\(\int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx\) [163]

Optimal result

Integrand size = 21, antiderivative size = 163 \[ \int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=-\frac {e \sqrt {b x+c x^2}}{2 d (c d-b e) (d+e x)^2}-\frac {3 e (2 c d-b e) \sqrt {b x+c x^2}}{4 d^2 (c d-b e)^2 (d+e x)}+\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \text {arctanh}\left (\frac {\sqrt {c d-b e} x}{\sqrt {d} \sqrt {b x+c x^2}}\right )}{4 d^{5/2} (c d-b e)^{5/2}} \] Output:

-1/2*e*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^2-3/4*e*(-b*e+2*c*d)*(c*x^2+ 
b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e*x+d)+1/4*(3*b^2*e^2-8*b*c*d*e+8*c^2*d^2)*ar 
ctanh((-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 = 0.85 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.10 \[ \int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} \left (-\frac {\sqrt {d} e \sqrt {x} (b+c x) (2 c d (4 d+3 e x)-b e (5 d+3 e x))}{(c d-b e)^2 (d+e x)^2}-\frac {\left (8 c^2 d^2-8 b c d e+3 b^2 e^2\right ) \sqrt {b+c x} \arctan \left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}\right )}{4 d^{5/2} \sqrt {x (b+c x)}} \] Input:

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

Output:

(Sqrt[x]*(-((Sqrt[d]*e*Sqrt[x]*(b + c*x)*(2*c*d*(4*d + 3*e*x) - b*e*(5*d + 
 3*e*x)))/((c*d - b*e)^2*(d + e*x)^2)) - ((8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e 
^2)*Sqrt[b + c*x]*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)))/(4*d^(5/2)*Sqrt[x*(b 
 + c*x)])
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.20, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1167, 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.

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

\(\Big \downarrow \) 1167

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\left (3 b^2 e^2-8 b c d e+8 c^2 d^2\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{2 d^{3/2} (c d-b e)^{3/2}}-\frac {3 e \sqrt {b x+c x^2} (2 c d-b e)}{d (d+e x) (c d-b e)}}{4 d (c d-b e)}-\frac {e \sqrt {b x+c x^2}}{2 d (d+e x)^2 (c d-b e)}\)

Input:

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

Output:

-1/2*(e*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x)^2) + ((-3*e*(2*c*d - b 
*e)*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*(d + e*x)) + ((8*c^2*d^2 - 8*b*c*d*e 
 + 3*b^2*e^2)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*S 
qrt[b*x + c*x^2])])/(2*d^(3/2)*(c*d - b*e)^(3/2)))/(4*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 1167

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)/((m + 1)*(c*d 
^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2))   Int[ 
(d + e*x)^(m + 1)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, 
 x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m 
, -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp 
lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 0])
 

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 = 0.85 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {-\frac {3 \left (e x +d \right )^{2} \left (b^{2} e^{2}-\frac {8}{3} b c d e +\frac {8}{3} c^{2} d^{2}\right ) \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{4}+\frac {5 e \sqrt {x \left (c x +b \right )}\, \left (-\frac {8 c \,d^{2}}{5}+e \left (-\frac {6 c x}{5}+b \right ) d +\frac {3 x b \,e^{2}}{5}\right ) \sqrt {d \left (b e -c d \right )}}{4}}{\left (b e -c d \right )^{2} \sqrt {d \left (b e -c d \right )}\, d^{2} \left (e x +d \right )^{2}}\)	\(140\)
default	\(\frac {\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{2 d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{2}}+\frac {3 \left (b e -2 c d \right ) e \left (\frac {e^{2} \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )}-\frac {\left (b e -2 c d \right ) e \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}\right )}{4 d \left (b e -c d \right )}-\frac {c \,e^{2} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 d \left (b e -c d \right ) \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}}{e^{3}}\)	\(473\)

Input:

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

Output:

5/4/(b*e-c*d)^2/(d*(b*e-c*d))^(1/2)*(-3/5*(e*x+d)^2*(b^2*e^2-8/3*b*c*d*e+8 
/3*c^2*d^2)*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d))^(1/2))+e*(x*(c*x+b) 
)^(1/2)*(-8/5*c*d^2+e*(-6/5*c*x+b)*d+3/5*x*b*e^2)*(d*(b*e-c*d))^(1/2))/d^2 
/(e*x+d)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 361 vs. \(2 (143) = 286\).

Time = 0.10 (sec) , antiderivative size = 738, normalized size of antiderivative = 4.53 \[ \int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=\left [\frac {{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + {\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \, {\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} x\right )} \sqrt {c d^{2} - b d e} \log \left (\frac {b d + {\left (2 \, c d - b e\right )} x + 2 \, \sqrt {c d^{2} - b d e} \sqrt {c x^{2} + b x}}{e x + d}\right ) - 2 \, {\left (8 \, c^{2} d^{4} e - 13 \, b c d^{3} e^{2} + 5 \, b^{2} d^{2} e^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{8 \, {\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} + {\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}, -\frac {{\left (8 \, c^{2} d^{4} - 8 \, b c d^{3} e + 3 \, b^{2} d^{2} e^{2} + {\left (8 \, c^{2} d^{2} e^{2} - 8 \, b c d e^{3} + 3 \, b^{2} e^{4}\right )} x^{2} + 2 \, {\left (8 \, c^{2} d^{3} e - 8 \, b c d^{2} e^{2} + 3 \, b^{2} d e^{3}\right )} x\right )} \sqrt {-c d^{2} + b d e} \arctan \left (\frac {\sqrt {-c d^{2} + b d e} \sqrt {c x^{2} + b x}}{c d x + b d}\right ) + {\left (8 \, c^{2} d^{4} e - 13 \, b c d^{3} e^{2} + 5 \, b^{2} d^{2} e^{3} + 3 \, {\left (2 \, c^{2} d^{3} e^{2} - 3 \, b c d^{2} e^{3} + b^{2} d e^{4}\right )} x\right )} \sqrt {c x^{2} + b x}}{4 \, {\left (c^{3} d^{8} - 3 \, b c^{2} d^{7} e + 3 \, b^{2} c d^{6} e^{2} - b^{3} d^{5} e^{3} + {\left (c^{3} d^{6} e^{2} - 3 \, b c^{2} d^{5} e^{3} + 3 \, b^{2} c d^{4} e^{4} - b^{3} d^{3} e^{5}\right )} x^{2} + 2 \, {\left (c^{3} d^{7} e - 3 \, b c^{2} d^{6} e^{2} + 3 \, b^{2} c d^{5} e^{3} - b^{3} d^{4} e^{4}\right )} x\right )}}\right ] \] Input:

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

Output:

[1/8*((8*c^2*d^4 - 8*b*c*d^3*e + 3*b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d* 
e^3 + 3*b^2*e^4)*x^2 + 2*(8*c^2*d^3*e - 8*b*c*d^2*e^2 + 3*b^2*d*e^3)*x)*sq 
rt(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*(8*c^2*d^4*e - 13*b*c*d^3*e^2 + 5*b^2*d^2*e^3 
 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)*x)*sqrt(c*x^2 + b*x))/(c^ 
3*d^8 - 3*b*c^2*d^7*e + 3*b^2*c*d^6*e^2 - b^3*d^5*e^3 + (c^3*d^6*e^2 - 3*b 
*c^2*d^5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*x^2 + 2*(c^3*d^7*e - 3*b*c^2 
*d^6*e^2 + 3*b^2*c*d^5*e^3 - b^3*d^4*e^4)*x), -1/4*((8*c^2*d^4 - 8*b*c*d^3 
*e + 3*b^2*d^2*e^2 + (8*c^2*d^2*e^2 - 8*b*c*d*e^3 + 3*b^2*e^4)*x^2 + 2*(8* 
c^2*d^3*e - 8*b*c*d^2*e^2 + 3*b^2*d*e^3)*x)*sqrt(-c*d^2 + b*d*e)*arctan(sq 
rt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x + b*d)) + (8*c^2*d^4*e - 13*b* 
c*d^3*e^2 + 5*b^2*d^2*e^3 + 3*(2*c^2*d^3*e^2 - 3*b*c*d^2*e^3 + b^2*d*e^4)* 
x)*sqrt(c*x^2 + b*x))/(c^3*d^8 - 3*b*c^2*d^7*e + 3*b^2*c*d^6*e^2 - b^3*d^5 
*e^3 + (c^3*d^6*e^2 - 3*b*c^2*d^5*e^3 + 3*b^2*c*d^4*e^4 - b^3*d^3*e^5)*x^2 
 + 2*(c^3*d^7*e - 3*b*c^2*d^6*e^2 + 3*b^2*c*d^5*e^3 - b^3*d^4*e^4)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate(1/(e*x+d)^3/(c*x^2+b*x)^(1/2),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. 483 vs. \(2 (143) = 286\).

Time = 0.19 (sec) , antiderivative size = 483, normalized size of antiderivative = 2.96 \[ \int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx=-\frac {{\left (8 \, c^{2} d^{2} - 8 \, b c d e + 3 \, b^{2} e^{2}\right )} \arctan \left (\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e}}\right )}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt {-c d^{2} + b d e}} - \frac {8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} c^{2} d^{2} e - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c d e^{2} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b^{2} e^{3} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} c^{\frac {5}{2}} d^{3} - 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b c^{\frac {3}{2}} d^{2} e + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} \sqrt {c} d e^{2} + 24 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b c^{2} d^{3} - 20 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{2} c d^{2} e + 5 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} d e^{2} + 6 \, b^{2} c^{\frac {3}{2}} d^{3} - 3 \, b^{3} \sqrt {c} d^{2} e}{4 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} e + 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} d + b d\right )}^{2}} \] Input:

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

Output:

-1/4*(8*c^2*d^2 - 8*b*c*d*e + 3*b^2*e^2)*arctan(((sqrt(c)*x - sqrt(c*x^2 + 
 b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^2*d^4 - 2*b*c*d^3*e + b^2* 
d^2*e^2)*sqrt(-c*d^2 + b*d*e)) - 1/4*(8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3* 
c^2*d^2*e - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*b*c*d*e^2 + 3*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x))^3*b^2*e^3 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*c^(5/ 
2)*d^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*b*c^(3/2)*d^2*e + 9*(sqrt(c) 
*x - sqrt(c*x^2 + b*x))^2*b^2*sqrt(c)*d*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + 
 b*x))*b*c^2*d^3 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x))*b^2*c*d^2*e + 5*(sqr 
t(c)*x - sqrt(c*x^2 + b*x))*b^3*d*e^2 + 6*b^2*c^(3/2)*d^3 - 3*b^3*sqrt(c)* 
d^2*e)/((c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2)*((sqrt(c)*x - sqrt(c*x^2 + b 
*x))^2*e + 2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c)*d + b*d)^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.86 (sec) , antiderivative size = 1950, normalized size of antiderivative = 11.96 \[ \int \frac {1}{(d+e x)^3 \sqrt {b x+c x^2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*sqrt(d)*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**2*e**3 - 12*sqrt(d 
)*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**4*x - 6*sqrt(d)*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**5*x**2 + 28*sqrt(d)*sqrt(b*e - c*d)*atan((sq 
rt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)* 
sqrt(c)))*b**2*c*d**3*e**2 + 56*sqrt(d)*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**3*x + 28*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sq 
rt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b**2*c*d 
*e**4*x**2 - 48*sqrt(d)*sqrt(b*e - c*d)*atan((sqrt(b*e - c*d) - sqrt(e)*sq 
rt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(sqrt(d)*sqrt(c)))*b*c**2*d**4*e - 
96*sqrt(d)*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*c**2*d**3*e**2*x - 48*sqrt( 
d)*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*c**2*d**2*e**3*x**2 + 32*sqrt(d)*sq 
rt(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)))*c**3*d**5 + 64*sqrt(d)*sqrt(b*e - c*d)*ata 
n((sqrt(b*e - c*d) - sqrt(e)*sqrt(b + c*x) - sqrt(x)*sqrt(e)*sqrt(c))/(...