\(\int \frac {(d+e x)^{3/2}}{(b x+c x^2)^{3/2}} \, dx\) [215]

Optimal result

Integrand size = 23, antiderivative size = 268 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {2 (c d-b e) \sqrt {d+e x}}{b c \sqrt {b x+c x^2}}-\frac {2 (2 c d-b e) \sqrt {d+e x}}{b c \sqrt {b x+c x^2}}-\frac {2 (2 c d-b e) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{b^{3/2} \sqrt {c} \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}+\frac {2 e \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{\sqrt {b} \sqrt {c} \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 9.79 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.78 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {-2 i \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+2 (c d-b e) \left (b (d+e x)-i \sqrt {\frac {b}{c}} c e \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{b^2 c \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:

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

Output:

((-2*I)*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*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + 2*(c*d - b*e 
)*(b*(d + e*x) - I*Sqrt[b/c]*c*e*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)]))/(b^2*c*Sqrt[x*(b 
 + c*x)]*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.97, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {1164, 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.

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

\(\Big \downarrow \) 1164

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1169

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

\(\Big \downarrow \) 122

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

\(\Big \downarrow \) 120

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

\(\Big \downarrow \) 127

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

\(\Big \downarrow \) 126

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

Input:

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

Output:

(-2*Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(b^2*Sqrt[b*x + c*x^2]) + (e*(( 
2*Sqrt[-b]*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE 
[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sqrt[c]*e*Sqrt[1 + (e* 
x)/d]*Sqrt[b*x + c*x^2]) - (4*Sqrt[-b]*d*(c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x 
)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e) 
/(c*d)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/b^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 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 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[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]
 

Maple [A] (verified)

Time = 2.18 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {2 \left (\sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{2}-d^{2} b \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) e c -\sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} d \,e^{2}+3 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b c \,d^{2} e -2 \sqrt {\frac {e x +d}{d}}\, \sqrt {\frac {e \left (c x +b \right )}{b e -c d}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticE}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) c^{2} d^{3}+e^{3} x^{2} c b -2 d \,e^{2} c^{2} x^{2}-2 d^{2} e \,c^{2} x -d^{2} e b c \right ) \sqrt {x \left (c x +b \right )}}{x \left (c x +b \right ) e \,b^{2} c \sqrt {e x +d}}\)	\(454\)
elliptic	\(\frac {\sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 \left (c e \,x^{2}+c d x \right ) \left (b e -c d \right )}{b^{2} c \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) d}{b^{2} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (\frac {e^{2}}{c}-\frac {\left (b e -c d \right )^{2}}{c \,b^{2}}-\frac {d \left (b e -c d \right )}{b^{2}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {\left (b e -c d \right ) e}{b^{2}}+\frac {d e c}{b^{2}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(476\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.69 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left ({\left (2 \, c^{3} d^{2} - 2 \, b c^{2} d e - b^{2} c e^{2}\right )} x^{2} + {\left (2 \, b c^{2} d^{2} - 2 \, b^{2} c d e - b^{3} e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left ({\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} x^{2} + {\left (2 \, b c^{2} d e - b^{2} c e^{2}\right )} x\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) + 3 \, {\left (b c^{2} d e + {\left (2 \, c^{3} d e - b c^{2} e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{3 \, {\left (b^{2} c^{3} e x^{2} + b^{3} c^{2} e x\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{3/2}} \, dx=\frac {-4 \sqrt {e x +d}\, \sqrt {c x +b}\, b d +2 \sqrt {e x +d}\, \sqrt {c x +b}\, b e x -2 \sqrt {e x +d}\, \sqrt {c x +b}\, c d x +3 \sqrt {x}\, \left (\int \frac {\sqrt {e x +d}\, \sqrt {c x +b}}{\sqrt {x}\, b^{2} d +\sqrt {x}\, b^{2} e x +2 \sqrt {x}\, b c d x +2 \sqrt {x}\, b c e \,x^{2}+\sqrt {x}\, c^{2} d \,x^{2}+\sqrt {x}\, c^{2} e \,x^{3}}d x \right ) b^{3} d e -3 \sqrt {x}\, \left (\int \frac {\sqrt {e x +d}\, \sqrt {c x +b}}{\sqrt {x}\, b^{2} d +\sqrt {x}\, b^{2} e x +2 \sqrt {x}\, b c d x +2 \sqrt {x}\, b c e \,x^{2}+\sqrt {x}\, c^{2} d \,x^{2}+\sqrt {x}\, c^{2} e \,x^{3}}d x \right ) b^{2} c \,d^{2}+3 \sqrt {x}\, \left (\int \frac {\sqrt {e x +d}\, \sqrt {c x +b}}{\sqrt {x}\, b^{2} d +\sqrt {x}\, b^{2} e x +2 \sqrt {x}\, b c d x +2 \sqrt {x}\, b c e \,x^{2}+\sqrt {x}\, c^{2} d \,x^{2}+\sqrt {x}\, c^{2} e \,x^{3}}d x \right ) b^{2} c d e x -3 \sqrt {x}\, \left (\int \frac {\sqrt {e x +d}\, \sqrt {c x +b}}{\sqrt {x}\, b^{2} d +\sqrt {x}\, b^{2} e x +2 \sqrt {x}\, b c d x +2 \sqrt {x}\, b c e \,x^{2}+\sqrt {x}\, c^{2} d \,x^{2}+\sqrt {x}\, c^{2} e \,x^{3}}d x \right ) b \,c^{2} d^{2} x -\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c^{2} e \,x^{3}+2 b c e \,x^{2}+c^{2} d \,x^{2}+b^{2} e x +2 b c d x +b^{2} d}d x \right ) b^{2} c \,e^{2}+\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c^{2} e \,x^{3}+2 b c e \,x^{2}+c^{2} d \,x^{2}+b^{2} e x +2 b c d x +b^{2} d}d x \right ) b \,c^{2} d e -\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c^{2} e \,x^{3}+2 b c e \,x^{2}+c^{2} d \,x^{2}+b^{2} e x +2 b c d x +b^{2} d}d x \right ) b \,c^{2} e^{2} x +\sqrt {x}\, \left (\int \frac {\sqrt {x}\, \sqrt {e x +d}\, \sqrt {c x +b}\, x}{c^{2} e \,x^{3}+2 b c e \,x^{2}+c^{2} d \,x^{2}+b^{2} e x +2 b c d x +b^{2} d}d x \right ) c^{3} d e x}{2 \sqrt {x}\, b^{2} \left (c x +b \right )} \] Input:

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

Output:

( - 4*sqrt(d + e*x)*sqrt(b + c*x)*b*d + 2*sqrt(d + e*x)*sqrt(b + c*x)*b*e* 
x - 2*sqrt(d + e*x)*sqrt(b + c*x)*c*d*x + 3*sqrt(x)*int((sqrt(d + e*x)*sqr 
t(b + c*x))/(sqrt(x)*b**2*d + sqrt(x)*b**2*e*x + 2*sqrt(x)*b*c*d*x + 2*sqr 
t(x)*b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + sqrt(x)*c**2*e*x**3),x)*b**3*d*e - 
 3*sqrt(x)*int((sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**2*d + sqrt(x)*b** 
2*e*x + 2*sqrt(x)*b*c*d*x + 2*sqrt(x)*b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + s 
qrt(x)*c**2*e*x**3),x)*b**2*c*d**2 + 3*sqrt(x)*int((sqrt(d + e*x)*sqrt(b + 
 c*x))/(sqrt(x)*b**2*d + sqrt(x)*b**2*e*x + 2*sqrt(x)*b*c*d*x + 2*sqrt(x)* 
b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + sqrt(x)*c**2*e*x**3),x)*b**2*c*d*e*x - 
3*sqrt(x)*int((sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**2*d + sqrt(x)*b**2 
*e*x + 2*sqrt(x)*b*c*d*x + 2*sqrt(x)*b*c*e*x**2 + sqrt(x)*c**2*d*x**2 + sq 
rt(x)*c**2*e*x**3),x)*b*c**2*d**2*x - sqrt(x)*int((sqrt(x)*sqrt(d + e*x)*s 
qrt(b + c*x)*x)/(b**2*d + b**2*e*x + 2*b*c*d*x + 2*b*c*e*x**2 + c**2*d*x** 
2 + c**2*e*x**3),x)*b**2*c*e**2 + sqrt(x)*int((sqrt(x)*sqrt(d + e*x)*sqrt( 
b + c*x)*x)/(b**2*d + b**2*e*x + 2*b*c*d*x + 2*b*c*e*x**2 + c**2*d*x**2 + 
c**2*e*x**3),x)*b*c**2*d*e - sqrt(x)*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c 
*x)*x)/(b**2*d + b**2*e*x + 2*b*c*d*x + 2*b*c*e*x**2 + c**2*d*x**2 + c**2* 
e*x**3),x)*b*c**2*e**2*x + sqrt(x)*int((sqrt(x)*sqrt(d + e*x)*sqrt(b + c*x 
)*x)/(b**2*d + b**2*e*x + 2*b*c*d*x + 2*b*c*e*x**2 + c**2*d*x**2 + c**2*e* 
x**3),x)*c**3*d*e*x)/(2*sqrt(x)*b**2*(b + c*x))