\(\int \frac {(d+e x)^{3/2}}{(b x+c x^2)^{5/2}} \, dx\) [223]

Optimal result

Integrand size = 23, antiderivative size = 329 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 (c d-b e) \sqrt {d+e x}}{3 b c \left (b x+c x^2\right )^{3/2}}+\frac {2 (8 c d-5 b e) \sqrt {d+e x}}{3 b^3 \sqrt {b x+c x^2}}-\frac {2 (2 c d-b e) \sqrt {d+e x}}{3 b^2 c x \sqrt {b x+c x^2}}+\frac {16 \sqrt {c} (2 c d-b e) \sqrt {x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|1-\frac {b e}{c d}\right )}{3 b^{7/2} \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}}-\frac {2 e (8 c d-3 b e) \sqrt {x} \sqrt {d+e x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),1-\frac {b e}{c d}\right )}{3 b^{5/2} \sqrt {c} d \sqrt {\frac {b (d+e x)}{d (b+c x)}} \sqrt {b x+c x^2}} \] Output:

2/3*(-b*e+c*d)*(e*x+d)^(1/2)/b/c/(c*x^2+b*x)^(3/2)+2/3*(-5*b*e+8*c*d)*(e*x 
+d)^(1/2)/b^3/(c*x^2+b*x)^(1/2)-2/3*(-b*e+2*c*d)*(e*x+d)^(1/2)/b^2/c/x/(c* 
x^2+b*x)^(1/2)+16/3*c^(1/2)*(-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^(7/2)/(b*(e*x+ 
d)/d/(c*x+b))^(1/2)/(c*x^2+b*x)^(1/2)-2/3*e*(-3*b*e+8*c*d)*x^(1/2)*(e*x+d) 
^(1/2)*InverseJacobiAM(arctan(c^(1/2)*x^(1/2)/b^(1/2)),(1-b*e/c/d)^(1/2))/ 
b^(5/2)/c^(1/2)/d/(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 = 18.79 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.88 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (-8 (2 c d-b e) x (b+c x) (d+e x)+\frac {(d+e x) \left (16 c^3 d x^3+b^2 c x (6 d-13 e x)-8 b c^2 x^2 (-3 d+e x)-b^3 (d+4 e x)\right )}{b+c x}+8 i \sqrt {\frac {b}{c}} c e (-2 c d+b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{5/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )+i \sqrt {\frac {b}{c}} c e (8 c d-5 b e) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{5/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )}{3 b^4 x \sqrt {x (b+c x)} \sqrt {d+e x}} \] Input:

Integrate[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]
 

Output:

(2*(-8*(2*c*d - b*e)*x*(b + c*x)*(d + e*x) + ((d + e*x)*(16*c^3*d*x^3 + b^ 
2*c*x*(6*d - 13*e*x) - 8*b*c^2*x^2*(-3*d + e*x) - b^3*(d + 4*e*x)))/(b + c 
*x) + (8*I)*Sqrt[b/c]*c*e*(-2*c*d + b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x 
)]*x^(5/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] + I*Sqrt[b 
/c]*c*e*(8*c*d - 5*b*e)*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(5/2)*Ellipt 
icF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)]))/(3*b^4*x*Sqrt[x*(b + c*x) 
]*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1164, 27, 1235, 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.

\(\displaystyle \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {2 \int \frac {d (8 c d-5 b e)+3 e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {d (8 c d-5 b e)+3 e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x\right )^{3/2}}dx}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {-\frac {2 \int -\frac {d e (b (8 c d-3 b e) (c d-b e)+8 c (2 c d-b e) x (c d-b e))}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2 d (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {e \int \frac {b (8 c d-3 b e) (c d-b e)+8 c (2 c d-b e) x (c d-b e)}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {e \left (\frac {8 c (c d-b e) (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {(4 c d-3 b e) (c d-b e) (4 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 1169

\(\displaystyle -\frac {\frac {e \left (\frac {8 c \sqrt {x} \sqrt {b+c x} (c d-b e) (2 c d-b e) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x} (4 c d-3 b e) (c d-b e) (4 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 122

\(\displaystyle -\frac {\frac {e \left (\frac {8 c \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-b e) (2 c d-b e) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {b+c x} (4 c d-3 b e) (c d-b e) (4 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 120

\(\displaystyle -\frac {\frac {e \left (\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-b e) (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {b+c x} (4 c d-3 b e) (c d-b e) (4 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 127

\(\displaystyle -\frac {\frac {e \left (\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-b e) (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (4 c d-3 b e) (c d-b e) (4 c d-b e) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

\(\Big \downarrow \) 126

\(\displaystyle -\frac {\frac {e \left (\frac {16 \sqrt {-b} \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} (c d-b e) (2 c d-b e) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (4 c d-3 b e) (c d-b e) (4 c d-b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{b^2 (c d-b e)}-\frac {2 \sqrt {d+e x} (8 c x (2 c d-b e) (c d-b e)+b (8 c d-5 b e) (c d-b e))}{b^2 \sqrt {b x+c x^2} (c d-b e)}}{3 b^2}-\frac {2 \sqrt {d+e x} (x (2 c d-b e)+b d)}{3 b^2 \left (b x+c x^2\right )^{3/2}}\)

Input:

Int[(d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x]
 

Output:

(-2*Sqrt[d + e*x]*(b*d + (2*c*d - b*e)*x))/(3*b^2*(b*x + c*x^2)^(3/2)) - ( 
(-2*Sqrt[d + e*x]*(b*(8*c*d - 5*b*e)*(c*d - b*e) + 8*c*(c*d - b*e)*(2*c*d 
- b*e)*x))/(b^2*(c*d - b*e)*Sqrt[b*x + c*x^2]) + (e*((16*Sqrt[-b]*Sqrt[c]* 
(c*d - b*e)*(2*c*d - b*e)*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*Elliptic 
E[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(e*Sqrt[1 + (e*x)/d]*S 
qrt[b*x + c*x^2]) - (2*Sqrt[-b]*(4*c*d - 3*b*e)*(c*d - b*e)*(4*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*(c*d - b*e)))/(3*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 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(569\) vs. \(2(284)=568\).

Time = 4.02 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.73

method	result	size
elliptic	\(\frac {\sqrt {\left (c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 \left (b e -c d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} c \left (\frac {b}{c}+x \right )^{2}}-\frac {8 \left (c e \,x^{2}+c d x \right ) \left (b e -2 c d \right )}{3 b^{4} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 d \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 b^{3} x^{2}}-\frac {8 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (b e -2 c d \right )}{3 b^{4} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (-\frac {e \left (b e -c d \right )}{3 b^{3}}+\frac {4 \left (b e -c d \right ) \left (b e -2 c d \right )}{3 b^{4}}+\frac {4 c d \left (b e -2 c d \right )}{3 b^{4}}-\frac {c d e}{3 b^{3}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{e \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {16 c \left (b e -2 c d \right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {\frac {b}{c}+x}{-\frac {d}{e}+\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}+\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )-\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}+\frac {b}{c}\right )}}\right )}{c}\right )}{3 b^{4} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(570\)
default	\(-\frac {2 \left (5 x^{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 {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b^{2} c d \,e^{2}-8 x^{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 {EllipticF}\left (\sqrt {\frac {e x +d}{d}}, \sqrt {-\frac {d c}{b e -c d}}\right ) b \,c^{2} d^{2} e -8 x^{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 ) b^{2} c d \,e^{2}+24 x^{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 ) b \,c^{2} d^{2} e -16 x^{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^{3} d^{3}+5 x \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^{3} d \,e^{2}-8 x \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} c \,d^{2} e -8 x \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^{3} d \,e^{2}+24 x \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} c \,d^{2} e -16 x \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^{2} d^{3}+8 b \,c^{2} e^{3} x^{4}-16 x^{4} c^{3} d \,e^{2}+13 b^{2} c \,e^{3} x^{3}-16 b \,c^{2} d \,e^{2} x^{3}-16 c^{3} d^{2} e \,x^{3}+4 b^{3} e^{3} x^{2}+7 b^{2} c d \,e^{2} x^{2}-24 b \,c^{2} d^{2} e \,x^{2}+5 b^{3} d \,e^{2} x -6 b^{2} c \,d^{2} e x +b^{3} d^{2} e \right ) \sqrt {x \left (c x +b \right )}}{3 x^{2} e \,b^{4} \left (c x +b \right )^{2} \sqrt {e x +d}}\)	\(948\)

Input:

int((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

((c*x+b)*x*(e*x+d))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(-2/3*(b*e-c*d)/ 
b^3/c*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(b/c+x)^2-8/3*(c*e*x^2+c*d*x)/ 
b^4*(b*e-2*c*d)/((b/c+x)*(c*e*x^2+c*d*x))^(1/2)-2/3*d/b^3*(c*e*x^3+b*e*x^2 
+c*d*x^2+b*d*x)^(1/2)/x^2-8/3*(c*e*x^2+b*e*x+c*d*x+b*d)/b^4*(b*e-2*c*d)/(x 
*(c*e*x^2+b*e*x+c*d*x+b*d))^(1/2)+2*(-1/3*e*(b*e-c*d)/b^3+4/3*(b*e-c*d)*(b 
*e-2*c*d)/b^4+4/3*c*d/b^4*(b*e-2*c*d)-1/3/b^3*c*d*e)*d/e*((x+d/e)/d*e)^(1/ 
2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d* 
x)^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))+16/3*c*(b* 
e-2*c*d)/b^4*d*((x+d/e)/d*e)^(1/2)*((b/c+x)/(-d/e+b/c))^(1/2)*(-e*x/d)^(1/ 
2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*((-d/e+b/c)*EllipticE(((x+d/e)/d* 
e)^(1/2),(-d/e/(-d/e+b/c))^(1/2))-b/c*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/ 
(-d/e+b/c))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (284) = 568\).

Time = 0.10 (sec) , antiderivative size = 581, normalized size of antiderivative = 1.77 \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (16 \, c^{4} d^{2} - 16 \, b c^{3} d e + b^{2} c^{2} e^{2}\right )} x^{4} + 2 \, {\left (16 \, b c^{3} d^{2} - 16 \, b^{2} c^{2} d e + b^{3} c e^{2}\right )} x^{3} + {\left (16 \, b^{2} c^{2} d^{2} - 16 \, b^{3} c d e + b^{4} e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 24 \, {\left ({\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} x^{4} + 2 \, {\left (2 \, b c^{3} d e - b^{2} c^{2} e^{2}\right )} x^{3} + {\left (2 \, b^{2} c^{2} d e - b^{3} c e^{2}\right )} x^{2}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (b^{3} c d e - 8 \, {\left (2 \, c^{4} d e - b c^{3} e^{2}\right )} x^{3} - {\left (24 \, b c^{3} d e - 13 \, b^{2} c^{2} e^{2}\right )} x^{2} - 2 \, {\left (3 \, b^{2} c^{2} d e - 2 \, b^{3} c e^{2}\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{9 \, {\left (b^{4} c^{3} e x^{4} + 2 \, b^{5} c^{2} e x^{3} + b^{6} c e x^{2}\right )}} \] Input:

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")
 

Output:

2/9*(((16*c^4*d^2 - 16*b*c^3*d*e + b^2*c^2*e^2)*x^4 + 2*(16*b*c^3*d^2 - 16 
*b^2*c^2*d*e + b^3*c*e^2)*x^3 + (16*b^2*c^2*d^2 - 16*b^3*c*d*e + b^4*e^2)* 
x^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)) + 24*((2*c^4*d*e - b*c^3*e^2)*x^4 + 
2*(2*b*c^3*d*e - b^2*c^2*e^2)*x^3 + (2*b^2*c^2*d*e - b^3*c*e^2)*x^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), weierstra 
ssPInverse(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^3*c*d*e - 8*(2*c^4*d*e - b*c^3*e^2)*x^3 - (24*b*c^3* 
d*e - 13*b^2*c^2*e^2)*x^2 - 2*(3*b^2*c^2*d*e - 2*b^3*c*e^2)*x)*sqrt(c*x^2 
+ b*x)*sqrt(e*x + d))/(b^4*c^3*e*x^4 + 2*b^5*c^2*e*x^3 + b^6*c*e*x^2)
 

Sympy [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {3}{2}}}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)
 

Output:

Integral((d + e*x)**(3/2)/(x*(b + c*x))**(5/2), x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2), x)
 

Giac [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}}}{{\left (c x^{2} + b x\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((e*x + d)^(3/2)/(c*x^2 + b*x)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{3/2}}{{\left (c\,x^2+b\,x\right )}^{5/2}} \,d x \] Input:

int((d + e*x)^(3/2)/(b*x + c*x^2)^(5/2),x)
 

Output:

int((d + e*x)^(3/2)/(b*x + c*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^{3/2}}{\left (b x+c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:

int((e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x)
 

Output:

( - 2*sqrt(d + e*x)*sqrt(b + c*x)*d + 4*sqrt(x)*int((sqrt(d + e*x)*sqrt(b 
+ c*x))/(sqrt(x)*b**4*d*e*x + sqrt(x)*b**4*e**2*x**2 + 3*sqrt(x)*b**3*c*d* 
*2*x + 6*sqrt(x)*b**3*c*d*e*x**2 + 3*sqrt(x)*b**3*c*e**2*x**3 + 9*sqrt(x)* 
b**2*c**2*d**2*x**2 + 12*sqrt(x)*b**2*c**2*d*e*x**3 + 3*sqrt(x)*b**2*c**2* 
e**2*x**4 + 9*sqrt(x)*b*c**3*d**2*x**3 + 10*sqrt(x)*b*c**3*d*e*x**4 + sqrt 
(x)*b*c**3*e**2*x**5 + 3*sqrt(x)*c**4*d**2*x**4 + 3*sqrt(x)*c**4*d*e*x**5) 
,x)*b**4*d*e**2*x + 6*sqrt(x)*int((sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b 
**4*d*e*x + sqrt(x)*b**4*e**2*x**2 + 3*sqrt(x)*b**3*c*d**2*x + 6*sqrt(x)*b 
**3*c*d*e*x**2 + 3*sqrt(x)*b**3*c*e**2*x**3 + 9*sqrt(x)*b**2*c**2*d**2*x** 
2 + 12*sqrt(x)*b**2*c**2*d*e*x**3 + 3*sqrt(x)*b**2*c**2*e**2*x**4 + 9*sqrt 
(x)*b*c**3*d**2*x**3 + 10*sqrt(x)*b*c**3*d*e*x**4 + sqrt(x)*b*c**3*e**2*x* 
*5 + 3*sqrt(x)*c**4*d**2*x**4 + 3*sqrt(x)*c**4*d*e*x**5),x)*b**3*c*d**2*e* 
x + 8*sqrt(x)*int((sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**4*d*e*x + sqrt 
(x)*b**4*e**2*x**2 + 3*sqrt(x)*b**3*c*d**2*x + 6*sqrt(x)*b**3*c*d*e*x**2 + 
 3*sqrt(x)*b**3*c*e**2*x**3 + 9*sqrt(x)*b**2*c**2*d**2*x**2 + 12*sqrt(x)*b 
**2*c**2*d*e*x**3 + 3*sqrt(x)*b**2*c**2*e**2*x**4 + 9*sqrt(x)*b*c**3*d**2* 
x**3 + 10*sqrt(x)*b*c**3*d*e*x**4 + sqrt(x)*b*c**3*e**2*x**5 + 3*sqrt(x)*c 
**4*d**2*x**4 + 3*sqrt(x)*c**4*d*e*x**5),x)*b**3*c*d*e**2*x**2 - 18*sqrt(x 
)*int((sqrt(d + e*x)*sqrt(b + c*x))/(sqrt(x)*b**4*d*e*x + sqrt(x)*b**4*e** 
2*x**2 + 3*sqrt(x)*b**3*c*d**2*x + 6*sqrt(x)*b**3*c*d*e*x**2 + 3*sqrt(x...