\(\int \frac {\sqrt {c+d x^2}}{x^6 (a+b x^2)} \, dx\) [929]

Optimal result

Integrand size = 24, antiderivative size = 155 \[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=-\frac {\sqrt {c+d x^2}}{5 a x^5}+\frac {(5 b c-a d) \sqrt {c+d x^2}}{15 a^2 c x^3}-\frac {\left (15 b^2 c^2-5 a b c d-2 a^2 d^2\right ) \sqrt {c+d x^2}}{15 a^3 c^2 x}-\frac {b^2 \sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{7/2}} \] Output:

-1/5*(d*x^2+c)^(1/2)/a/x^5+1/15*(-a*d+5*b*c)*(d*x^2+c)^(1/2)/a^2/c/x^3-1/1 
5*(-2*a^2*d^2-5*a*b*c*d+15*b^2*c^2)*(d*x^2+c)^(1/2)/a^3/c^2/x-b^2*(-a*d+b* 
c)^(1/2)*arctan((-a*d+b*c)^(1/2)*x/a^(1/2)/(d*x^2+c)^(1/2))/a^(7/2)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=-\frac {\sqrt {c+d x^2} \left (15 b^2 c^2 x^4-5 a b c x^2 \left (c+d x^2\right )+a^2 \left (3 c^2+c d x^2-2 d^2 x^4\right )\right )}{15 a^3 c^2 x^5}+\frac {b^2 \sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x \left (\sqrt {d} x-\sqrt {c+d x^2}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{a^{7/2}} \] Input:

Integrate[Sqrt[c + d*x^2]/(x^6*(a + b*x^2)),x]
 

Output:

-1/15*(Sqrt[c + d*x^2]*(15*b^2*c^2*x^4 - 5*a*b*c*x^2*(c + d*x^2) + a^2*(3* 
c^2 + c*d*x^2 - 2*d^2*x^4)))/(a^3*c^2*x^5) + (b^2*Sqrt[b*c - a*d]*ArcTan[( 
a*Sqrt[d] + b*x*(Sqrt[d]*x - Sqrt[c + d*x^2]))/(Sqrt[a]*Sqrt[b*c - a*d])]) 
/a^(7/2)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {377, 25, 445, 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.

\(\displaystyle \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx\)

\(\Big \downarrow \) 377

\(\displaystyle \frac {\int -\frac {4 b d x^2+5 b c-a d}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int \frac {4 b d x^2+5 b c-a d}{x^4 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {-\frac {\int \frac {15 b^2 c^2-5 a b d c-2 a^2 d^2+2 b d (5 b c-a d) x^2}{x^2 \left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{3 a c}-\frac {\sqrt {c+d x^2} (5 b c-a d)}{3 a c x^3}}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

\(\Big \downarrow \) 445

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {15 b^2 c^2 (b c-a d)}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a c}-\frac {\sqrt {c+d x^2} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x}}{3 a c}-\frac {\sqrt {c+d x^2} (5 b c-a d)}{3 a c x^3}}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {15 b^2 c (b c-a d) \int \frac {1}{\left (b x^2+a\right ) \sqrt {d x^2+c}}dx}{a}-\frac {\sqrt {c+d x^2} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x}}{3 a c}-\frac {\sqrt {c+d x^2} (5 b c-a d)}{3 a c x^3}}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

\(\Big \downarrow \) 291

\(\displaystyle -\frac {-\frac {-\frac {15 b^2 c (b c-a d) \int \frac {1}{a-\frac {(a d-b c) x^2}{d x^2+c}}d\frac {x}{\sqrt {d x^2+c}}}{a}-\frac {\sqrt {c+d x^2} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x}}{3 a c}-\frac {\sqrt {c+d x^2} (5 b c-a d)}{3 a c x^3}}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

\(\Big \downarrow \) 218

\(\displaystyle -\frac {-\frac {-\frac {15 b^2 c \sqrt {b c-a d} \arctan \left (\frac {x \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{a^{3/2}}-\frac {\sqrt {c+d x^2} \left (\frac {15 b^2 c}{a}-\frac {2 a d^2}{c}-5 b d\right )}{x}}{3 a c}-\frac {\sqrt {c+d x^2} (5 b c-a d)}{3 a c x^3}}{5 a}-\frac {\sqrt {c+d x^2}}{5 a x^5}\)

Input:

Int[Sqrt[c + d*x^2]/(x^6*(a + b*x^2)),x]
 

Output:

-1/5*Sqrt[c + d*x^2]/(a*x^5) - (-1/3*((5*b*c - a*d)*Sqrt[c + d*x^2])/(a*c* 
x^3) - (-((((15*b^2*c)/a - 5*b*d - (2*a*d^2)/c)*Sqrt[c + d*x^2])/x) - (15* 
b^2*c*Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2]) 
])/a^(3/2))/(3*a*c))/(5*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 = 0.78 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(-\frac {\sqrt {\left (a d -b c \right ) a}\, \left (\left (-\frac {2 x^{2} d}{3}+c \right ) \left (x^{2} d +c \right ) a^{2}-\frac {5 \left (x^{2} d +c \right ) a b c \,x^{2}}{3}+5 b^{2} c^{2} x^{4}\right ) \sqrt {x^{2} d +c}-5 b^{2} c^{2} x^{5} \left (a d -b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {x^{2} d +c}\, a}{x \sqrt {\left (a d -b c \right ) a}}\right )}{5 \sqrt {\left (a d -b c \right ) a}\, a^{3} x^{5} c^{2}}\)	\(139\)
risch	\(-\frac {\sqrt {x^{2} d +c}\, \left (-2 a^{2} d^{2} x^{4}-5 a b c d \,x^{4}+15 b^{2} c^{2} x^{4}+a^{2} c d \,x^{2}-5 a b \,c^{2} x^{2}+3 a^{2} c^{2}\right )}{15 c^{2} a^{3} x^{5}}+\frac {\left (a d -b c \right ) b^{2} \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}+\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 \sqrt {-a b}\, \sqrt {-\frac {a d -b c}{b}}}\right )}{a^{3}}\)	\(401\)
default	\(\frac {-\frac {\left (x^{2} d +c \right )^{\frac {3}{2}}}{5 c \,x^{5}}+\frac {2 d \left (x^{2} d +c \right )^{\frac {3}{2}}}{15 c^{2} x^{3}}}{a}+\frac {b^{2} \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^{3}}+\frac {b \left (x^{2} d +c \right )^{\frac {3}{2}}}{3 a^{2} c \,x^{3}}-\frac {b^{3} \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 \sqrt {-a b}\, a^{3}}+\frac {b^{3} \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 \sqrt {-a b}\, a^{3}}\)	\(792\)

Input:

int((d*x^2+c)^(1/2)/x^6/(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-1/5*(((a*d-b*c)*a)^(1/2)*((-2/3*x^2*d+c)*(d*x^2+c)*a^2-5/3*(d*x^2+c)*a*b* 
c*x^2+5*b^2*c^2*x^4)*(d*x^2+c)^(1/2)-5*b^2*c^2*x^5*(a*d-b*c)*arctanh((d*x^ 
2+c)^(1/2)/x*a/((a*d-b*c)*a)^(1/2)))/((a*d-b*c)*a)^(1/2)/a^3/x^5/c^2
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 409, normalized size of antiderivative = 2.64 \[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=\left [\frac {15 \, b^{2} c^{2} x^{5} \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{2} + 4 \, {\left (a^{2} c x - {\left (a b c - 2 \, a^{2} d\right )} x^{3}\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left ({\left (15 \, b^{2} c^{2} - 5 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} - {\left (5 \, a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, a^{3} c^{2} x^{5}}, -\frac {15 \, b^{2} c^{2} x^{5} \sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{2} - a c\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{3} + {\left (b c^{2} - a c d\right )} x\right )}}\right ) + 2 \, {\left ({\left (15 \, b^{2} c^{2} - 5 \, a b c d - 2 \, a^{2} d^{2}\right )} x^{4} + 3 \, a^{2} c^{2} - {\left (5 \, a b c^{2} - a^{2} c d\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, a^{3} c^{2} x^{5}}\right ] \] Input:

integrate((d*x^2+c)^(1/2)/x^6/(b*x^2+a),x, algorithm="fricas")
 

Output:

[1/60*(15*b^2*c^2*x^5*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*((15*b^2*c^2 - 5*a*b*c*d - 2*a^2*d^2)*x^4 + 3*a^2*c^2 - (5* 
a*b*c^2 - a^2*c*d)*x^2)*sqrt(d*x^2 + c))/(a^3*c^2*x^5), -1/30*(15*b^2*c^2* 
x^5*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*((15 
*b^2*c^2 - 5*a*b*c*d - 2*a^2*d^2)*x^4 + 3*a^2*c^2 - (5*a*b*c^2 - a^2*c*d)* 
x^2)*sqrt(d*x^2 + c))/(a^3*c^2*x^5)]
 

Sympy [F]

\[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {c + d x^{2}}}{x^{6} \left (a + b x^{2}\right )}\, dx \] Input:

integrate((d*x**2+c)**(1/2)/x**6/(b*x**2+a),x)
 

Output:

Integral(sqrt(c + d*x**2)/(x**6*(a + b*x**2)), x)
 

Maxima [F]

\[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=\int { \frac {\sqrt {d x^{2} + c}}{{\left (b x^{2} + a\right )} x^{6}} \,d x } \] Input:

integrate((d*x^2+c)^(1/2)/x^6/(b*x^2+a),x, algorithm="maxima")
 

Output:

integrate(sqrt(d*x^2 + c)/((b*x^2 + a)*x^6), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (133) = 266\).

Time = 0.32 (sec) , antiderivative size = 465, normalized size of antiderivative = 3.00 \[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=\frac {{\left (b^{3} c \sqrt {d} - a b^{2} d^{\frac {3}{2}}\right )} \arctan \left (\frac {{\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt {a b c d - a^{2} d^{2}}}\right )}{\sqrt {a b c d - a^{2} d^{2}} a^{3}} + \frac {2 \, {\left (15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} b^{2} c \sqrt {d} - 15 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{8} a b d^{\frac {3}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} b^{2} c^{2} \sqrt {d} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a b c d^{\frac {3}{2}} + 30 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{6} a^{2} d^{\frac {5}{2}} + 90 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} b^{2} c^{3} \sqrt {d} - 20 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a b c^{2} d^{\frac {3}{2}} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{4} a^{2} c d^{\frac {5}{2}} - 60 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} b^{2} c^{4} \sqrt {d} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a b c^{3} d^{\frac {3}{2}} + 10 \, {\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} a^{2} c^{2} d^{\frac {5}{2}} + 15 \, b^{2} c^{5} \sqrt {d} - 5 \, a b c^{4} d^{\frac {3}{2}} - 2 \, a^{2} c^{3} d^{\frac {5}{2}}\right )}}{15 \, {\left ({\left (\sqrt {d} x - \sqrt {d x^{2} + c}\right )}^{2} - c\right )}^{5} a^{3}} \] Input:

integrate((d*x^2+c)^(1/2)/x^6/(b*x^2+a),x, algorithm="giac")
 

Output:

(b^3*c*sqrt(d) - a*b^2*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^3) 
+ 2/15*(15*(sqrt(d)*x - sqrt(d*x^2 + c))^8*b^2*c*sqrt(d) - 15*(sqrt(d)*x - 
 sqrt(d*x^2 + c))^8*a*b*d^(3/2) - 60*(sqrt(d)*x - sqrt(d*x^2 + c))^6*b^2*c 
^2*sqrt(d) + 30*(sqrt(d)*x - sqrt(d*x^2 + c))^6*a*b*c*d^(3/2) + 30*(sqrt(d 
)*x - sqrt(d*x^2 + c))^6*a^2*d^(5/2) + 90*(sqrt(d)*x - sqrt(d*x^2 + c))^4* 
b^2*c^3*sqrt(d) - 20*(sqrt(d)*x - sqrt(d*x^2 + c))^4*a*b*c^2*d^(3/2) + 10* 
(sqrt(d)*x - sqrt(d*x^2 + c))^4*a^2*c*d^(5/2) - 60*(sqrt(d)*x - sqrt(d*x^2 
 + c))^2*b^2*c^4*sqrt(d) + 10*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c^3*d^(3 
/2) + 10*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a^2*c^2*d^(5/2) + 15*b^2*c^5*sqrt 
(d) - 5*a*b*c^4*d^(3/2) - 2*a^2*c^3*d^(5/2))/(((sqrt(d)*x - sqrt(d*x^2 + c 
))^2 - c)^5*a^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=\int \frac {\sqrt {d\,x^2+c}}{x^6\,\left (b\,x^2+a\right )} \,d x \] Input:

int((c + d*x^2)^(1/2)/(x^6*(a + b*x^2)),x)
 

Output:

int((c + d*x^2)^(1/2)/(x^6*(a + b*x^2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {c+d x^2}}{x^6 \left (a+b x^2\right )} \, dx=\frac {15 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b^{2} c^{2} x^{5}+15 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{2}+c}+\sqrt {d}\, \sqrt {b}\, x \right ) b^{2} c^{2} x^{5}-15 \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +2 a d +2 b d \,x^{2}\right ) b^{2} c^{2} x^{5}-6 \sqrt {d \,x^{2}+c}\, a^{3} c^{2}-2 \sqrt {d \,x^{2}+c}\, a^{3} c d \,x^{2}+4 \sqrt {d \,x^{2}+c}\, a^{3} d^{2} x^{4}+10 \sqrt {d \,x^{2}+c}\, a^{2} b \,c^{2} x^{2}+10 \sqrt {d \,x^{2}+c}\, a^{2} b c d \,x^{4}-30 \sqrt {d \,x^{2}+c}\, a \,b^{2} c^{2} x^{4}-4 \sqrt {d}\, a^{3} d^{2} x^{5}+2 \sqrt {d}\, a^{2} b c d \,x^{5}+18 \sqrt {d}\, a \,b^{2} c^{2} x^{5}}{30 a^{4} c^{2} x^{5}} \] Input:

int((d*x^2+c)^(1/2)/x^6/(b*x^2+a),x)
 

Output:

(15*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**2*c**2*x 
**5 + 15*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**2*c**2 
*x**5 - 15*sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 
 2*sqrt(d)*sqrt(c + d*x**2)*b*x + 2*a*d + 2*b*d*x**2)*b**2*c**2*x**5 - 6*s 
qrt(c + d*x**2)*a**3*c**2 - 2*sqrt(c + d*x**2)*a**3*c*d*x**2 + 4*sqrt(c + 
d*x**2)*a**3*d**2*x**4 + 10*sqrt(c + d*x**2)*a**2*b*c**2*x**2 + 10*sqrt(c 
+ d*x**2)*a**2*b*c*d*x**4 - 30*sqrt(c + d*x**2)*a*b**2*c**2*x**4 - 4*sqrt( 
d)*a**3*d**2*x**5 + 2*sqrt(d)*a**2*b*c*d*x**5 + 18*sqrt(d)*a*b**2*c**2*x** 
5)/(30*a**4*c**2*x**5)