\(\int \frac {1}{x^2 (c+d x) \sqrt {a+b x^2}} \, dx\) [1188]

Optimal result

Integrand size = 22, antiderivative size = 111 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a+b x^2}} \, dx=-\frac {\sqrt {a+b x^2}}{a c x}-\frac {d^2 \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^2 \sqrt {b c^2+a d^2}}+\frac {d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a} c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a+b x^2}} \, dx=-\frac {\frac {c \sqrt {a+b x^2}}{a x}+\frac {2 d^2 \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {2 d \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}}{c^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 111, 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.

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

\(\Big \downarrow \) 617

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Time = 0.42 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.62

method	result	size
default	\(-\frac {\sqrt {b \,x^{2}+a}}{a c x}-\frac {d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{2} \sqrt {a}}\)	\(180\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}}{a c x}-\frac {d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{2} \sqrt {a}}\)	\(180\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 773, normalized size of antiderivative = 6.96 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a+b x^2}} \, dx=\left [\frac {\sqrt {b c^{2} + a d^{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 ) + {\left (b c^{2} d + a d^{3}\right )} \sqrt {a} x \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) - 2 \, {\left (b c^{3} + a c d^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b c^{4} + a^{2} c^{2} d^{2}\right )} x}, -\frac {2 \, \sqrt {-b c^{2} - a d^{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 ) - {\left (b c^{2} d + a d^{3}\right )} \sqrt {a} x \log \left (-\frac {b x^{2} + 2 \, \sqrt {b x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right ) + 2 \, {\left (b c^{3} + a c d^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b c^{4} + a^{2} c^{2} d^{2}\right )} x}, \frac {\sqrt {b c^{2} + a d^{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 \, {\left (b c^{2} d + a d^{3}\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) - 2 \, {\left (b c^{3} + a c d^{2}\right )} \sqrt {b x^{2} + a}}{2 \, {\left (a b c^{4} + a^{2} c^{2} d^{2}\right )} x}, -\frac {\sqrt {-b c^{2} - a d^{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 ) + {\left (b c^{2} d + a d^{3}\right )} \sqrt {-a} x \arctan \left (\frac {\sqrt {b x^{2} + a} \sqrt {-a}}{a}\right ) + {\left (b c^{3} + a c d^{2}\right )} \sqrt {b x^{2} + a}}{{\left (a b c^{4} + a^{2} c^{2} d^{2}\right )} x}\right ] \] Input:

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

Output:

[1/2*(sqrt(b*c^2 + a*d^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)) + (b*c^2*d + a*d^3)*sqrt(a)*x*log(-(b*x 
^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(b*c^3 + a*c*d^2)*sqrt(b*x^ 
2 + a))/((a*b*c^4 + a^2*c^2*d^2)*x), -1/2*(2*sqrt(-b*c^2 - a*d^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)) - (b*c^2*d + a*d^3)*sqrt(a)*x*log(-(b*x^2 + 
 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(b*c^3 + a*c*d^2)*sqrt(b*x^2 + 
a))/((a*b*c^4 + a^2*c^2*d^2)*x), 1/2*(sqrt(b*c^2 + a*d^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*(b* 
c^2*d + a*d^3)*sqrt(-a)*x*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - 2*(b*c^3 + 
a*c*d^2)*sqrt(b*x^2 + a))/((a*b*c^4 + a^2*c^2*d^2)*x), -(sqrt(-b*c^2 - a*d 
^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)) + (b*c^2*d + a*d^3)*sqrt(-a)*x* 
arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) + (b*c^3 + a*c*d^2)*sqrt(b*x^2 + a))/(( 
a*b*c^4 + a^2*c^2*d^2)*x)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.29 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a+b x^2}} \, dx=2 \, b {\left (\frac {d^{2} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} b c^{2}} - \frac {d \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} b c^{2}} + \frac {1}{{\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )} \sqrt {b} c}\right )} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 1208, normalized size of antiderivative = 10.88 \[ \int \frac {1}{x^2 (c+d x) \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

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