\(\int \frac {\sqrt {a+b x+c x^2}}{x^3 (d+e x)} \, dx\) [18]

Optimal result

Integrand size = 25, antiderivative size = 216 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 (d+e x)} \, dx=-\frac {\sqrt {a+b x+c x^2}}{2 d x^2}-\frac {(b d-4 a e) \sqrt {a+b x+c x^2}}{4 a d^2 x}+\frac {\left (b^2 d^2+4 a b d e-4 a \left (c d^2+2 a e^2\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{8 a^{3/2} d^3}-\frac {e \sqrt {c d^2-b d e+a e^2} \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^3} \] Output:

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

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.03 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 (d+e x)} \, dx=\frac {\sqrt {a} \left (d (-2 a d-b d x+4 a e x) \sqrt {a+x (b+c x)}-8 a e \sqrt {-c d^2+b d e-a e^2} x^2 \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+x (b+c x)}}{\sqrt {-c d^2+e (b d-a e)}}\right )\right )+8 a^2 e^2 x^2 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )+d \left (b^2 d-4 a c d+4 a b e\right ) x^2 \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{4 a^{3/2} d^3 x^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {1289, 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.

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

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 1289

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + 
 g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( 
IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
 

rule 2009

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

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.33

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (-4 a e x +b d x +2 a d \right )}{4 a \,d^{2} x^{2}}+\frac {-\frac {\left (8 e^{2} a^{2}-4 a b d e +4 a \,d^{2} c -b^{2} d^{2}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {8 a \left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{8 a \,d^{2}}\)	\(288\)
default	\(\frac {-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{2 a \,x^{2}}-\frac {b \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )}{4 a}+\frac {c \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}}{d}+\frac {e^{2} \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{d^{3}}-\frac {e \left (-\frac {\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}{a x}+\frac {b \left (\sqrt {c \,x^{2}+b x +a}+\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{2 \sqrt {c}}-\sqrt {a}\, \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )\right )}{2 a}+\frac {2 c \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x +a}}{4 c}+\frac {\left (4 a c -b^{2}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x +a}\right )}{8 c^{\frac {3}{2}}}\right )}{a}\right )}{d^{2}}-\frac {e^{2} \left (\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 {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}+\frac {\left (b e -2 c d \right ) \ln \left (\frac {\frac {b e -2 c d}{2 e}+c \left (x +\frac {d}{e}\right )}{\sqrt {c}}+\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 {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\right )}{2 e \sqrt {c}}-\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{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 {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}\right )}{d^{3}}\)	\(890\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 1217, normalized size of antiderivative = 5.63 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 (d+e x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/16*(8*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*e*x^2*log((8*a*b*d*e - 8*a^2*e^2 
- (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4* 
sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - 
b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2* 
d*e*x + d^2)) - (4*a*b*d*e - 8*a^2*e^2 + (b^2 - 4*a*c)*d^2)*sqrt(a)*x^2*lo 
g(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqrt 
(a) + 8*a^2)/x^2) - 4*(2*a^2*d^2 + (a*b*d^2 - 4*a^2*d*e)*x)*sqrt(c*x^2 + b 
*x + a))/(a^2*d^3*x^2), -1/16*(16*sqrt(-c*d^2 + b*d*e - a*e^2)*a^2*e*x^2*a 
rctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e 
 + (2*c*d - b*e)*x)/(a*c*d^2 - a*b*d*e + a^2*e^2 + (c^2*d^2 - b*c*d*e + a* 
c*e^2)*x^2 + (b*c*d^2 - b^2*d*e + a*b*e^2)*x)) + (4*a*b*d*e - 8*a^2*e^2 + 
(b^2 - 4*a*c)*d^2)*sqrt(a)*x^2*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt( 
c*x^2 + b*x + a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(2*a^2*d^2 + (a*b*d 
^2 - 4*a^2*d*e)*x)*sqrt(c*x^2 + b*x + a))/(a^2*d^3*x^2), 1/8*(4*sqrt(c*d^2 
 - b*d*e + a*e^2)*a^2*e*x^2*log((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 
 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e 
+ a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c* 
d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e*x + d^2)) - (4* 
a*b*d*e - 8*a^2*e^2 + (b^2 - 4*a*c)*d^2)*sqrt(-a)*x^2*arctan(1/2*sqrt(c*x^ 
2 + b*x + a)*(b*x + 2*a)*sqrt(-a)/(a*c*x^2 + a*b*x + a^2)) - 2*(2*a^2*d...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 432 vs. \(2 (190) = 380\).

Time = 1.22 (sec) , antiderivative size = 432, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 (d+e x)} \, dx=-\frac {2 \, {\left (c d^{2} e - b d e^{2} + a e^{3}\right )} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{\sqrt {-c d^{2} + b d e - a e^{2}} d^{3}} - \frac {{\left (b^{2} d^{2} - 4 \, a c d^{2} + 4 \, a b d e - 8 \, a^{2} e^{2}\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a d^{3}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} b^{2} d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a c d - 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{3} a b e + 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a b \sqrt {c} d - 8 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} a^{2} \sqrt {c} e + {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a b^{2} d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} c d + 4 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} a^{2} b e + 8 \, a^{3} \sqrt {c} e}{4 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )}^{2} a d^{2}} \] Input:

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

Output:

-2*(c*d^2*e - b*d*e^2 + a*e^3)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/(sqrt(-c*d^2 + b*d*e - a*e^ 
2)*d^3) - 1/4*(b^2*d^2 - 4*a*c*d^2 + 4*a*b*d*e - 8*a^2*e^2)*arctan(-(sqrt( 
c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a*d^3) + 1/4*((sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^3*b^2*d + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
3*a*c*d - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*b*e + 8*(sqrt(c)*x - s 
qrt(c*x^2 + b*x + a))^2*a*b*sqrt(c)*d - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + 
a))^2*a^2*sqrt(c)*e + (sqrt(c)*x - sqrt(c*x^2 + b*x + a))*a*b^2*d + 4*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))*a^2*c*d + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))*a^2*b*e + 8*a^3*sqrt(c)*e)/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - 
 a)^2*a*d^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.75 \[ \int \frac {\sqrt {a+b x+c x^2}}{x^3 (d+e x)} \, dx=\frac {8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (-2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a^{2} e \,x^{2}-8 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a^{2} e \,x^{2}-4 \sqrt {c \,x^{2}+b x +a}\, a^{2} d^{2}+8 \sqrt {c \,x^{2}+b x +a}\, a^{2} d e x -2 \sqrt {c \,x^{2}+b x +a}\, a b \,d^{2} x +8 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} e^{2} x^{2}-4 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a b d e \,x^{2}+4 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a c \,d^{2} x^{2}-\sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{2} d^{2} x^{2}-8 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} e^{2} x^{2}+4 \sqrt {a}\, \mathrm {log}\left (x \right ) a b d e \,x^{2}-4 \sqrt {a}\, \mathrm {log}\left (x \right ) a c \,d^{2} x^{2}+\sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} d^{2} x^{2}}{8 a^{2} d^{3} x^{2}} \] Input:

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

Output:

(8*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e* 
*2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*e*x**2 - 8*sqrt 
(a*e**2 - b*d*e + c*d**2)*log(d + e*x)*a**2*e*x**2 - 4*sqrt(a + b*x + c*x* 
*2)*a**2*d**2 + 8*sqrt(a + b*x + c*x**2)*a**2*d*e*x - 2*sqrt(a + b*x + c*x 
**2)*a*b*d**2*x + 8*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a - b 
*x)*a**2*e**2*x**2 - 4*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a 
- b*x)*a*b*d*e*x**2 + 4*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a 
 - b*x)*a*c*d**2*x**2 - sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a 
 - b*x)*b**2*d**2*x**2 - 8*sqrt(a)*log(x)*a**2*e**2*x**2 + 4*sqrt(a)*log(x 
)*a*b*d*e*x**2 - 4*sqrt(a)*log(x)*a*c*d**2*x**2 + sqrt(a)*log(x)*b**2*d**2 
*x**2)/(8*a**2*d**3*x**2)