\(\int \frac {\sqrt {a+b x^2}}{x^2 (c+d x)} \, dx\) [1002]

Optimal result

Integrand size = 22, antiderivative size = 105 \[ \int \frac {\sqrt {a+b x^2}}{x^2 (c+d x)} \, dx=-\frac {\sqrt {a+b x^2}}{c x}-\frac {\sqrt {b c^2+a d^2} \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^2}+\frac {\sqrt {a} d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {a+b x^2}}{x^2 (c+d x)} \, dx=\frac {-c \sqrt {a+b x^2}+2 \sqrt {-b c^2-a d^2} x \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )+\sqrt {a} d x \log (x)-\sqrt {a} d x \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c^2 x} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {617, 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.

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

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 617

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p, x^m*(c + d*x)^n, x], x] /; FreeQ[{ 
a, b, c, d, p}, x] && ILtQ[n, 0] && IntegerQ[m] && IntegerQ[2*p]
 

rule 2009

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(194\) vs. \(2(91)=182\).

Time = 0.42 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.86

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}}{c x}+\frac {-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c d \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\sqrt {a}\, d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c}}{c}\)	\(195\)
default	\(\frac {-\frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}}}{a x}+\frac {2 b \left (\frac {x \sqrt {b \,x^{2}+a}}{2}+\frac {a \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{2 \sqrt {b}}\right )}{a}}{c}+\frac {d \left (\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}-\frac {\sqrt {b}\, c \ln \left (\frac {-\frac {b c}{d}+b \left (x +\frac {c}{d}\right )}{\sqrt {b}}+\sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\right )}{d}-\frac {\left (a \,d^{2}+b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{c^{2}}-\frac {d \left (\sqrt {b \,x^{2}+a}-\sqrt {a}\, \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )\right )}{c^{2}}\)	\(370\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 605, normalized size of antiderivative = 5.76 \[ \int \frac {\sqrt {a+b x^2}}{x^2 (c+d x)} \, dx=\left [\frac {\sqrt {a} d x \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + \sqrt {b c^{2} + a d^{2}} x \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, \sqrt {b x^{2} + a} c}{2 \, c^{2} x}, \frac {\sqrt {a} d x \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, \sqrt {-b c^{2} - a d^{2}} x \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) - 2 \, \sqrt {b x^{2} + a} c}{2 \, c^{2} x}, -\frac {2 \, \sqrt {-a} d x \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - \sqrt {b c^{2} + a d^{2}} x \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + 2 \, \sqrt {b x^{2} + a} c}{2 \, c^{2} x}, -\frac {\sqrt {-a} d x \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + \sqrt {-b c^{2} - a d^{2}} x \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) + \sqrt {b x^{2} + a} c}{c^{2} x}\right ] \] Input:

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

Output:

[1/2*(sqrt(a)*d*x*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + sq 
rt(b*c^2 + a*d^2)*x*log((2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + 
a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x 
^2 + 2*c*d*x + c^2)) - 2*sqrt(b*x^2 + a)*c)/(c^2*x), 1/2*(sqrt(a)*d*x*log( 
-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*sqrt(-b*c^2 - a*d^2)*x 
*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + a^2* 
d^2 + (b^2*c^2 + a*b*d^2)*x^2)) - 2*sqrt(b*x^2 + a)*c)/(c^2*x), -1/2*(2*sq 
rt(-a)*d*x*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - sqrt(b*c^2 + a*d^2)*x*log( 
(2*a*b*c*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b* 
c^2 + a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 2 
*sqrt(b*x^2 + a)*c)/(c^2*x), -(sqrt(-a)*d*x*arctan(sqrt(b*x^2 + a)*sqrt(-a 
)/a) + sqrt(-b*c^2 - a*d^2)*x*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sq 
rt(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + sqrt(b*x^2 
+ a)*c)/(c^2*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {a+b x^2}}{x^2 (c+d x)} \, dx=-\frac {2 \, a d \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{2}} + \frac {2 \, a \sqrt {b}}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} c} + \frac {2 \, {\left (b c^{2} + a d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} c^{2}} \] Input:

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

Output:

-2*a*d*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*c^2) + 2* 
a*sqrt(b)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)*c) + 2*(b*c^2 + a*d^2)*ar 
ctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/ 
(sqrt(-b*c^2 - a*d^2)*c^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 1111, normalized size of antiderivative = 10.58 \[ \int \frac {\sqrt {a+b x^2}}{x^2 (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)* 
sqrt(a*d**2 + b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt( 
b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*c*x - 2*sqrt(2*sqrt(b)*sq 
rt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan((sqrt(a + b*x**2)*d + sqrt 
(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2))*a*d* 
*2*x - 2*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c - a*d**2 - 2*b*c**2)*atan( 
(sqrt(a + b*x**2)*d + sqrt(b)*d*x)/sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c 
- a*d**2 - 2*b*c**2))*b*c**2*x - sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c* 
*2)*c + a*d**2 + 2*b*c**2)*sqrt(a*d**2 + b*c**2)*log( - sqrt(2*sqrt(b)*sqr 
t(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + sqrt(a + b*x**2)*d + sqrt(b)*d 
*x)*c*x + sqrt(b)*sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c* 
*2)*sqrt(a*d**2 + b*c**2)*log(sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d 
**2 + 2*b*c**2) + sqrt(a + b*x**2)*d + sqrt(b)*d*x)*c*x - sqrt(a*d**2 + b* 
c**2)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + 
 sqrt(a + b*x**2)*d + sqrt(b)*d*x)*a*d**2*x - sqrt(a*d**2 + b*c**2)*log(sq 
rt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) + sqrt(a + b*x** 
2)*d + sqrt(b)*d*x)*a*d**2*x + sqrt(a*d**2 + b*c**2)*log(2*sqrt(b)*sqrt(a* 
d**2 + b*c**2)*c + 2*sqrt(b)*sqrt(a + b*x**2)*d**2*x - 2*b*c**2 + 2*b*d**2 
*x**2)*a*d**2*x + sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c* 
*2)*log( - sqrt(2*sqrt(b)*sqrt(a*d**2 + b*c**2)*c + a*d**2 + 2*b*c**2) ...