\(\int \frac {1}{x^2 (d+e x) \sqrt {a+b x+c x^2}} \, dx\) [36]

Optimal result

Integrand size = 25, antiderivative size = 162 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a+b x+c x^2}} \, dx=-\frac {\sqrt {a+b x+c x^2}}{a d x}+\frac {(b d+2 a e) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2} d^2}+\frac {e^2 \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{d^2 \sqrt {c d^2-b d e+a e^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.29 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.19 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {-\frac {2 d \sqrt {a+x (b+c x)}}{a x}+\frac {4 e^2 \sqrt {-c d^2+b d e-a e^2} \arctan \left (\frac {\sqrt {-c d^2+b d e-a e^2} x}{\sqrt {a} (d+e x)-d \sqrt {a+x (b+c x)}}\right )}{c d^2+e (-b d+a e)}+\frac {(b d+2 a e) \log (x)}{a^{3/2}}-\frac {(b d+2 a e) \log \left (a d^2 \left (2 a+b x-2 \sqrt {a} \sqrt {a+x (b+c x)}\right )\right )}{a^{3/2}}}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.21, 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.

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

\(\Big \downarrow \) 1289

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

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 = 1.60 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.46

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}}{a d x}-\frac {-\frac {\left (2 a e +b d \right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d \sqrt {a}}+\frac {2 e a \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}}{2 a d}\)	\(237\)
default	\(\frac {-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}}{d}-\frac {e \ln \left (\frac {\frac {2 a \,e^{2}-2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}\, \sqrt {c \left (x +\frac {d}{e}\right )^{2}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d^{2} \sqrt {\frac {a \,e^{2}-b d e +c \,d^{2}}{e^{2}}}}+\frac {e \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{d^{2} \sqrt {a}}\)	\(256\)

Input:

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

Output:

-(c*x^2+b*x+a)^(1/2)/a/d/x-1/2/a/d*(-(2*a*e+b*d)/d/a^(1/2)*ln((2*a+b*x+2*a 
^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+2*e/d*a/((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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (142) = 284\).

Time = 0.50 (sec) , antiderivative size = 1340, normalized size of antiderivative = 8.27 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*sqrt(c*d^2 - b*d*e + a*e^2)*a^2*e^2*x*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*s 
qrt(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)) + (b*c*d^3 - a*b*d*e^2 + 2*a^2*e^3 - (b^2 - 2*a*c)*d^2*e)*sqr 
t(a)*x*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*(a*c*d^3 - a*b*d^2*e + a^2*d*e^2)*sqrt(c*x^ 
2 + b*x + a))/((a^2*c*d^4 - a^2*b*d^3*e + a^3*d^2*e^2)*x), 1/4*(4*sqrt(-c* 
d^2 + b*d*e - a*e^2)*a^2*e^2*x*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*sq 
rt(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)) + (b*c*d^3 - a*b*d*e^2 + 2*a^2*e^3 - (b^2 - 2*a*c)*d^2*e)*sqrt(a)*x*l 
og(-(8*a*b*x + (b^2 + 4*a*c)*x^2 + 4*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*sqr 
t(a) + 8*a^2)/x^2) - 4*(a*c*d^3 - a*b*d^2*e + a^2*d*e^2)*sqrt(c*x^2 + b*x 
+ a))/((a^2*c*d^4 - a^2*b*d^3*e + a^3*d^2*e^2)*x), 1/2*(sqrt(c*d^2 - b*d*e 
 + a*e^2)*a^2*e^2*x*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)) - (b*c*d^3 - 
a*b*d*e^2 + 2*a^2*e^3 - (b^2 - 2*a*c)*d^2*e)*sqrt(-a)*x*arctan(1/2*sqrt...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \, e^{2} \arctan \left (-\frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} e + \sqrt {c} d}{\sqrt {-c d^{2} + b d e - a e^{2}}}\right )}{\sqrt {-c d^{2} + b d e - a e^{2}} d^{2}} - \frac {{\left (b d + 2 \, a e\right )} \arctan \left (-\frac {\sqrt {c} x - \sqrt {c x^{2} + b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a d^{2}} + \frac {{\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} b + 2 \, a \sqrt {c}}{{\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )}^{2} - a\right )} a d} \] Input:

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

Output:

2*e^2*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^2) - (b*d + 2*a*e)*a 
rctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a*d^2) + (( 
sqrt(c)*x - sqrt(c*x^2 + b*x + a))*b + 2*a*sqrt(c))/(((sqrt(c)*x - sqrt(c* 
x^2 + b*x + a))^2 - a)*a*d)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.66 \[ \int \frac {1}{x^2 (d+e x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (2 \sqrt {c \,x^{2}+b x +a}\, \sqrt {a \,e^{2}-b d e +c \,d^{2}}-2 a e +b d -b e x +2 c d x \right ) a^{2} e^{2} x -2 \sqrt {a \,e^{2}-b d e +c \,d^{2}}\, \mathrm {log}\left (e x +d \right ) a^{2} e^{2} x -2 \sqrt {c \,x^{2}+b x +a}\, a^{2} d \,e^{2}+2 \sqrt {c \,x^{2}+b x +a}\, a b \,d^{2} e -2 \sqrt {c \,x^{2}+b x +a}\, a c \,d^{3}+2 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} e^{3} x -\sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a b d \,e^{2} x +2 \sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a c \,d^{2} e x -\sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{2} d^{2} e x +\sqrt {a}\, \mathrm {log}\left (-2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b c \,d^{3} x -2 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} e^{3} x +\sqrt {a}\, \mathrm {log}\left (x \right ) a b d \,e^{2} x -2 \sqrt {a}\, \mathrm {log}\left (x \right ) a c \,d^{2} e x +\sqrt {a}\, \mathrm {log}\left (x \right ) b^{2} d^{2} e x -\sqrt {a}\, \mathrm {log}\left (x \right ) b c \,d^{3} x}{2 a^{2} d^{2} x \left (a \,e^{2}-b d e +c \,d^{2}\right )} \] Input:

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

Output:

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