\(\int \frac {\sqrt {d+e x}}{x^3 (a+b x+c x^2)} \, dx\) [65]

Optimal result

Integrand size = 25, antiderivative size = 462 \[ \int \frac {\sqrt {d+e x}}{x^3 \left (a+b x+c x^2\right )} \, dx=-\frac {\sqrt {d+e x}}{2 a x^2}+\frac {(4 b d-a e) \sqrt {d+e x}}{4 a^2 d x}-\frac {\left (8 b^2 d^2-4 a b d e-a \left (8 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 a^3 d^{3/2}}+\frac {\sqrt {2} \sqrt {c} \left (b^3 d-a c \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 \left (\sqrt {b^2-4 a c} d-a e\right )-a b \left (3 c d+\sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}-\frac {\sqrt {2} \sqrt {c} \left (b^3 d-b^2 \left (\sqrt {b^2-4 a c} d+a e\right )+a c \left (\sqrt {b^2-4 a c} d+2 a e\right )-a b \left (3 c d-\sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{a^3 \sqrt {b^2-4 a c} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \] Output:

-1/2*(e*x+d)^(1/2)/a/x^2+1/4*(-a*e+4*b*d)*(e*x+d)^(1/2)/a^2/d/x-1/4*(8*b^2 
*d^2-4*a*b*d*e-a*(a*e^2+8*c*d^2))*arctanh((e*x+d)^(1/2)/d^(1/2))/a^3/d^(3/ 
2)+2^(1/2)*c^(1/2)*(b^3*d-a*c*((-4*a*c+b^2)^(1/2)*d-2*a*e)+b^2*((-4*a*c+b^ 
2)^(1/2)*d-a*e)-a*b*(3*c*d+(-4*a*c+b^2)^(1/2)*e))*arctanh(2^(1/2)*c^(1/2)* 
(e*x+d)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2))/a^3/(-4*a*c+b^2)^(1/ 
2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)-2^(1/2)*c^(1/2)*(b^3*d-b^2*((-4* 
a*c+b^2)^(1/2)*d+a*e)+a*c*((-4*a*c+b^2)^(1/2)*d+2*a*e)-a*b*(3*c*d-(-4*a*c+ 
b^2)^(1/2)*e))*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-(b+(-4*a*c+b^2 
)^(1/2))*e)^(1/2))/a^3/(-4*a*c+b^2)^(1/2)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e) 
^(1/2)
 

Mathematica [A] (verified)

Time = 2.76 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {d+e x}}{x^3 \left (a+b x+c x^2\right )} \, dx=\frac {\frac {a \sqrt {d+e x} (4 b d x-a (2 d+e x))}{d x^2}+\frac {4 \sqrt {2} \sqrt {c} \left (-b^3 d+a c \left (\sqrt {b^2-4 a c} d-2 a e\right )+b^2 \left (-\sqrt {b^2-4 a c} d+a e\right )+a b \left (3 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+b e-\sqrt {b^2-4 a c} e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b-\sqrt {b^2-4 a c}\right ) e}}+\frac {4 \sqrt {2} \sqrt {c} \left (b^3 d-b^2 \left (\sqrt {b^2-4 a c} d+a e\right )+a c \left (\sqrt {b^2-4 a c} d+2 a e\right )+a b \left (-3 c d+\sqrt {b^2-4 a c} e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {b^2-4 a c} \sqrt {-2 c d+\left (b+\sqrt {b^2-4 a c}\right ) e}}+\frac {\left (-8 b^2 d^2+4 a b d e+a \left (8 c d^2+a e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}}{4 a^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.09 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1199, 2009}

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.

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

\(\Big \downarrow \) 1199

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1199

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Denominator[m]}, Simp[q/e   Subs 
t[Int[ExpandIntegrand[x^(q*(m + 1) - 1)*(((e*f - d*g)/e + g*(x^q/e))^n/((c* 
d^2 - b*d*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))), x], 
 x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Integer 
Q[n] && FractionQ[m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.75 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.06

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \left (a e x -4 b d x +2 a d \right )}{4 d \,a^{2} x^{2}}-\frac {e \left (-\frac {\left (e^{2} a^{2}+4 a b d e +8 a \,d^{2} c -8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{e a \sqrt {d}}-\frac {32 d c \left (-\frac {\left (-2 a^{2} c \,e^{2}+a \,b^{2} e^{2}+3 a b c d e -b^{3} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a^{2} c \,e^{2}-a \,b^{2} e^{2}-3 a b c d e +b^{3} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{a e}\right )}{4 a^{2} d}\)	\(489\)
derivativedivides	\(2 e^{4} \left (-\frac {\frac {\frac {a e \left (a e -4 b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d}+\left (\frac {1}{2} a b d e +\frac {1}{8} e^{2} a^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}-\frac {\left (e^{2} a^{2}+4 a b d e +8 a \,d^{2} c -8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{e^{4} a^{3}}+\frac {4 c \left (-\frac {\left (-2 a^{2} c \,e^{2}+a \,b^{2} e^{2}+3 a b c d e -b^{3} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a^{2} c \,e^{2}-a \,b^{2} e^{2}-3 a b c d e +b^{3} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{4} a^{3}}\right )\)	\(507\)
default	\(2 e^{4} \left (-\frac {\frac {\frac {a e \left (a e -4 b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d}+\left (\frac {1}{2} a b d e +\frac {1}{8} e^{2} a^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}-\frac {\left (e^{2} a^{2}+4 a b d e +8 a \,d^{2} c -8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{e^{4} a^{3}}+\frac {4 c \left (-\frac {\left (-2 a^{2} c \,e^{2}+a \,b^{2} e^{2}+3 a b c d e -b^{3} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}+\frac {\left (2 a^{2} c \,e^{2}-a \,b^{2} e^{2}-3 a b c d e +b^{3} d e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a b e +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, a c d -\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, b^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{8 \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{e^{4} a^{3}}\right )\)	\(507\)
pseudoelliptic	\(-\frac {-8 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\frac {\left (-a \,d^{\frac {3}{2}} b e -d^{\frac {5}{2}} \left (a c -b^{2}\right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (a e \left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}-\frac {3 d^{\frac {5}{2}} b \left (a c -\frac {b^{2}}{3}\right )}{2}\right ) e \right ) x^{2} \sqrt {2}\, c \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (-8 x^{2} \sqrt {2}\, c \left (\frac {\left (a \,d^{\frac {3}{2}} b e +d^{\frac {5}{2}} \left (a c -b^{2}\right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}}{2}+\left (a e \left (a c -\frac {b^{2}}{2}\right ) d^{\frac {3}{2}}-\frac {3 d^{\frac {5}{2}} b \left (a c -\frac {b^{2}}{3}\right )}{2}\right ) e \right ) \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (-x^{2} \left (e^{2} a^{2}+4 \left (b d e +2 c \,d^{2}\right ) a -8 b^{2} d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\sqrt {e x +d}\, a \left (2 \left (-2 b x +a \right ) d^{\frac {3}{2}}+a \sqrt {d}\, e x \right )\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right )}{4 \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, d^{\frac {3}{2}} \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, x^{2} a^{3}}\)	\(515\)

Input:

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

Output:

-1/4*(e*x+d)^(1/2)*(a*e*x-4*b*d*x+2*a*d)/d/a^2/x^2-1/4/a^2/d*e*(-1/e*(a^2* 
e^2+4*a*b*d*e+8*a*c*d^2-8*b^2*d^2)/a/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2) 
)-32*d/a/e*c*(-1/8*(-2*a^2*c*e^2+a*b^2*e^2+3*a*b*c*d*e-b^3*d*e+(-e^2*(4*a* 
c-b^2))^(1/2)*a*b*e+(-e^2*(4*a*c-b^2))^(1/2)*a*c*d-(-e^2*(4*a*c-b^2))^(1/2 
)*b^2*d)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c-b^2))^ 
(1/2))*c)^(1/2)*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b*e+2*c*d+(-e^2*(4*a*c- 
b^2))^(1/2))*c)^(1/2))+1/8*(2*a^2*c*e^2-a*b^2*e^2-3*a*b*c*d*e+b^3*d*e+(-e^ 
2*(4*a*c-b^2))^(1/2)*a*b*e+(-e^2*(4*a*c-b^2))^(1/2)*a*c*d-(-e^2*(4*a*c-b^2 
))^(1/2)*b^2*d)/(-e^2*(4*a*c-b^2))^(1/2)*2^(1/2)/((b*e-2*c*d+(-e^2*(4*a*c- 
b^2))^(1/2))*c)^(1/2)*arctan((e*x+d)^(1/2)*c*2^(1/2)/((b*e-2*c*d+(-e^2*(4* 
a*c-b^2))^(1/2))*c)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3707 vs. \(2 (395) = 790\).

Time = 78.99 (sec) , antiderivative size = 7422, normalized size of antiderivative = 16.06 \[ \int \frac {\sqrt {d+e x}}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1043 vs. \(2 (395) = 790\).

Time = 0.33 (sec) , antiderivative size = 1043, normalized size of antiderivative = 2.26 \[ \int \frac {\sqrt {d+e x}}{x^3 \left (a+b x+c x^2\right )} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/4*(sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e)*((b^4 - 5*a*b^2*c + 
 4*a^2*c^2)*d - (a*b^3 - 4*a^2*b*c)*e)*a^2*e^2 - 2*((a*b^2*c - a^2*c^2)*sq 
rt(b^2 - 4*a*c)*d^2 - (a*b^3 - a^2*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a^2*b^2 - 
 a^3*c)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c - sqrt(b^2 - 4*a*c)* 
c)*e)*abs(a)*abs(e) - (2*(a^2*b^3*c - 3*a^3*b*c^2)*d^2*e - (a^2*b^4 - a^3* 
b^2*c - 4*a^4*c^2)*d*e^2 + (a^3*b^3 - 2*a^4*b*c)*e^3)*sqrt(-4*c^2*d + 2*(b 
*c - sqrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*a^ 
3*c*d - a^3*b*e + sqrt(-4*(a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*a^3*c + (2*a^3 
*c*d - a^3*b*e)^2))/(a^3*c)))/((sqrt(b^2 - 4*a*c)*a^4*c*d^2 - sqrt(b^2 - 4 
*a*c)*a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs(a)*abs(c)*abs(e)) + 1/4*( 
sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)*((b^4 - 5*a*b^2*c + 4*a^2 
*c^2)*d - (a*b^3 - 4*a^2*b*c)*e)*a^2*e^2 + 2*((a*b^2*c - a^2*c^2)*sqrt(b^2 
 - 4*a*c)*d^2 - (a*b^3 - a^2*b*c)*sqrt(b^2 - 4*a*c)*d*e + (a^2*b^2 - a^3*c 
)*sqrt(b^2 - 4*a*c)*e^2)*sqrt(-4*c^2*d + 2*(b*c + sqrt(b^2 - 4*a*c)*c)*e)* 
abs(a)*abs(e) - (2*(a^2*b^3*c - 3*a^3*b*c^2)*d^2*e - (a^2*b^4 - a^3*b^2*c 
- 4*a^4*c^2)*d*e^2 + (a^3*b^3 - 2*a^4*b*c)*e^3)*sqrt(-4*c^2*d + 2*(b*c + s 
qrt(b^2 - 4*a*c)*c)*e))*arctan(2*sqrt(1/2)*sqrt(e*x + d)/sqrt(-(2*a^3*c*d 
- a^3*b*e - sqrt(-4*(a^3*c*d^2 - a^3*b*d*e + a^4*e^2)*a^3*c + (2*a^3*c*d - 
 a^3*b*e)^2))/(a^3*c)))/((sqrt(b^2 - 4*a*c)*a^4*c*d^2 - sqrt(b^2 - 4*a*c)* 
a^4*b*d*e + sqrt(b^2 - 4*a*c)*a^5*e^2)*abs(a)*abs(c)*abs(e)) + 1/4*(8*b...
 

Mupad [B] (verification not implemented)

Time = 15.94 (sec) , antiderivative size = 33838, normalized size of antiderivative = 73.24 \[ \int \frac {\sqrt {d+e x}}{x^3 \left (a+b x+c x^2\right )} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((((128*a^12*c^4*d*e^12 + 768*a^10*c^6*d^5*e^8 + 896*a^11*c^5*d^3*e 
^10 + 128*a^8*b^4*c^4*d^5*e^8 - 96*a^8*b^5*c^3*d^4*e^9 - 32*a^8*b^6*c^2*d^ 
3*e^10 - 704*a^9*b^2*c^5*d^5*e^8 + 448*a^9*b^3*c^4*d^4*e^9 + 392*a^9*b^4*c 
^3*d^3*e^10 + 24*a^9*b^5*c^2*d^2*e^11 - 1280*a^10*b^2*c^4*d^3*e^10 - 192*a 
^10*b^3*c^3*d^2*e^11 - 256*a^10*b*c^5*d^4*e^9 + 8*a^10*b^4*c^2*d*e^12 + 38 
4*a^11*b*c^4*d^2*e^11 - 64*a^11*b^2*c^3*d*e^12)/(2*a^8*d^2) - ((d + e*x)^( 
1/2)*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33 
*a^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c 
- b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2* 
b^5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b* 
c^2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/( 
2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))^(1/2)*(1536*a^12*c^5*d^4*e^8 + 10 
24*a^13*c^4*d^2*e^10 + 128*a^10*b^4*c^3*d^4*e^8 - 128*a^10*b^5*c^2*d^3*e^9 
 - 896*a^11*b^2*c^4*d^4*e^8 + 960*a^11*b^3*c^3*d^3*e^9 + 64*a^11*b^4*c^2*d 
^2*e^10 - 512*a^12*b^2*c^3*d^2*e^10 - 1792*a^12*b*c^4*d^3*e^9))/(2*a^8*d^2 
))*((b^8*d + 8*a^4*c^4*d - b^5*d*(-(4*a*c - b^2)^3)^(1/2) - a*b^7*e + 33*a 
^2*b^4*c^2*d - 38*a^3*b^2*c^3*d - 25*a^3*b^3*c^2*e + a^3*c^2*e*(-(4*a*c - 
b^2)^3)^(1/2) - 10*a*b^6*c*d + a*b^4*e*(-(4*a*c - b^2)^3)^(1/2) + 9*a^2*b^ 
5*c*e + 20*a^4*b*c^3*e + 4*a*b^3*c*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b*c^ 
2*d*(-(4*a*c - b^2)^3)^(1/2) - 3*a^2*b^2*c*e*(-(4*a*c - b^2)^3)^(1/2))/...
 

Reduce [F]

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

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

Output:

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