3.5.8 \(\int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx\) [408]

3.5.8.1 Optimal result

Integrand size = 23, antiderivative size = 241 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 e \sqrt {d+e x} \sqrt {b x+c x^2}}{3 c}+\frac {4 \sqrt {-b} (2 c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{3 c^{3/2} \sqrt {d+e x} \sqrt {b x+c x^2}} \]

output

4/3*(-b*e+2*c*d)*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b 
)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/c^(3/2)/(1+e*x/d)^(1/2)/(c*x 
^2+b*x)^(1/2)-2/3*d*(-b*e+c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c 
/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(3/2)/(e*x 
+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*e*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c
 

3.5.8.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {-2 b (b+c x) (d+e x) (2 b e-c (4 d+e x))-4 i b \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 i \sqrt {\frac {b}{c}} c \left (3 c^2 d^2-5 b c d e+2 b^2 e^2\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{3 b c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]

input

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

output

(-2*b*(b + c*x)*(d + e*x)*(2*b*e - c*(4*d + e*x)) - (4*I)*b*Sqrt[b/c]*c*e* 
(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticE[I*Arc 
Sinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + (2*I)*Sqrt[b/c]*c*(3*c^2*d^2 - 5*b 
*c*d*e + 2*b^2*e^2)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[ 
I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])/(3*b*c^2*Sqrt[x*(b + c*x)]*Sqr 
t[d + e*x])
 

3.5.8.3 Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1166, 27, 1269, 1169, 122, 120, 127, 126}

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.

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

\(\Big \downarrow \) 1166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

\(\displaystyle \frac {2 (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx-d (c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 1169

\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {b+c x} (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{\sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 122

\(\displaystyle \frac {\frac {2 \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{\sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 120

\(\displaystyle \frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{\sqrt {b x+c x^2}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 127

\(\displaystyle \frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{\sqrt {b x+c x^2} \sqrt {d+e x}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 126

\(\displaystyle \frac {\frac {4 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} \sqrt {b x+c x^2} \sqrt {d+e x}}}{3 c}+\frac {2 e \sqrt {b x+c x^2} \sqrt {d+e x}}{3 c}\)

input

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

output

(2*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])/(3*c) + ((4*Sqrt[-b]*(2*c*d - b*e)*S 
qrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/ 
Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - ( 
2*Sqrt[-b]*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[1 + (e*x)/d]*Ellip 
ticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*Sqrt[d + e 
*x]*Sqrt[b*x + c*x^2]))/(3*c)
 

3.5.8.3.1 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 120

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- 
b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt 
Q[e, 0] &&  !LtQ[-b/d, 0]
 

rule 122

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] 
 :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) 
)   Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b 
, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 126

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* 
Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & 
& GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
 

rule 127

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x 
_] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x 
]))   Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free 
Q[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])
 

rule 1166

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 
 1))), x] + Simp[1/(c*(m + 2*p + 1))   Int[(d + e*x)^(m - 2)*Simp[c*d^2*(m 
+ 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]* 
(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && If[Ration 
alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat 
icQ[a, b, c, d, e, m, p, x]
 

rule 1169

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> 
 Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2])   Int[(d + e*x)^m/(Sqrt[x]* 
Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq 
Q[m^2, 1/4]
 

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]
 

3.5.8.4 Maple [A] (verified)

Time = 2.08 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.58

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c}+\frac {2 \left (d^{2}-\frac {e b d}{3 c}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (2 d e -\frac {2 e \left (b e +c d \right )}{3 c}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(381\)
default	\(\frac {2 \sqrt {e x +d}\, \sqrt {x \left (c x +b \right )}\, \left (b^{2} d \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) e c -\sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}+2 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{3} e^{2}-6 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b^{2} c d e +4 \sqrt {\frac {c x +b}{b}}\, \sqrt {-\frac {c \left (e x +d \right )}{b e -c d}}\, \sqrt {-\frac {c x}{b}}\, E\left (\sqrt {\frac {c x +b}{b}}, \sqrt {\frac {b e}{b e -c d}}\right ) b \,c^{2} d^{2}+c^{3} e^{2} x^{3}+b \,c^{2} e^{2} x^{2}+c^{3} d e \,x^{2}+b \,c^{2} d e x \right )}{3 x \left (c e \,x^{2}+b e x +c d x +b d \right ) c^{3}}\)	\(460\)

input

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

output

(x*(e*x+d)*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/3/c*e*(c*e*x^ 
3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(d^2-1/3/c*e*b*d)/c*b*((1/c*b+x)*c/b)^(1/ 
2)*((x+d/e)/(-1/c*b+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b* 
d*x)^(1/2)*EllipticF(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))+2* 
(2*d*e-2/3/c*e*(b*e+c*d))/c*b*((1/c*b+x)*c/b)^(1/2)*((x+d/e)/(-1/c*b+d/e)) 
^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-1/c*b+d/e)* 
EllipticE(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))-d/e*EllipticF 
(((1/c*b+x)*c/b)^(1/2),(-1/c*b/(-1/c*b+d/e))^(1/2))))
 

3.5.8.5 Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.16 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {c x^{2} + b x} \sqrt {e x + d} c^{2} e^{2} + {\left (5 \, c^{2} d^{2} - 5 \, b c d e + 2 \, b^{2} e^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) - 6 \, {\left (2 \, c^{2} d e - b c e^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right )\right )}}{9 \, c^{3} e} \]

input

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

output

2/9*(3*sqrt(c*x^2 + b*x)*sqrt(e*x + d)*c^2*e^2 + (5*c^2*d^2 - 5*b*c*d*e + 
2*b^2*e^2)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2) 
/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/ 
(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) - 6*(2*c^2*d*e - b*c*e^2)*sqrt 
(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*( 
2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), weierstr 
assPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 
- 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d 
 + b*e)/(c*e))))/(c^3*e)
 

3.5.8.6 Sympy [F]

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

input

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

output

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

3.5.8.7 Maxima [F]

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

input

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

output

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

3.5.8.8 Giac [F]

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

input

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

output

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

3.5.8.9 Mupad [F(-1)]

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

input

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

output

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