\(\int \frac {(a x+b x^2)^{3/2}}{(c+d x)^2} \, dx\) [102]

Optimal result

Integrand size = 21, antiderivative size = 196 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{(c+d x)^2} \, dx=-\frac {3 (4 b c-3 a d) \sqrt {a x+b x^2}}{4 d^3}+\frac {3 b x \sqrt {a x+b x^2}}{2 d^2}-\frac {\left (a x+b x^2\right )^{3/2}}{d (c+d x)}+\frac {3 \left (8 b^2 c^2-8 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{4 \sqrt {b} d^4}-\frac {3 \sqrt {c} \sqrt {b c-a d} (2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^4} \] Output:

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

Mathematica [A] (verified)

Time = 10.51 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.05 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{(c+d x)^2} \, dx=\frac {\sqrt {x (a+b x)} \left (\frac {d \sqrt {x} \left (a d (9 c+5 d x)-2 b \left (6 c^2+3 c d x-d^2 x^2\right )\right )}{c+d x}+\frac {3 \left (8 b^2 c^2-8 a b c d+a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} \sqrt {1+\frac {b x}{a}}}-\frac {12 \sqrt {c} \sqrt {b c-a d} (2 b c-a d) \text {arctanh}\left (\frac {\sqrt {b c-a d} \sqrt {x}}{\sqrt {c} \sqrt {a+b x}}\right )}{\sqrt {a+b x}}\right )}{4 d^4 \sqrt {x}} \] Input:

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

Output:

(Sqrt[x*(a + b*x)]*((d*Sqrt[x]*(a*d*(9*c + 5*d*x) - 2*b*(6*c^2 + 3*c*d*x - 
 d^2*x^2)))/(c + d*x) + (3*(8*b^2*c^2 - 8*a*b*c*d + a^2*d^2)*ArcSinh[(Sqrt 
[b]*Sqrt[x])/Sqrt[a]])/(Sqrt[a]*Sqrt[b]*Sqrt[1 + (b*x)/a]) - (12*Sqrt[c]*S 
qrt[b*c - a*d]*(2*b*c - a*d)*ArcTanh[(Sqrt[b*c - a*d]*Sqrt[x])/(Sqrt[c]*Sq 
rt[a + b*x])])/Sqrt[a + b*x]))/(4*d^4*Sqrt[x])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1161, 1231, 25, 27, 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.

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

\(\Big \downarrow \) 1161

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 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 1161

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 1))), x] - Si 
mp[p/(e*(m + 1))   Int[(d + e*x)^(m + 1)*(b + 2*c*x)*(a + b*x + c*x^2)^(p - 
 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && GtQ[p, 0] && (IntegerQ[p] || 
 LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, 
 c, d, e, m, p, x]
 

rule 1231

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) 
 - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ 
(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 
 2*p + 2))   Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* 
a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* 
c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c 
^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x 
] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  !R 
ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Integer 
Q[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 = 0.65 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.94

method	result	size
pseudoelliptic	\(\frac {6 \left (d x +c \right ) \left (-a d +b c \right ) \left (b c -\frac {a d}{2}\right ) \sqrt {b}\, c \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )-3 \left (-\frac {\left (a^{2} d^{2}-8 a b c d +8 b^{2} c^{2}\right ) \left (d x +c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{4}+d \left (b \,c^{2}-\frac {3 d \left (-\frac {2 b x}{3}+a \right ) c}{4}-\frac {5 d^{2} x \left (\frac {2 b x}{5}+a \right )}{12}\right ) \sqrt {b}\, \sqrt {x \left (b x +a \right )}\right ) \sqrt {c \left (a d -b c \right )}}{\sqrt {b}\, d^{4} \left (d x +c \right ) \sqrt {c \left (a d -b c \right )}}\)	\(185\)
risch	\(\frac {\left (2 b d x +5 a d -8 b c \right ) x \left (b x +a \right )}{4 d^{3} \sqrt {x \left (b x +a \right )}}+\frac {\frac {3 \left (a^{2} d^{2}-8 a b c d +8 b^{2} c^{2}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {16 c \left (a^{2} d^{2}-3 a b c d +2 b^{2} c^{2}\right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}+\frac {8 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{c \left (a d -b c \right ) \left (x +\frac {c}{d}\right )}-\frac {\left (a d -2 b c \right ) d \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 c \left (a d -b c \right ) \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{3}}}{8 d^{3}}\)	\(503\)
default	\(\text {Expression too large to display}\)	\(966\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 1008, normalized size of antiderivative = 5.14 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/8*(3*(8*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2 + (8*b^2*c^2*d - 8*a*b*c*d^2 
+ a^2*d^3)*x)*sqrt(b)*log(2*b*x + a + 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 12*(2 
*b^2*c^2 - a*b*c*d + (2*b^2*c*d - a*b*d^2)*x)*sqrt(b*c^2 - a*c*d)*log((a*c 
 + (2*b*c - a*d)*x + 2*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) + 
 2*(2*b^2*d^3*x^2 - 12*b^2*c^2*d + 9*a*b*c*d^2 - (6*b^2*c*d^2 - 5*a*b*d^3) 
*x)*sqrt(b*x^2 + a*x))/(b*d^5*x + b*c*d^4), 1/8*(24*(2*b^2*c^2 - a*b*c*d + 
 (2*b^2*c*d - a*b*d^2)*x)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d) 
*sqrt(b*x^2 + a*x)/(b*c*x + a*c)) + 3*(8*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2 
 + (8*b^2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*x)*sqrt(b)*log(2*b*x + a + 2*sqrt 
(b*x^2 + a*x)*sqrt(b)) + 2*(2*b^2*d^3*x^2 - 12*b^2*c^2*d + 9*a*b*c*d^2 - ( 
6*b^2*c*d^2 - 5*a*b*d^3)*x)*sqrt(b*x^2 + a*x))/(b*d^5*x + b*c*d^4), -1/4*( 
3*(8*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2 + (8*b^2*c^2*d - 8*a*b*c*d^2 + a^2* 
d^3)*x)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) + 6*(2*b^2*c 
^2 - a*b*c*d + (2*b^2*c*d - a*b*d^2)*x)*sqrt(b*c^2 - a*c*d)*log((a*c + (2* 
b*c - a*d)*x + 2*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a*x))/(d*x + c)) - (2*b^ 
2*d^3*x^2 - 12*b^2*c^2*d + 9*a*b*c*d^2 - (6*b^2*c*d^2 - 5*a*b*d^3)*x)*sqrt 
(b*x^2 + a*x))/(b*d^5*x + b*c*d^4), 1/4*(12*(2*b^2*c^2 - a*b*c*d + (2*b^2* 
c*d - a*b*d^2)*x)*sqrt(-b*c^2 + a*c*d)*arctan(sqrt(-b*c^2 + a*c*d)*sqrt(b* 
x^2 + a*x)/(b*c*x + a*c)) - 3*(8*b^2*c^3 - 8*a*b*c^2*d + a^2*c*d^2 + (8*b^ 
2*c^2*d - 8*a*b*c*d^2 + a^2*d^3)*x)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*s...
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((b*x^2+a*x)^(3/2)/(d*x+c)^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 [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 719, normalized size of antiderivative = 3.67 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{(c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*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)))*a*b*c*d + 12*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)))*a*b*d**2*x - 24*sqrt(c)*sqrt(a*d - b*c)*atan((s 
qrt(a*d - b*c) - sqrt(d)*sqrt(a + b*x) - sqrt(x)*sqrt(d)*sqrt(b))/(sqrt(c) 
*sqrt(b)))*b**2*c**2 - 24*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**2*c 
*d*x + 12*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)))*a*b*c*d + 12*sqrt(c)*s 
qrt(a*d - b*c)*atan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqr 
t(d)*sqrt(b))/(sqrt(c)*sqrt(b)))*a*b*d**2*x - 24*sqrt(c)*sqrt(a*d - b*c)*a 
tan((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(s 
qrt(c)*sqrt(b)))*b**2*c**2 - 24*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**2*c*d*x + 9*sqrt(x)*sqrt(a + b*x)*a*b*c*d**2 + 5*sqrt(x)*sqrt(a + b*x)* 
a*b*d**3*x - 12*sqrt(x)*sqrt(a + b*x)*b**2*c**2*d - 6*sqrt(x)*sqrt(a + b*x 
)*b**2*c*d**2*x + 2*sqrt(x)*sqrt(a + b*x)*b**2*d**3*x**2 + 3*sqrt(b)*log(( 
sqrt(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*c*d**2 + 3*sqrt(b)*log((sqr 
t(a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a**2*d**3*x - 24*sqrt(b)*log((sqrt( 
a + b*x) + sqrt(x)*sqrt(b))/sqrt(a))*a*b*c**2*d - 24*sqrt(b)*log((sqrt(...