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

Optimal result

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.85 (sec) , antiderivative size = 500, normalized size of antiderivative = 2.34 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {(x (b+c x))^{3/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (3 b^2 e^2+2 b c e (-15 d+7 e x)+4 c^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+48 \sqrt {c} \sqrt {d} (c d-b e) \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+48 \sqrt {c} \sqrt {d} (c d-b e) \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (\sqrt {b}-\sqrt {b+c x}\right )}\right )+6 \left (16 c^3 d^3-24 b c^2 d^2 e+6 b^2 c d e^2+b^3 e^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}-\sqrt {b+c x}}\right )\right )}{24 c^{3/2} e^4 x^{3/2} (b+c x)^{3/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.80 (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.

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

\(\Big \downarrow \) 1162

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

\(\Big \downarrow \) 1231

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1091

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

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

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 = 0.78 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86

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

Input:

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

Output:

-1/e^4*(-1/24*e*(x*(c*x+b))^(1/2)*(8*c^2*e^2*x^2+14*b*c*e^2*x-12*c^2*d*e*x 
+3*b^2*e^2-30*b*c*d*e+24*c^2*d^2)/c+1/8*(b^3*e^3+6*b^2*c*d*e^2-24*b*c^2*d^ 
2*e+16*c^3*d^3)/c^(3/2)*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+2*(b*e-c*d)^2 
*d^2/(d*(b*e-c*d))^(1/2)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2)) 
)
 

Fricas [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 896, normalized size of antiderivative = 4.19 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx =\text {Too large to display} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((c*x^2+b*x)^(3/2)/(e*x+d),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(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.13 \[ \int \frac {\left (b x+c x^2\right )^{3/2}}{d+e x} \, dx=\frac {-48 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b \,c^{2} d e +48 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}-\sqrt {e}\, \sqrt {c x +b}-\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c^{3} d^{2}-48 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) b \,c^{2} d e +48 \sqrt {d}\, \sqrt {b e -c d}\, \mathit {atan} \left (\frac {\sqrt {b e -c d}+\sqrt {e}\, \sqrt {c x +b}+\sqrt {x}\, \sqrt {e}\, \sqrt {c}}{\sqrt {d}\, \sqrt {c}}\right ) c^{3} d^{2}+3 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c \,e^{3}-30 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{2} d \,e^{2}+14 \sqrt {x}\, \sqrt {c x +b}\, b \,c^{2} e^{3} x +24 \sqrt {x}\, \sqrt {c x +b}\, c^{3} d^{2} e -12 \sqrt {x}\, \sqrt {c x +b}\, c^{3} d \,e^{2} x +8 \sqrt {x}\, \sqrt {c x +b}\, c^{3} e^{3} x^{2}-3 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{3} e^{3}-18 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b^{2} c d \,e^{2}+72 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) b \,c^{2} d^{2} e -48 \sqrt {c}\, \mathrm {log}\left (\frac {\sqrt {c x +b}+\sqrt {x}\, \sqrt {c}}{\sqrt {b}}\right ) c^{3} d^{3}}{24 c^{2} e^{4}} \] Input:

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

Output:

( - 48*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c* 
x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**2*d*e + 48*sqrt(d)*s 
qrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqr 
t(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*c**3*d**2 - 48*sqrt(d)*sqrt(b*e - c*d)*at 
an((sqrt(b*e - c*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sq 
rt(d)*sqrt(c)))*b*c**2*d*e + 48*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c 
*d) + sqrt(e)*sqrt(b + c*x) + sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))* 
c**3*d**2 + 3*sqrt(x)*sqrt(b + c*x)*b**2*c*e**3 - 30*sqrt(x)*sqrt(b + c*x) 
*b*c**2*d*e**2 + 14*sqrt(x)*sqrt(b + c*x)*b*c**2*e**3*x + 24*sqrt(x)*sqrt( 
b + c*x)*c**3*d**2*e - 12*sqrt(x)*sqrt(b + c*x)*c**3*d*e**2*x + 8*sqrt(x)* 
sqrt(b + c*x)*c**3*e**3*x**2 - 3*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt 
(c))/sqrt(b))*b**3*e**3 - 18*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c)) 
/sqrt(b))*b**2*c*d*e**2 + 72*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c)) 
/sqrt(b))*b*c**2*d**2*e - 48*sqrt(c)*log((sqrt(b + c*x) + sqrt(x)*sqrt(c)) 
/sqrt(b))*c**3*d**3)/(24*c**2*e**4)