\(\int \frac {1}{(d+e x)^2 (b x+c x^2)^{3/2}} \, dx\) [169]

Optimal result

Integrand size = 21, antiderivative size = 197 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=-\frac {2 c d-3 b e}{b d^2 (c d-b e) \sqrt {b x+c x^2}}-\frac {c \left (4 c^2 d^2-4 b c d e+3 b^2 e^2\right ) x}{b^2 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {e}{d (c d-b e) (d+e x) \sqrt {b x+c x^2}}+\frac {3 e^2 (2 c d-b e) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{d^{5/2} (c d-b e)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\frac {x \left (-\frac {\sqrt {d} (b+c x) \left (4 c^3 d^2 x (d+e x)+b^3 e^2 (2 d+3 e x)+2 b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-4 d^2-2 d e x+3 e^2 x^2\right )\right )}{b^2 (c d-b e)^2 (d+e x)}-\frac {3 e^2 (2 c d-b e) \sqrt {x} (b+c x)^{3/2} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}\right )}{d^{5/2} (x (b+c x))^{3/2}} \] Input:

Integrate[1/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]
 

Output:

(x*(-((Sqrt[d]*(b + c*x)*(4*c^3*d^2*x*(d + e*x) + b^3*e^2*(2*d + 3*e*x) + 
2*b*c^2*d*(d^2 - d*e*x - 2*e^2*x^2) + b^2*c*e*(-4*d^2 - 2*d*e*x + 3*e^2*x^ 
2)))/(b^2*(c*d - b*e)^2*(d + e*x))) - (3*e^2*(2*c*d - b*e)*Sqrt[x]*(b + c* 
x)^(3/2)*ArcTan[(-(e*Sqrt[x]*Sqrt[b + c*x]) + Sqrt[c]*(d + e*x))/(Sqrt[d]* 
Sqrt[-(c*d) + b*e])])/(-(c*d) + b*e)^(5/2)))/(d^(5/2)*(x*(b + c*x))^(3/2))
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1165, 27, 1228, 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.

\(\displaystyle \int \frac {1}{\left (b x+c x^2\right )^{3/2} (d+e x)^2} \, dx\)

\(\Big \downarrow \) 1165

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

Input:

Int[1/((d + e*x)^2*(b*x + c*x^2)^(3/2)),x]
 

Output:

(-2*(b*(c*d - b*e) + c*(2*c*d - b*e)*x))/(b^2*d*(c*d - b*e)*(d + e*x)*Sqrt 
[b*x + c*x^2]) - (e*(((4*c^2*d^2 - 4*b*c*d*e + 3*b^2*e^2)*Sqrt[b*x + c*x^2 
])/(d*(c*d - b*e)*(d + e*x)) - (3*b^2*e*(2*c*d - b*e)*ArcTanh[(b*d + (2*c* 
d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(2*d^(3/2)*(c* 
d - b*e)^(3/2))))/(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 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 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 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 1228

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + 
 b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e 
*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^ 
(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
] && EqQ[Simplify[m + 2*p + 3], 0]
 

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.95

method	result	size
pseudoelliptic	\(-\frac {2 \left (-\frac {3 \sqrt {x \left (c x +b \right )}\, b^{2} e^{2} \left (e x +d \right ) \left (b e -2 c d \right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{2}+\left (c^{2} \left (2 c x +b \right ) d^{3}-2 e c \left (-c^{2} x^{2}+\frac {1}{2} c b x +b^{2}\right ) d^{2}+b \,e^{2} \left (c x +b \right ) \left (-2 c x +b \right ) d +\frac {3 b^{2} e^{3} x \left (c x +b \right )}{2}\right ) \sqrt {d \left (b e -c d \right )}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {d \left (b e -c d \right )}\, d^{2} \left (b e -c d \right )^{2} \left (e x +d \right ) b^{2}}\)	\(187\)
risch	\(-\frac {2 \left (c x +b \right )}{b^{2} d^{2} \sqrt {x \left (c x +b \right )}}-\frac {2 c^{2} \sqrt {c \left (\frac {b}{c}+x \right )^{2}-\left (\frac {b}{c}+x \right ) b}}{b^{2} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )}-\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d^{2} \left (b e -c d \right )^{2} \left (x +\frac {d}{e}\right )}+\frac {3 b \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d^{2} \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}-\frac {3 e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right ) c}{d \left (b e -c d \right )^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\)	\(437\)
default	\(\frac {\frac {e^{2}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {3 \left (b e -2 c d \right ) e \left (-\frac {e^{2}}{d \left (b e -c d \right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {\left (b e -2 c d \right ) e \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{2 d \left (b e -c d \right )}+\frac {4 c \,e^{2} \left (2 c \left (x +\frac {d}{e}\right )+\frac {b e -2 c d}{e}\right )}{d \left (b e -c d \right ) \left (-\frac {4 c d \left (b e -c d \right )}{e^{2}}-\frac {\left (b e -2 c d \right )^{2}}{e^{2}}\right ) \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{2}}\)	\(560\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (181) = 362\).

Time = 0.10 (sec) , antiderivative size = 910, normalized size of antiderivative = 4.62 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

[-1/2*(3*((2*b^2*c^2*d*e^3 - b^3*c*e^4)*x^3 + (2*b^2*c^2*d^2*e^2 + b^3*c*d 
*e^3 - b^4*e^4)*x^2 + (2*b^3*c*d^2*e^2 - b^4*d*e^3)*x)*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)) + 2*(2*b*c^3*d^5 - 6*b^2*c^2*d^4*e + 6*b^3*c*d^3*e^2 - 2*b^4*d^2*e 
^3 + (4*c^4*d^4*e - 8*b*c^3*d^3*e^2 + 7*b^2*c^2*d^2*e^3 - 3*b^3*c*d*e^4)*x 
^2 + (4*c^4*d^5 - 6*b*c^3*d^4*e + 5*b^3*c*d^2*e^3 - 3*b^4*d*e^4)*x)*sqrt(c 
*x^2 + b*x))/((b^2*c^4*d^6*e - 3*b^3*c^3*d^5*e^2 + 3*b^4*c^2*d^4*e^3 - b^5 
*c*d^3*e^4)*x^3 + (b^2*c^4*d^7 - 2*b^3*c^3*d^6*e + 2*b^5*c*d^4*e^3 - b^6*d 
^3*e^4)*x^2 + (b^3*c^3*d^7 - 3*b^4*c^2*d^6*e + 3*b^5*c*d^5*e^2 - b^6*d^4*e 
^3)*x), -(3*((2*b^2*c^2*d*e^3 - b^3*c*e^4)*x^3 + (2*b^2*c^2*d^2*e^2 + b^3* 
c*d*e^3 - b^4*e^4)*x^2 + (2*b^3*c*d^2*e^2 - b^4*d*e^3)*x)*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)) + (2*b*c 
^3*d^5 - 6*b^2*c^2*d^4*e + 6*b^3*c*d^3*e^2 - 2*b^4*d^2*e^3 + (4*c^4*d^4*e 
- 8*b*c^3*d^3*e^2 + 7*b^2*c^2*d^2*e^3 - 3*b^3*c*d*e^4)*x^2 + (4*c^4*d^5 - 
6*b*c^3*d^4*e + 5*b^3*c*d^2*e^3 - 3*b^4*d*e^4)*x)*sqrt(c*x^2 + b*x))/((b^2 
*c^4*d^6*e - 3*b^3*c^3*d^5*e^2 + 3*b^4*c^2*d^4*e^3 - b^5*c*d^3*e^4)*x^3 + 
(b^2*c^4*d^7 - 2*b^3*c^3*d^6*e + 2*b^5*c*d^4*e^3 - b^6*d^3*e^4)*x^2 + (b^3 
*c^3*d^7 - 3*b^4*c^2*d^6*e + 3*b^5*c*d^5*e^2 - b^6*d^4*e^3)*x)]
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{2}}\, dx \] Input:

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

Output:

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

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(e*x+d)^2/(c*x^2+b*x)^(3/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (181) = 362\).

Time = 0.49 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.17 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

1/2*((6*b^2*c*d*e^5*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*sqrt(c 
)*abs(e))) - 3*b^3*e^6*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*sqr 
t(c)*abs(e))) + 8*sqrt(c*d^2 - b*d*e)*c^(5/2)*d^2*e^2*abs(e) - 8*sqrt(c*d^ 
2 - b*d*e)*b*c^(3/2)*d*e^3*abs(e) + 6*sqrt(c*d^2 - b*d*e)*b^2*sqrt(c)*e^4* 
abs(e))*sgn(1/(e*x + d))*sgn(e)/(sqrt(c*d^2 - b*d*e)*b^2*c^2*d^4*abs(e) - 
2*sqrt(c*d^2 - b*d*e)*b^3*c*d^3*e*abs(e) + sqrt(c*d^2 - b*d*e)*b^4*d^2*e^2 
*abs(e)) - 2*((4*c^3*d^2*e^7 - 4*b*c^2*d*e^8 + 3*b^2*c*e^9)/(b^2*c^2*d^4*e 
^5*sgn(1/(e*x + d))*sgn(e) - 2*b^3*c*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + b^4 
*d^2*e^7*sgn(1/(e*x + d))*sgn(e)) - ((4*c^3*d^3*e^8 - 6*b*c^2*d^2*e^9 + 8* 
b^2*c*d*e^10 - 3*b^3*e^11)/(b^2*c^2*d^4*e^5*sgn(1/(e*x + d))*sgn(e) - 2*b^ 
3*c*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + b^4*d^2*e^7*sgn(1/(e*x + d))*sgn(e)) 
 - (b^2*c*d^2*e^11 - b^3*d*e^12)/((b^2*c^2*d^4*e^5*sgn(1/(e*x + d))*sgn(e) 
 - 2*b^3*c*d^3*e^6*sgn(1/(e*x + d))*sgn(e) + b^4*d^2*e^7*sgn(1/(e*x + d))* 
sgn(e))*(e*x + d)*e))/((e*x + d)*e))/sqrt(c - 2*c*d/(e*x + d) + c*d^2/(e*x 
 + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2) - 3*(2*c*d*e^5 - b*e^6)*log(a 
bs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e)*(sqrt(c - 2*c*d/(e*x + d) + c*d 
^2/(e*x + d)^2 + b*e/(e*x + d) - b*d*e/(e*x + d)^2) + sqrt(c*d^2*e^2 - b*d 
*e^3)/((e*x + d)*e))*abs(e)))/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*sqrt( 
c*d^2 - b*d*e)*abs(e)*sgn(1/(e*x + d))*sgn(e)))/e^2
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (c\,x^2+b\,x\right )}^{3/2}\,{\left (d+e\,x\right )}^2} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 919, normalized size of antiderivative = 4.66 \[ \int \frac {1}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*s 
qrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**3*d*e**3*x + 
 3*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*s 
qrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**3*e**4*x**2 
- 6*sqrt(d)*sqrt(b + c*x)*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**2*c*d**2*e* 
*2*x - 6*sqrt(d)*sqrt(b + c*x)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqr 
t(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c*d* 
e**3*x**2 + 3*sqrt(d)*sqrt(b + c*x)*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**3 
*d*e**3*x + 3*sqrt(d)*sqrt(b + c*x)*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**3 
*e**4*x**2 - 6*sqrt(d)*sqrt(b + c*x)*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** 
2*c*d**2*e**2*x - 6*sqrt(d)*sqrt(b + c*x)*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**2*c*d*e**3*x**2 - 4*sqrt(c)*sqrt(b + c*x)*b**2*c*d**3*e**2*x - 4*sqrt 
(c)*sqrt(b + c*x)*b**2*c*d**2*e**3*x**2 + 8*sqrt(c)*sqrt(b + c*x)*b*c**2*d 
**4*e*x + 8*sqrt(c)*sqrt(b + c*x)*b*c**2*d**3*e**2*x**2 - 4*sqrt(c)*sqrt(b 
 + c*x)*c**3*d**5*x - 4*sqrt(c)*sqrt(b + c*x)*c**3*d**4*e*x**2 - 2*sqrt...