\(\int \frac {(a x+b x^2)^{3/2}}{c+d x} \, dx\) [93]

Optimal result

Integrand size = 21, antiderivative size = 214 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{c+d x} \, dx=-\frac {\left (10 a c-\frac {8 b c^2}{d}-\frac {a^2 d}{b}\right ) \sqrt {a x+b x^2}}{8 d^2}-\frac {(2 b c-a d) x \sqrt {a x+b x^2}}{4 d^2}+\frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {(2 b c-a d) \left (8 b^2 c^2-8 a b c d-a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right )}{8 b^{3/2} d^4}+\frac {2 c^{3/2} (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a x+b x^2}}\right )}{d^4} \] Output:

-1/8*(10*a*c-8*b*c^2/d-a^2*d/b)*(b*x^2+a*x)^(1/2)/d^2-1/4*(-a*d+2*b*c)*x*( 
b*x^2+a*x)^(1/2)/d^2+1/3*(b*x^2+a*x)^(3/2)/d-1/8*(-a*d+2*b*c)*(-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^(3/2)/d^4+2*c^(3 
/2)*(-a*d+b*c)^(3/2)*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a*x)^(1/2)) 
/d^4
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.81 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.34 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{c+d x} \, dx=\frac {(x (a+b x))^{3/2} \left (\sqrt {b} d \sqrt {x} \sqrt {a+b x} \left (3 a^2 d^2+2 a b d (-15 c+7 d x)+4 b^2 \left (6 c^2-3 c d x+2 d^2 x^2\right )\right )+48 \sqrt {b} \sqrt {c} (b c-a d) \left (b c-a d-i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d-2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )+48 \sqrt {b} \sqrt {c} (b c-a d) \left (b c-a d+i \sqrt {a} \sqrt {d} \sqrt {b c-a d}\right ) \sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \arctan \left (\frac {\sqrt {-b c+2 a d+2 i \sqrt {a} \sqrt {d} \sqrt {b c-a d}} \sqrt {x}}{\sqrt {c} \left (\sqrt {a}-\sqrt {a+b x}\right )}\right )+6 \left (16 b^3 c^3-24 a b^2 c^2 d+6 a^2 b c d^2+a^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}-\sqrt {a+b x}}\right )\right )}{24 b^{3/2} d^4 x^{3/2} (a+b x)^{3/2}} \] Input:

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

Output:

((x*(a + b*x))^(3/2)*(Sqrt[b]*d*Sqrt[x]*Sqrt[a + b*x]*(3*a^2*d^2 + 2*a*b*d 
*(-15*c + 7*d*x) + 4*b^2*(6*c^2 - 3*c*d*x + 2*d^2*x^2)) + 48*Sqrt[b]*Sqrt[ 
c]*(b*c - a*d)*(b*c - a*d - 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]]*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]))] + 48*Sqrt[b]*Sqrt[c]*(b*c - a*d)*(b*c - a*d + I*Sqrt[a]*S 
qrt[d]*Sqrt[b*c - a*d])*Sqrt[-(b*c) + 2*a*d + (2*I)*Sqrt[a]*Sqrt[d]*Sqrt[b 
*c - a*d]]*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]))] + 6*(16*b^3*c^3 - 24*a 
*b^2*c^2*d + 6*a^2*b*c*d^2 + a^3*d^3)*ArcTanh[(Sqrt[b]*Sqrt[x])/(Sqrt[a] - 
 Sqrt[a + b*x])]))/(24*b^(3/2)*d^4*x^(3/2)*(a + b*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1162, 1231, 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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1231

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {-\frac {\int -\frac {a c \left (8 b^2 c^2-10 a b d c+a^2 d^2\right )+(2 b c-a d) \left (8 b^2 c^2-8 a b d c-a^2 d^2\right ) x}{2 (c+d x) \sqrt {b x^2+a x}}dx}{4 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {\frac {\int \frac {a c \left (8 b^2 c^2-10 a b d c+a^2 d^2\right )+(2 b c-a d) \left (8 b^2 c^2-8 a b d c-a^2 d^2\right ) x}{(c+d x) \sqrt {b x^2+a x}}dx}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {\frac {\frac {(2 b c-a d) \left (-a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a x}}dx}{d}-\frac {16 b c^2 (b c-a d)^2 \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

\(\Big \downarrow \) 1091

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {\frac {\frac {2 (2 b c-a d) \left (-a^2 d^2-8 a b c d+8 b^2 c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a x}}d\frac {x}{\sqrt {b x^2+a x}}}{d}-\frac {16 b c^2 (b c-a d)^2 \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) \left (-a^2 d^2-8 a b c d+8 b^2 c^2\right )}{\sqrt {b} d}-\frac {16 b c^2 (b c-a d)^2 \int \frac {1}{(c+d x) \sqrt {b x^2+a x}}dx}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {\frac {\frac {32 b c^2 (b c-a d)^2 \int \frac {1}{4 c (b c-a d)-\frac {(a c+(2 b c-a d) x)^2}{b x^2+a x}}d\left (-\frac {a c+(2 b c-a d) x}{\sqrt {b x^2+a x}}\right )}{d}+\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) \left (-a^2 d^2-8 a b c d+8 b^2 c^2\right )}{\sqrt {b} d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\left (a x+b x^2\right )^{3/2}}{3 d}-\frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a x+b x^2}}\right ) (2 b c-a d) \left (-a^2 d^2-8 a b c d+8 b^2 c^2\right )}{\sqrt {b} d}-\frac {16 b c^{3/2} (b c-a d)^{3/2} \text {arctanh}\left (\frac {x (2 b c-a d)+a c}{2 \sqrt {c} \sqrt {a x+b x^2} \sqrt {b c-a d}}\right )}{d}}{8 b d^2}-\frac {\sqrt {a x+b x^2} \left (a^2 d^2-2 b d x (2 b c-a d)-10 a b c d+8 b^2 c^2\right )}{4 b d^2}}{2 d}\)

Input:

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

Output:

(a*x + b*x^2)^(3/2)/(3*d) - (-1/4*((8*b^2*c^2 - 10*a*b*c*d + a^2*d^2 - 2*b 
*d*(2*b*c - a*d)*x)*Sqrt[a*x + b*x^2])/(b*d^2) + ((2*(2*b*c - a*d)*(8*b^2* 
c^2 - 8*a*b*c*d - a^2*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a*x + b*x^2]])/(Sqrt[b 
]*d) - (16*b*c^(3/2)*(b*c - a*d)^(3/2)*ArcTanh[(a*c + (2*b*c - a*d)*x)/(2* 
Sqrt[c]*Sqrt[b*c - a*d]*Sqrt[a*x + b*x^2])])/d)/(8*b*d^2))/(2*d)
 

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 1162

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 + 2*p + 1))), x 
] - Simp[p/(e*(m + 2*p + 1))   Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*d - 
b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x 
] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || LtQ[m, 1]) && 
!ILtQ[m + 2*p, 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 = 0.67 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(-\frac {-\frac {d \sqrt {x \left (b x +a \right )}\, \left (8 b^{2} d^{2} x^{2}+14 a b \,d^{2} x -12 b^{2} c x d +3 a^{2} d^{2}-30 a b c d +24 b^{2} c^{2}\right )}{24 b}+\frac {\left (a^{3} d^{3}+6 a^{2} b c \,d^{2}-24 a \,b^{2} c^{2} d +16 b^{3} c^{3}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (b x +a \right )}}{x \sqrt {b}}\right )}{8 b^{\frac {3}{2}}}+\frac {2 \left (a d -b c \right )^{2} c^{2} \arctan \left (\frac {\sqrt {x \left (b x +a \right )}\, c}{x \sqrt {c \left (a d -b c \right )}}\right )}{\sqrt {c \left (a d -b c \right )}}}{d^{4}}\)	\(184\)
risch	\(\frac {\left (8 b^{2} d^{2} x^{2}+14 a b \,d^{2} x -12 b^{2} c x d +3 a^{2} d^{2}-30 a b c d +24 b^{2} c^{2}\right ) x \left (b x +a \right )}{24 b \,d^{3} \sqrt {x \left (b x +a \right )}}-\frac {\frac {\left (a^{3} d^{3}+6 a^{2} b c \,d^{2}-24 a \,b^{2} c^{2} d +16 b^{3} c^{3}\right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right )}{d \sqrt {b}}+\frac {16 c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}}{16 d^{3} b}\)	\(307\)
default	\(\frac {\frac {\left (b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}\right )^{\frac {3}{2}}}{3}+\frac {\left (a d -2 b c \right ) \left (\frac {\left (2 b \left (x +\frac {c}{d}\right )+\frac {a d -2 b c}{d}\right ) \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{4 b}+\frac {\left (-\frac {4 b c \left (a d -b c \right )}{d^{2}}-\frac {\left (a d -2 b c \right )^{2}}{d^{2}}\right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{8 b^{\frac {3}{2}}}\right )}{2 d}-\frac {c \left (a d -b c \right ) \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}+\frac {\left (a d -2 b c \right ) \ln \left (\frac {\frac {a d -2 b c}{2 d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}\right )}{2 d \sqrt {b}}+\frac {c \left (a d -b c \right ) \ln \left (\frac {-\frac {2 c \left (a d -b c \right )}{d^{2}}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}+\frac {\left (a d -2 b c \right ) \left (x +\frac {c}{d}\right )}{d}-\frac {c \left (a d -b c \right )}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {-\frac {c \left (a d -b c \right )}{d^{2}}}}\right )}{d^{2}}}{d}\)	\(549\)

Input:

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

Output:

-1/d^4*(-1/24*d*(x*(b*x+a))^(1/2)*(8*b^2*d^2*x^2+14*a*b*d^2*x-12*b^2*c*d*x 
+3*a^2*d^2-30*a*b*c*d+24*b^2*c^2)/b+1/8*(a^3*d^3+6*a^2*b*c*d^2-24*a*b^2*c^ 
2*d+16*b^3*c^3)/b^(3/2)*arctanh((x*(b*x+a))^(1/2)/x/b^(1/2))+2*(a*d-b*c)^2 
*c^2/(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 = 0.36 (sec) , antiderivative size = 896, normalized size of antiderivative = 4.19 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{c+d x} \, dx =\text {Too large to display} \] Input:

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

Output:

[1/48*(3*(16*b^3*c^3 - 24*a*b^2*c^2*d + 6*a^2*b*c*d^2 + a^3*d^3)*sqrt(b)*l 
og(2*b*x + a - 2*sqrt(b*x^2 + a*x)*sqrt(b)) - 48*(b^3*c^2 - a*b^2*c*d)*sqr 
t(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*(8*b^3*d^3*x^2 + 24*b^3*c^2*d - 30*a*b^2*c*d^2 
 + 3*a^2*b*d^3 - 2*(6*b^3*c*d^2 - 7*a*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/(b^2* 
d^4), -1/48*(96*(b^3*c^2 - a*b^2*c*d)*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*(16*b^3*c^3 - 24*a*b^2*c 
^2*d + 6*a^2*b*c*d^2 + a^3*d^3)*sqrt(b)*log(2*b*x + a - 2*sqrt(b*x^2 + a*x 
)*sqrt(b)) - 2*(8*b^3*d^3*x^2 + 24*b^3*c^2*d - 30*a*b^2*c*d^2 + 3*a^2*b*d^ 
3 - 2*(6*b^3*c*d^2 - 7*a*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/(b^2*d^4), 1/24*(3 
*(16*b^3*c^3 - 24*a*b^2*c^2*d + 6*a^2*b*c*d^2 + a^3*d^3)*sqrt(-b)*arctan(s 
qrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a)) - 24*(b^3*c^2 - a*b^2*c*d)*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)) + (8*b^3*d^3*x^2 + 24*b^3*c^2*d - 30*a*b^2*c*d^2 + 3*a^2* 
b*d^3 - 2*(6*b^3*c*d^2 - 7*a*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/(b^2*d^4), -1/ 
24*(48*(b^3*c^2 - a*b^2*c*d)*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*(16*b^3*c^3 - 24*a*b^2*c^2*d + 6* 
a^2*b*c*d^2 + a^3*d^3)*sqrt(-b)*arctan(sqrt(b*x^2 + a*x)*sqrt(-b)/(b*x + a 
)) - (8*b^3*d^3*x^2 + 24*b^3*c^2*d - 30*a*b^2*c*d^2 + 3*a^2*b*d^3 - 2*(6*b 
^3*c*d^2 - 7*a*b^2*d^3)*x)*sqrt(b*x^2 + a*x))/(b^2*d^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.13 \[ \int \frac {\left (a x+b x^2\right )^{3/2}}{c+d x} \, dx=\frac {-48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c d +48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b x +a}-\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}-48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) a \,b^{2} c d +48 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b x +a}+\sqrt {x}\, \sqrt {d}\, \sqrt {b}}{\sqrt {c}\, \sqrt {b}}\right ) b^{3} c^{2}+3 \sqrt {x}\, \sqrt {b x +a}\, a^{2} b \,d^{3}-30 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} c \,d^{2}+14 \sqrt {x}\, \sqrt {b x +a}\, a \,b^{2} d^{3} x +24 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c^{2} d -12 \sqrt {x}\, \sqrt {b x +a}\, b^{3} c \,d^{2} x +8 \sqrt {x}\, \sqrt {b x +a}\, b^{3} d^{3} x^{2}-3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{3} d^{3}-18 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a^{2} b c \,d^{2}+72 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) a \,b^{2} c^{2} d -48 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b x +a}+\sqrt {x}\, \sqrt {b}}{\sqrt {a}}\right ) b^{3} c^{3}}{24 b^{2} d^{4}} \] Input:

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

Output:

( - 48*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**2*c*d + 48*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)))*b**3*c**2 - 48*sqrt(c)*sqrt(a*d - b*c)*at 
an((sqrt(a*d - b*c) + sqrt(d)*sqrt(a + b*x) + sqrt(x)*sqrt(d)*sqrt(b))/(sq 
rt(c)*sqrt(b)))*a*b**2*c*d + 48*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 + 3*sqrt(x)*sqrt(a + b*x)*a**2*b*d**3 - 30*sqrt(x)*sqrt(a + b*x) 
*a*b**2*c*d**2 + 14*sqrt(x)*sqrt(a + b*x)*a*b**2*d**3*x + 24*sqrt(x)*sqrt( 
a + b*x)*b**3*c**2*d - 12*sqrt(x)*sqrt(a + b*x)*b**3*c*d**2*x + 8*sqrt(x)* 
sqrt(a + b*x)*b**3*d**3*x**2 - 3*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt 
(b))/sqrt(a))*a**3*d**3 - 18*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*a**2*b*c*d**2 + 72*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*a*b**2*c**2*d - 48*sqrt(b)*log((sqrt(a + b*x) + sqrt(x)*sqrt(b)) 
/sqrt(a))*b**3*c**3)/(24*b**2*d**4)