\(\int \frac {1}{(d+e x)^{5/2} (b x+c x^2)^2} \, dx\) [120]

Optimal result

Integrand size = 21, antiderivative size = 277 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=-\frac {e \left (6 c^2 d^2-6 b c d e+5 b^2 e^2\right )}{3 b^2 d^2 (c d-b e)^2 (d+e x)^{3/2}}-\frac {c (2 c d-b e)}{b^2 d (c d-b e) (b+c x) (d+e x)^{3/2}}-\frac {1}{b d x (b+c x) (d+e x)^{3/2}}-\frac {e (2 c d-b e) \left (c^2 d^2-b c d e+5 b^2 e^2\right )}{b^2 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {(4 c d+5 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{7/2}}-\frac {c^{7/2} (4 c d-9 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{7/2}} \] Output:

-1/3*e*(5*b^2*e^2-6*b*c*d*e+6*c^2*d^2)/b^2/d^2/(-b*e+c*d)^2/(e*x+d)^(3/2)- 
c*(-b*e+2*c*d)/b^2/d/(-b*e+c*d)/(c*x+b)/(e*x+d)^(3/2)-1/b/d/x/(c*x+b)/(e*x 
+d)^(3/2)-e*(-b*e+2*c*d)*(5*b^2*e^2-b*c*d*e+c^2*d^2)/b^2/d^3/(-b*e+c*d)^3/ 
(e*x+d)^(1/2)+(5*b*e+4*c*d)*arctanh((e*x+d)^(1/2)/d^(1/2))/b^3/d^(7/2)-c^( 
7/2)*(-9*b*e+4*c*d)*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(- 
b*e+c*d)^(7/2)
 

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \left (-6 c^4 d^3 x (d+e x)^2-3 b c^3 d^2 (d-3 e x) (d+e x)^2+b^4 e^3 \left (3 d^2+20 d e x+15 e^2 x^2\right )+b^2 c^2 d e \left (9 d^3+9 d^2 e x-35 d e^2 x^2-33 e^3 x^3\right )+b^3 c e^2 \left (-9 d^3-41 d^2 e x-13 d e^2 x^2+15 e^3 x^3\right )\right )}{d^3 (c d-b e)^3 x (b+c x) (d+e x)^{3/2}}-\frac {3 c^{7/2} (4 c d-9 b e) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}+\frac {3 (4 c d+5 b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{7/2}}}{3 b^3} \] Input:

Integrate[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]
 

Output:

((b*(-6*c^4*d^3*x*(d + e*x)^2 - 3*b*c^3*d^2*(d - 3*e*x)*(d + e*x)^2 + b^4* 
e^3*(3*d^2 + 20*d*e*x + 15*e^2*x^2) + b^2*c^2*d*e*(9*d^3 + 9*d^2*e*x - 35* 
d*e^2*x^2 - 33*e^3*x^3) + b^3*c*e^2*(-9*d^3 - 41*d^2*e*x - 13*d*e^2*x^2 + 
15*e^3*x^3)))/(d^3*(c*d - b*e)^3*x*(b + c*x)*(d + e*x)^(3/2)) - (3*c^(7/2) 
*(4*c*d - 9*b*e)*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c* 
d) + b*e)^(7/2) + (3*(4*c*d + 5*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(7/ 
2))/(3*b^3)
 

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {1165, 27, 1198, 1198, 1197, 27, 1480, 221}

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.

\(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^2 (d+e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1198

\(\displaystyle -\frac {\frac {\int \frac {(4 c d+5 b e) (c d-b e)^2+c e \left (6 c^2 d^2-6 b c e d+5 b^2 e^2\right ) x}{(d+e x)^{3/2} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\)

\(\Big \downarrow \) 1198

\(\displaystyle -\frac {\frac {\frac {\int \frac {(4 c d+5 b e) (c d-b e)^3+c e (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {\frac {\frac {2 \int \frac {e \left (2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4+c (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\frac {2 e \int \frac {2 c^4 d^4-4 b c^3 e d^3-14 b^2 c^2 e^2 d^2+16 b^3 c e^3 d-5 b^4 e^4+c (2 c d-b e) \left (c^2 d^2-b c e d+5 b^2 e^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {\frac {\frac {2 e \left (\frac {c (c d-b e)^3 (5 b e+4 c d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^4 d^3 (4 c d-9 b e) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\frac {2 e \left (\frac {c^{7/2} d^3 (4 c d-9 b e) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^3 (5 b e+4 c d)}{b \sqrt {d} e}\right )}{d (c d-b e)}+\frac {2 e (2 c d-b e) \left (5 b^2 e^2-b c d e+c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{d (c d-b e)}+\frac {2 e \left (5 b^2 e^2-6 b c d e+6 c^2 d^2\right )}{3 d (d+e x)^{3/2} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 c d-b e)+b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{3/2} (c d-b e)}\)

Input:

Int[1/((d + e*x)^(5/2)*(b*x + c*x^2)^2),x]
 

Output:

-((b*(c*d - b*e) + c*(2*c*d - b*e)*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(3/2)*( 
b*x + c*x^2))) - ((2*e*(6*c^2*d^2 - 6*b*c*d*e + 5*b^2*e^2))/(3*d*(c*d - b* 
e)*(d + e*x)^(3/2)) + ((2*e*(2*c*d - b*e)*(c^2*d^2 - b*c*d*e + 5*b^2*e^2)) 
/(d*(c*d - b*e)*Sqrt[d + e*x]) + (2*e*(-(((c*d - b*e)^3*(4*c*d + 5*b*e)*Ar 
cTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (c^(7/2)*d^3*(4*c*d - 9*b*e 
)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])) 
)/(d*(c*d - b*e)))/(d*(c*d - b*e)))/(2*b^2*d*(c*d - b*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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1165

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

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1198

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

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(2 e^{3} \left (\frac {c^{4} \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{3}}-\frac {1}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b e -4 c d}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{3} e^{3}}\right )\)	\(204\)
default	\(2 e^{3} \left (\frac {c^{4} \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{b^{3} e^{3} \left (b e -c d \right )^{3}}-\frac {1}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 b e -4 c d}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {-\frac {b \sqrt {e x +d}}{2 x}+\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{3} e^{3}}\right )\)	\(204\)
risch	\(-\frac {\sqrt {e x +d}}{d^{3} b^{2} x}-\frac {e \left (-\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {4 b^{2} e^{2} \left (b e -2 c d \right )}{\left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {2 d \,e^{2} b^{2}}{3 \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 c^{4} d^{3} \left (\frac {b e \sqrt {e x +d}}{2 \left (e x +d \right ) c +2 b e -2 c d}+\frac {\left (9 b e -4 c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{3} b e}\right )}{b^{2} d^{3}}\)	\(219\)
pseudoelliptic	\(e^{3} \left (\frac {-\frac {b \sqrt {e x +d}}{x}+\frac {\left (5 b e +4 c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{\sqrt {d}}}{b^{3} d^{3} e^{3}}-\frac {2}{3 d^{2} \left (b e -c d \right )^{2} \left (e x +d \right )^{\frac {3}{2}}}-\frac {4 \left (b e -2 c d \right )}{d^{3} \left (b e -c d \right )^{3} \sqrt {e x +d}}+\frac {\sqrt {e x +d}\, c^{4}}{b^{2} e^{3} \left (c x +b \right ) \left (b e -c d \right )^{3}}+\frac {9 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right ) c^{4}}{\sqrt {c \left (b e -c d \right )}\, b^{2} e^{2} \left (b e -c d \right )^{3}}-\frac {4 \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right ) c^{5} d}{\sqrt {c \left (b e -c d \right )}\, b^{3} e^{3} \left (b e -c d \right )^{3}}\right )\)	\(252\)

Input:

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

2*e^3*(c^4/b^3/e^3/(b*e-c*d)^3*(1/2*b*e*(e*x+d)^(1/2)/((e*x+d)*c+b*e-c*d)+ 
1/2*(9*b*e-4*c*d)/(c*(b*e-c*d))^(1/2)*arctan(c*(e*x+d)^(1/2)/(c*(b*e-c*d)) 
^(1/2)))-1/3/d^2/(b*e-c*d)^2/(e*x+d)^(3/2)-(2*b*e-4*c*d)/d^3/(b*e-c*d)^3/( 
e*x+d)^(1/2)+1/b^3/d^3/e^3*(-1/2*b*(e*x+d)^(1/2)/x+1/2*(5*b*e+4*c*d)/d^(1/ 
2)*arctanh((e*x+d)^(1/2)/d^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 939 vs. \(2 (251) = 502\).

Time = 1.39 (sec) , antiderivative size = 3829, normalized size of antiderivative = 13.82 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\int \frac {1}{x^{2} \left (b + c x\right )^{2} \left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x)**2,x)
 

Output:

Integral(1/(x**2*(b + c*x)**2*(d + e*x)**(5/2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,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 [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 470, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\frac {{\left (4 \, c^{5} d - 9 \, b c^{4} e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{3} d^{3} - 3 \, b^{4} c^{2} d^{2} e + 3 \, b^{5} c d e^{2} - b^{6} e^{3}\right )} \sqrt {-c^{2} d + b c e}} - \frac {2 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{4} d^{3} e - 2 \, \sqrt {e x + d} c^{4} d^{4} e - 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b c^{3} d^{2} e^{2} + 4 \, \sqrt {e x + d} b c^{3} d^{3} e^{2} + 3 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} c^{2} d e^{3} - 6 \, \sqrt {e x + d} b^{2} c^{2} d^{2} e^{3} - {\left (e x + d\right )}^{\frac {3}{2}} b^{3} c e^{4} + 4 \, \sqrt {e x + d} b^{3} c d e^{4} - \sqrt {e x + d} b^{4} e^{5}}{{\left (b^{2} c^{3} d^{6} - 3 \, b^{3} c^{2} d^{5} e + 3 \, b^{4} c d^{4} e^{2} - b^{5} d^{3} e^{3}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} + {\left (e x + d\right )} b e - b d e\right )}} - \frac {2 \, {\left (12 \, {\left (e x + d\right )} c d e^{3} + c d^{2} e^{3} - 6 \, {\left (e x + d\right )} b e^{4} - b d e^{4}\right )}}{3 \, {\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e + 3 \, b^{2} c d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}} - \frac {{\left (4 \, c d + 5 \, b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{3}} \] Input:

integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x, algorithm="giac")
 

Output:

(4*c^5*d - 9*b*c^4*e)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c 
^3*d^3 - 3*b^4*c^2*d^2*e + 3*b^5*c*d*e^2 - b^6*e^3)*sqrt(-c^2*d + b*c*e)) 
- (2*(e*x + d)^(3/2)*c^4*d^3*e - 2*sqrt(e*x + d)*c^4*d^4*e - 3*(e*x + d)^( 
3/2)*b*c^3*d^2*e^2 + 4*sqrt(e*x + d)*b*c^3*d^3*e^2 + 3*(e*x + d)^(3/2)*b^2 
*c^2*d*e^3 - 6*sqrt(e*x + d)*b^2*c^2*d^2*e^3 - (e*x + d)^(3/2)*b^3*c*e^4 + 
 4*sqrt(e*x + d)*b^3*c*d*e^4 - sqrt(e*x + d)*b^4*e^5)/((b^2*c^3*d^6 - 3*b^ 
3*c^2*d^5*e + 3*b^4*c*d^4*e^2 - b^5*d^3*e^3)*((e*x + d)^2*c - 2*(e*x + d)* 
c*d + c*d^2 + (e*x + d)*b*e - b*d*e)) - 2/3*(12*(e*x + d)*c*d*e^3 + c*d^2* 
e^3 - 6*(e*x + d)*b*e^4 - b*d*e^4)/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4 
*e^2 - b^3*d^3*e^3)*(e*x + d)^(3/2)) - (4*c*d + 5*b*e)*arctan(sqrt(e*x + d 
)/sqrt(-d))/(b^3*sqrt(-d)*d^3)
 

Mupad [B] (verification not implemented)

Time = 6.89 (sec) , antiderivative size = 5736, normalized size of antiderivative = 20.71 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

int(1/((b*x + c*x^2)^2*(d + e*x)^(5/2)),x)
 

Output:

((10*e^3*(b*e - 2*c*d)*(d + e*x))/(3*(c*d^2 - b*d*e)^2) - (2*e^3)/(3*(c*d^ 
2 - b*d*e)) + (e*(d + e*x)^2*(15*b^4*e^4 + 6*c^4*d^4 + 64*b^2*c^2*d^2*e^2 
- 12*b*c^3*d^3*e - 58*b^3*c*d*e^3))/(3*b^2*(c*d^2 - b*d*e)^3) + (e*(b*e - 
2*c*d)*(d + e*x)^3*(c^3*d^2 + 5*b^2*c*e^2 - b*c^2*d*e))/(b^2*(c*d^2 - b*d* 
e)^3))/(c*(d + e*x)^(7/2) + (c*d^2 - b*d*e)*(d + e*x)^(3/2) + (b*e - 2*c*d 
)*(d + e*x)^(5/2)) + (atan((((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(( 
d + e*x)^(1/2)*(64*b^6*c^20*d^26*e^2 - 832*b^7*c^19*d^25*e^3 + 4820*b^8*c^ 
18*d^24*e^4 - 16240*b^9*c^17*d^23*e^5 + 34490*b^10*c^16*d^22*e^6 - 45430*b 
^11*c^15*d^21*e^7 + 29414*b^12*c^14*d^20*e^8 + 10670*b^13*c^13*d^19*e^9 - 
39550*b^14*c^12*d^18*e^10 + 25730*b^15*c^11*d^17*e^11 + 19048*b^16*c^10*d^ 
16*e^12 - 53852*b^17*c^9*d^15*e^13 + 55510*b^18*c^8*d^14*e^14 - 35210*b^19 
*c^7*d^13*e^15 + 14830*b^20*c^6*d^12*e^16 - 4082*b^21*c^5*d^11*e^17 + 670* 
b^22*c^4*d^10*e^18 - 50*b^23*c^3*d^9*e^19) + ((-c^7*(b*e - c*d)^7)^(1/2)*( 
9*b*e - 4*c*d)*(8*b^10*c^18*d^28*e^3 - 112*b^11*c^17*d^27*e^4 + 664*b^12*c 
^16*d^26*e^5 - 2080*b^13*c^15*d^25*e^6 + 2996*b^14*c^14*d^24*e^7 + 2528*b^ 
15*c^13*d^23*e^8 - 23056*b^16*c^12*d^22*e^9 + 59312*b^17*c^11*d^21*e^10 - 
95700*b^18*c^10*d^20*e^11 + 109648*b^19*c^9*d^19*e^12 - 92840*b^20*c^8*d^1 
8*e^13 + 58688*b^21*c^7*d^17*e^14 - 27476*b^22*c^6*d^16*e^15 + 9280*b^23*c 
^5*d^15*e^16 - 2144*b^24*c^4*d^14*e^17 + 304*b^25*c^3*d^13*e^18 - 20*b^26* 
c^2*d^12*e^19 - ((-c^7*(b*e - c*d)^7)^(1/2)*(9*b*e - 4*c*d)*(d + e*x)^(...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 2148, normalized size of antiderivative = 7.75 \[ \int \frac {1}{(d+e x)^{5/2} \left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

int(1/(e*x+d)^(5/2)/(c*x^2+b*x)^2,x)
 

Output:

(54*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)* 
sqrt(b*e - c*d)))*b**2*c**3*d**5*e*x + 54*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - 
 c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d**4*e** 
2*x**2 - 24*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/( 
sqrt(c)*sqrt(b*e - c*d)))*b*c**4*d**6*x + 30*sqrt(c)*sqrt(d + e*x)*sqrt(b* 
e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b*c**4*d**5*e*x 
**2 + 54*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqr 
t(c)*sqrt(b*e - c*d)))*b*c**4*d**4*e**2*x**3 - 24*sqrt(c)*sqrt(d + e*x)*sq 
rt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*c**5*d**6* 
x**2 - 24*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sq 
rt(c)*sqrt(b*e - c*d)))*c**5*d**5*e*x**3 - 15*sqrt(d)*sqrt(d + e*x)*log(sq 
rt(d + e*x) - sqrt(d))*b**6*d*e**5*x - 15*sqrt(d)*sqrt(d + e*x)*log(sqrt(d 
 + e*x) - sqrt(d))*b**6*e**6*x**2 + 48*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + 
e*x) - sqrt(d))*b**5*c*d**2*e**4*x + 33*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + 
 e*x) - sqrt(d))*b**5*c*d*e**5*x**2 - 15*sqrt(d)*sqrt(d + e*x)*log(sqrt(d 
+ e*x) - sqrt(d))*b**5*c*e**6*x**3 - 42*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + 
 e*x) - sqrt(d))*b**4*c**2*d**3*e**3*x + 6*sqrt(d)*sqrt(d + e*x)*log(sqrt( 
d + e*x) - sqrt(d))*b**4*c**2*d**2*e**4*x**2 + 48*sqrt(d)*sqrt(d + e*x)*lo 
g(sqrt(d + e*x) - sqrt(d))*b**4*c**2*d*e**5*x**3 - 12*sqrt(d)*sqrt(d + e*x 
)*log(sqrt(d + e*x) - sqrt(d))*b**3*c**3*d**4*e**2*x - 54*sqrt(d)*sqrt(...