (a x+b x^2)^{3/2}}{(c+d x)^3} \, dx\) [109]

Optimal result

Integrand size = 22, antiderivative size = 266 \[ \int \frac {x \left (a x+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=-\frac {3 (2 b c-a d) \sqrt {a x+b x^2}}{d^4}+\frac {(12 b c-5 a d) x \sqrt {a x+b x^2}}{4 c d^3}-\frac {(8 b c-5 a d) x^2 \sqrt {a x+b x^2}}{4 c d^2 (c+d x)}-\frac {x \left (a x+b x^2\right )^{3/2}}{2 d (c+d x)^2}+\frac {3 \left (16 b^2 c^2-12 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 \sqrt {b} d^5}-\frac {3 \sqrt {c} \left (16 b^2 c^2-20 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{4 d^5 \sqrt {b c-a d}} \] Output:

-3*(-a*d+2*b*c)*(b*x^2+a*x)^(1/2)/d^4+1/4*(-5*a*d+12*b*c)*x*(b*x^2+a*x)^(1 
/2)/c/d^3-1/4*(-5*a*d+8*b*c)*x^2*(b*x^2+a*x)^(1/2)/c/d^2/(d*x+c)-1/2*x*(b* 
x^2+a*x)^(3/2)/d/(d*x+c)^2+3/4*(a^2*d^2-12*a*b*c*d+16*b^2*c^2)*arctanh(b^( 
1/2)*x/(b*x^2+a*x)^(1/2))/b^(1/2)/d^5-3/4*c^(1/2)*(5*a^2*d^2-20*a*b*c*d+16 
*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)^(1/2)
 

Mathematica [A] (verified)

Time = 10.94 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.03 \[ \int \frac {x \left (a x+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {d (-b c+a d) \sqrt {x} \left (a d \left (12 c^2+19 c d x+5 d^2 x^2\right )-2 b \left (12 c^3+18 c^2 d x+4 c d^2 x^2-d^3 x^3\right )\right )}{(c+d x)^2}+\frac {3 \left (-16 b^3 c^3+28 a b^2 c^2 d-13 a^2 b c d^2+a^3 d^3\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {1+\frac {b x}{a}}}+\frac {3 \sqrt {c} \sqrt {b c-a d} \left (16 b^2 c^2-20 a b c d+5 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {a+b x}}\right )}{4 d^5 (-b c+a d) \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(a + b*x)]*((d*(-(b*c) + a*d)*Sqrt[x]*(a*d*(12*c^2 + 19*c*d*x + 5* 
d^2*x^2) - 2*b*(12*c^3 + 18*c^2*d*x + 4*c*d^2*x^2 - d^3*x^3)))/(c + d*x)^2 
 + (3*(-16*b^3*c^3 + 28*a*b^2*c^2*d - 13*a^2*b*c*d^2 + a^3*d^3)*ArcSinh[(S 
qrt[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*x)/a]) + (3*Sqrt[c] 
*Sqrt[b*c - a*d]*(16*b^2*c^2 - 20*a*b*c*d + 5*a^2*d^2)*ArcTanh[(Sqrt[b*c - 
 a*d]*Sqrt[x])/(Sqrt[c]*Sqrt[a + b*x])])/Sqrt[a + b*x]))/(4*d^5*(-(b*c) + 
a*d)*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.92, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1230, 27, 1230, 1269, 1091, 219, 1154, 219}

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.

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1230

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

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} (2 c+d x)}{2 d^2 (c+d x)^2}-\frac {3 \left (\frac {\sqrt {a x+b x^2} (d x (4 b c-a d)+4 c (2 b c-a d))}{d^2 (c+d x)}-\frac {-\frac {c \left (5 a^2 d^2-20 a b c d+16 b^2 c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}-\frac {\left (4 a b c d-(a d-4 b c)^2\right ) \int \frac {1}{\sqrt {b x^2+a x}}dx}{d}}{2 d^2}\right )}{4 d^2}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} (2 c+d x)}{2 d^2 (c+d x)^2}-\frac {3 \left (\frac {\sqrt {a x+b x^2} (d x (4 b c-a d)+4 c (2 b c-a d))}{d^2 (c+d x)}-\frac {-\frac {c \left (5 a^2 d^2-20 a b c d+16 b^2 c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}-\frac {2 \left (4 a b c d-(a d-4 b c)^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{d}}{2 d^2}\right )}{4 d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} (2 c+d x)}{2 d^2 (c+d x)^2}-\frac {3 \left (\frac {\sqrt {a x+b x^2} (d x (4 b c-a d)+4 c (2 b c-a d))}{d^2 (c+d x)}-\frac {-\frac {c \left (5 a^2 d^2-20 a b c d+16 b^2 c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (4 a b c d-(a d-4 b c)^2\right )}{\sqrt {b} d}}{2 d^2}\right )}{4 d^2}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} (2 c+d x)}{2 d^2 (c+d x)^2}-\frac {3 \left (\frac {\sqrt {a x+b x^2} (d x (4 b c-a d)+4 c (2 b c-a d))}{d^2 (c+d x)}-\frac {\frac {2 c \left (5 a^2 d^2-20 a b c d+16 b^2 c^2\right ) \int \frac {1}{4 c (b c-a d)-\frac {(a c+(2 b c-a d) x)^2}{b x^2+a x}}d\left (-\frac {a c+(2 b c-a d) x}{\sqrt {b x^2+a x}}\right )}{d}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (4 a b c d-(a d-4 b c)^2\right )}{\sqrt {b} d}}{2 d^2}\right )}{4 d^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2} (2 c+d x)}{2 d^2 (c+d x)^2}-\frac {3 \left (\frac {\sqrt {a x+b x^2} (d x (4 b c-a d)+4 c (2 b c-a d))}{d^2 (c+d x)}-\frac {-\frac {\sqrt {c} \left (5 a^2 d^2-20 a b c d+16 b^2 c^2\right ) \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{d \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) \left (4 a b c d-(a d-4 b c)^2\right )}{\sqrt {b} d}}{2 d^2}\right )}{4 d^2}\)

Input:

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

Output:

((2*c + d*x)*(a*x + b*x^2)^(3/2))/(2*d^2*(c + d*x)^2) - (3*(((4*c*(2*b*c - 
 a*d) + d*(4*b*c - a*d)*x)*Sqrt[a*x + b*x^2])/(d^2*(c + d*x)) - ((-2*(4*a* 
b*c*d - (-4*b*c + a*d)^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b] 
*d) - (Sqrt[c]*(16*b^2*c^2 - 20*a*b*c*d + 5*a^2*d^2)*ArcTanh[(a*c + (2*b*c 
 - a*d)*x)/(2*Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/(d*Sqrt[b*c - a 
*d]))/(2*d^2)))/(4*d^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 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 1091

Int[1/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[Int[1/(1 
 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1230

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - 
 d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p 
+ 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a 
+ b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m 
+ 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, 
 x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 
1] || EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ 
[m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97

method	result	size
pseudoelliptic	\(\frac {12 \left (d x +c \right )^{2} \left (b^{2} c^{2}-\frac {5}{4} a b c d +\frac {5}{16} a^{2} d^{2}\right ) x \sqrt {b}\, a c \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )+3 \sqrt {c \left (a d -b c \right )}\, \left (\frac {a x \left (d x +c \right )^{2} \left (a^{2} d^{2}-12 a b c d +16 b^{2} c^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{4}+d \left (\left (-b \,c^{3}+\frac {7}{12} a \,c^{2} d \right ) \left (x \left (b x +a \right )\right )^{\frac {3}{2}}+\frac {5 \left (-\frac {12 \left (-b x +a \right ) b \,c^{3}}{5}+d a \left (-\frac {43 b x}{5}+a \right ) c^{2}+\frac {19 d^{2} x a \left (-\frac {8 b x}{19}+a \right ) c}{5}+a \,d^{3} x^{2} \left (\frac {2 b x}{5}+a \right )\right ) x \sqrt {x \left (b x +a \right )}}{12}\right ) \sqrt {b}\right )}{a x \sqrt {b}\, d^{5} \left (d x +c \right )^{2} \sqrt {c \left (a d -b c \right )}}\)	\(258\)
risch	\(\text {Expression too large to display}\)	\(1003\)
default	\(\text {Expression too large to display}\)	\(2596\)

Input:

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

Output:

3*(4*(d*x+c)^2*(b^2*c^2-5/4*a*b*c*d+5/16*a^2*d^2)*x*b^(1/2)*a*c*arctan((x* 
(b*x+a))^(1/2)/x*c/(c*(a*d-b*c))^(1/2))+(c*(a*d-b*c))^(1/2)*(1/4*a*x*(d*x+ 
c)^2*(a^2*d^2-12*a*b*c*d+16*b^2*c^2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))+ 
d*((-b*c^3+7/12*a*c^2*d)*(x*(b*x+a))^(3/2)+5/12*(-12/5*(-b*x+a)*b*c^3+d*a* 
(-43/5*b*x+a)*c^2+19/5*d^2*x*a*(-8/19*b*x+a)*c+a*d^3*x^2*(2/5*b*x+a))*x*(x 
*(b*x+a))^(1/2))*b^(1/2)))/b^(1/2)/(c*(a*d-b*c))^(1/2)/a/d^5/x/(d*x+c)^2
 

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1635, normalized size of antiderivative = 6.15 \[ \int \frac {x \left (a x+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/8*(3*(16*b^2*c^4 - 12*a*b*c^3*d + a^2*c^2*d^2 + (16*b^2*c^2*d^2 - 12*a* 
b*c*d^3 + a^2*d^4)*x^2 + 2*(16*b^2*c^3*d - 12*a*b*c^2*d^2 + a^2*c*d^3)*x)* 
sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) + 3*(16*b^3*c^4 - 20* 
a*b^2*c^3*d + 5*a^2*b*c^2*d^2 + (16*b^3*c^2*d^2 - 20*a*b^2*c*d^3 + 5*a^2*b 
*d^4)*x^2 + 2*(16*b^3*c^3*d - 20*a*b^2*c^2*d^2 + 5*a^2*b*c*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*(2*b^2*d^4*x^3 - 24*b^2*c^3*d + 12*a*b 
*c^2*d^2 - (8*b^2*c*d^3 - 5*a*b*d^4)*x^2 - (36*b^2*c^2*d^2 - 19*a*b*c*d^3) 
*x)*sqrt(b*x^2 + a*x))/(b*d^7*x^2 + 2*b*c*d^6*x + b*c^2*d^5), -1/8*(6*(16* 
b^3*c^4 - 20*a*b^2*c^3*d + 5*a^2*b*c^2*d^2 + (16*b^3*c^2*d^2 - 20*a*b^2*c* 
d^3 + 5*a^2*b*d^4)*x^2 + 2*(16*b^3*c^3*d - 20*a*b^2*c^2*d^2 + 5*a^2*b*c*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)) - 3*(16*b^2*c^4 - 12*a*b*c^3*d + a^2*c^2*d^2 + 
(16*b^2*c^2*d^2 - 12*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(16*b^2*c^3*d - 12*a*b*c 
^2*d^2 + a^2*c*d^3)*x)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b) 
) - 2*(2*b^2*d^4*x^3 - 24*b^2*c^3*d + 12*a*b*c^2*d^2 - (8*b^2*c*d^3 - 5*a* 
b*d^4)*x^2 - (36*b^2*c^2*d^2 - 19*a*b*c*d^3)*x)*sqrt(b*x^2 + a*x))/(b*d^7* 
x^2 + 2*b*c*d^6*x + b*c^2*d^5), -1/8*(6*(16*b^2*c^4 - 12*a*b*c^3*d + a^2*c 
^2*d^2 + (16*b^2*c^2*d^2 - 12*a*b*c*d^3 + a^2*d^4)*x^2 + 2*(16*b^2*c^3*d - 
 12*a*b*c^2*d^2 + a^2*c*d^3)*x)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(x*(b*x^2+a*x)^(3/2)/(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-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. 548 vs. \(2 (232) = 464\).

Time = 0.19 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.06 \[ \int \frac {x \left (a x+b x^2\right )^{3/2}}{(c+d x)^3} \, dx=\frac {1}{4} \, \sqrt {b x^{2} + a x} {\left (\frac {2 \, b x}{d^{3}} - \frac {12 \, b^{2} c d^{8} - 5 \, a b d^{9}}{b d^{12}}\right )} - \frac {3 \, {\left (16 \, b^{2} c^{3} - 20 \, a b c^{2} d + 5 \, a^{2} c d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} + a c d}}\right )}{4 \, \sqrt {-b c^{2} + a c d} d^{5}} - \frac {3 \, {\left (16 \, b^{2} c^{2} - 12 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} + a \right |}\right )}{8 \, \sqrt {b} d^{5}} - \frac {32 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} b^{2} c^{3} d - 36 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a b c^{2} d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{3} a^{2} c d^{3} + 56 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} b^{\frac {5}{2}} c^{4} - 44 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a b^{\frac {3}{2}} c^{3} d + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} a^{2} \sqrt {b} c^{2} d^{2} + 56 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a b^{2} c^{4} - 48 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{2} b c^{3} d + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} a^{3} c^{2} d^{2} + 14 \, a^{2} b^{\frac {3}{2}} c^{4} - 9 \, a^{3} \sqrt {b} c^{3} d}{4 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a x}\right )} \sqrt {b} c + a c\right )}^{2} d^{5}} \] Input:

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

Output:

1/4*sqrt(b*x^2 + a*x)*(2*b*x/d^3 - (12*b^2*c*d^8 - 5*a*b*d^9)/(b*d^12)) - 
3/4*(16*b^2*c^3 - 20*a*b*c^2*d + 5*a^2*c*d^2)*arctan(-((sqrt(b)*x - sqrt(b 
*x^2 + a*x))*d + sqrt(b)*c)/sqrt(-b*c^2 + a*c*d))/(sqrt(-b*c^2 + a*c*d)*d^ 
5) - 3/8*(16*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))/(sqrt(b)*d^5) - 1/4*(32*(sqrt(b)*x - sqrt(b*x^2 
 + a*x))^3*b^2*c^3*d - 36*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a*b*c^2*d^2 + 
9*(sqrt(b)*x - sqrt(b*x^2 + a*x))^3*a^2*c*d^3 + 56*(sqrt(b)*x - sqrt(b*x^2 
 + a*x))^2*b^(5/2)*c^4 - 44*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a*b^(3/2)*c^ 
3*d + 3*(sqrt(b)*x - sqrt(b*x^2 + a*x))^2*a^2*sqrt(b)*c^2*d^2 + 56*(sqrt(b 
)*x - sqrt(b*x^2 + a*x))*a*b^2*c^4 - 48*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^ 
2*b*c^3*d + 7*(sqrt(b)*x - sqrt(b*x^2 + a*x))*a^3*c^2*d^2 + 14*a^2*b^(3/2) 
*c^4 - 9*a^3*sqrt(b)*c^3*d)/(((sqrt(b)*x - sqrt(b*x^2 + a*x))^2*d + 2*(sqr 
t(b)*x - sqrt(b*x^2 + a*x))*sqrt(b)*c + a*c)^2*d^5)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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