\(\int \frac {\sqrt {c+d x^2}}{x^4 (a+b x^2)} \, dx\) [928]

Optimal result

Integrand size = 24, antiderivative size = 105 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=-\frac {\sqrt {c+d x^2}}{3 a x^3}+\frac {(3 b c-a d) \sqrt {c+d x^2}}{3 a^2 c x}+\frac {b \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{5/2}} \] Output:

-1/3*(d*x^2+c)^(1/2)/a/x^3+1/3*(-a*d+3*b*c)*(d*x^2+c)^(1/2)/a^2/c/x+b*(-a* 
d+b*c)^(1/2)*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2))/a^(5/2)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=\frac {\sqrt {c+d x^2} \left (3 b c x^2-a \left (c+d x^2\right )\right )}{3 a^2 c x^3}-\frac {b \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{5/2}} \] Input:

Integrate[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)),x]
 

Output:

(Sqrt[c + d*x^2]*(3*b*c*x^2 - a*(c + d*x^2)))/(3*a^2*c*x^3) - (b*Sqrt[b*c 
- a*d]*ArcTan[(a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqr 
t[b*c - a*d])])/a^(5/2)
 

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {377, 25, 445, 27, 291, 218}

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.

\(\displaystyle \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {3 b (b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {c+d x^2} (3 b c-a d)}{a c x}}{3 a}-\frac {\sqrt {c+d x^2}}{3 a x^3}\)

\(\Big \downarrow \) 291

\(\displaystyle -\frac {-\frac {3 b (b c-a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{a}-\frac {\sqrt {c+d x^2} (3 b c-a d)}{a c x}}{3 a}-\frac {\sqrt {c+d x^2}}{3 a x^3}\)

\(\Big \downarrow \) 218

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

Input:

Int[Sqrt[c + d*x^2]/(x^4*(a + b*x^2)),x]
 

Output:

-1/3*Sqrt[c + d*x^2]/(a*x^3) - (-(((3*b*c - a*d)*Sqrt[c + d*x^2])/(a*c*x)) 
 - (3*b*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2 
])])/a^(3/2))/(3*a)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 377

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_) 
, x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(a*e*( 
m + 1))), x] - Simp[1/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)*(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[b*c*(m + 1) + 2*(b*c*(p + 1) + a*d*q) + d*(b*(m + 1) 
+ 2*b*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b 
*c - a*d, 0] && LtQ[0, q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m 
, 2, p, q, x]
 

rule 445

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ 
.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p 
+ 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) 
 Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c 
+ a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 
2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.87

method	result	size
pseudoelliptic	\(\frac {-\frac {\sqrt {x^{2} d +c}\, \left (a d \,x^{2}-3 x^{2} b c +a c \right )}{3 x^{3}}-\frac {b c \left (a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{\sqrt {\left (a d -b c \right ) a}}}{a^{2} c}\)	\(91\)
risch	\(-\frac {\sqrt {x^{2} d +c}\, \left (a d \,x^{2}-3 x^{2} b c +a c \right )}{3 c \,a^{2} x^{3}}-\frac {\left (a d -b c \right ) b \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{2}}\)	\(358\)
default	\(-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{3 a c \,x^{3}}-\frac {b \left (-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{c x}+\frac {2 d \left (\frac {x \sqrt {x^{2} d +c}}{2}+\frac {c \ln \left (\sqrt {d}\, x +\sqrt {x^{2} d +c}\right )}{2 \sqrt {d}}\right )}{c}\right )}{a^{2}}+\frac {b^{2} \left (\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+d \left (x -\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 a^{2} \sqrt {-a b}}-\frac {b^{2} \left (\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+d \left (x +\frac {\sqrt {-a b}}{b}\right )}{\sqrt {d}}+\sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}\right )}{b}+\frac {\left (a d -b c \right ) \ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -b c}{b}}}\right )}{2 a^{2} \sqrt {-a b}}\)	\(750\)

Input:

int((d*x^2+c)^(1/2)/x^4/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 325, normalized size of antiderivative = 3.10 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=\left [\frac {3 \, b c x^{3} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left ({\left (3 \, b c - a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{12 \, a^{2} c x^{3}}, \frac {3 \, b c x^{3} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (3 \, b c - a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c}}{6 \, a^{2} c x^{3}}\right ] \] Input:

integrate((d*x^2+c)^(1/2)/x^4/(b*x^2+a),x, algorithm="fricas")
 

Output:

[1/12*(3*b*c*x^3*sqrt(-(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^ 
2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 - 4*(a^2*c*x - (a*b*c - 2 
*a^2*d)*x^3)*sqrt(d*x^2 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^4 + 2*a*b*x^2 + 
a^2)) + 4*((3*b*c - a*d)*x^2 - a*c)*sqrt(d*x^2 + c))/(a^2*c*x^3), 1/6*(3*b 
*c*x^3*sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 
 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^3 + (b*c^2 - a*c*d)*x)) + 2*( 
(3*b*c - a*d)*x^2 - a*c)*sqrt(d*x^2 + c))/(a^2*c*x^3)]
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x^{4} \left (a + b x^{2}\right )}\, dx \] Input:

integrate((d*x**2+c)**(1/2)/x**4/(b*x**2+a),x)
 

Output:

Integral(sqrt(c + d*x**2)/(x**4*(a + b*x**2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )} x^{4}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)/x^4/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (87) = 174\).

Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=-\frac {{\left (b^{2} c \sqrt {d} - a b d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} a^{2}} - \frac {2 \, {\left (3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b c \sqrt {d} - 3 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a d^{\frac {3}{2}} - 6 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b c^{2} \sqrt {d} + 3 \, b c^{3} \sqrt {d} - a c^{2} d^{\frac {3}{2}}\right )}}{3 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{3} a^{2}} \] Input:

integrate((d*x^2+c)^(1/2)/x^4/(b*x^2+a),x, algorithm="giac")
 

Output:

-(b^2*c*sqrt(d) - a*b*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2 
*b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*a^2) - 
 2/3*(3*(sqrt(d)*x - sqrt(d*x^2 + c))^4*b*c*sqrt(d) - 3*(sqrt(d)*x - sqrt( 
d*x^2 + c))^4*a*d^(3/2) - 6*(sqrt(d)*x - sqrt(d*x^2 + c))^2*b*c^2*sqrt(d) 
+ 3*b*c^3*sqrt(d) - a*c^2*d^(3/2))/(((sqrt(d)*x - sqrt(d*x^2 + c))^2 - c)^ 
3*a^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {d\,x^2+c}}{x^4\,\left (b\,x^2+a\right )} \,d x \] Input:

int((c + d*x^2)^(1/2)/(x^4*(a + b*x^2)),x)
 

Output:

int((c + d*x^2)^(1/2)/(x^4*(a + b*x^2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.51 \[ \int \frac {\sqrt {c+d x^2}}{x^4 \left (a+b x^2\right )} \, dx=\frac {-3 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c \,x^{3}-3 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b c \,x^{3}+3 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b c \,x^{3}-2 \sqrt {d \,x^{2}+c}\, a^{2} c -2 \sqrt {d \,x^{2}+c}\, a^{2} d \,x^{2}+6 \sqrt {d \,x^{2}+c}\, a b c \,x^{2}-2 \sqrt {d}\, a^{2} d \,x^{3}-2 \sqrt {d}\, a b c \,x^{3}}{6 a^{3} c \,x^{3}} \] Input:

int((d*x^2+c)^(1/2)/x^4/(b*x^2+a),x)
 

Output:

( - 3*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c 
) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x**3 
- 3*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) - 2 
*a*d + b*c) + sqrt(b)*sqrt(c + d*x**2) + sqrt(d)*sqrt(b)*x)*b*c*x**3 + 3*s 
qrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqrt(d)*s 
qrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*b*c*x**3 - 2*sqrt(c + d*x**2)*a* 
*2*c - 2*sqrt(c + d*x**2)*a**2*d*x**2 + 6*sqrt(c + d*x**2)*a*b*c*x**2 - 2* 
sqrt(d)*a**2*d*x**3 - 2*sqrt(d)*a*b*c*x**3)/(6*a**3*c*x**3)