\(\int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx\) [35]

Optimal result

Integrand size = 21, antiderivative size = 219 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 x^5}+\frac {7 b^2 c \sqrt {a+b \sqrt {c x^2}}}{240 a^2 x^3}+\frac {7 b^4 c^2 \sqrt {a+b \sqrt {c x^2}}}{128 a^4 x}-\frac {b \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{40 a c^2 x^9}-\frac {7 b^3 \left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}{192 a^3 c x^7}-\frac {7 b^5 \left (c x^2\right )^{5/2} \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{128 a^{9/2} x^5} \] Output:

-1/5*(a+b*(c*x^2)^(1/2))^(1/2)/x^5+7/240*b^2*c*(a+b*(c*x^2)^(1/2))^(1/2)/a 
^2/x^3+7/128*b^4*c^2*(a+b*(c*x^2)^(1/2))^(1/2)/a^4/x-1/40*b*(c*x^2)^(5/2)* 
(a+b*(c*x^2)^(1/2))^(1/2)/a/c^2/x^9-7/192*b^3*(c*x^2)^(5/2)*(a+b*(c*x^2)^( 
1/2))^(1/2)/a^3/c/x^7-7/128*b^5*(c*x^2)^(5/2)*arctanh((a+b*(c*x^2)^(1/2))^ 
(1/2)/a^(1/2))/a^(9/2)/x^5
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.29 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\frac {2 b^5 \left (c x^2\right )^{5/2} \left (a+b \sqrt {c x^2}\right )^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},6,\frac {5}{2},1+\frac {b \sqrt {c x^2}}{a}\right )}{3 a^6 x^5} \] Input:

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

Output:

(2*b^5*(c*x^2)^(5/2)*(a + b*Sqrt[c*x^2])^(3/2)*Hypergeometric2F1[3/2, 6, 5 
/2, 1 + (b*Sqrt[c*x^2])/a])/(3*a^6*x^5)
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {892, 51, 52, 52, 52, 52, 73, 221}

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.

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

\(\Big \downarrow \) 892

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \int \frac {\sqrt {a+b \sqrt {c x^2}}}{c^3 x^6}d\sqrt {c x^2}}{x^5}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \int \frac {1}{\left (c x^2\right )^{5/2} \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \left (-\frac {7 b \int \frac {1}{c^2 x^4 \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}}{8 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 a c^2 x^4}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \left (-\frac {7 b \left (-\frac {5 b \int \frac {1}{\left (c x^2\right )^{3/2} \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}}{6 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 a \left (c x^2\right )^{3/2}}\right )}{8 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 a c^2 x^4}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \int \frac {1}{c x^2 \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}}{4 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 a c x^2}\right )}{6 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 a \left (c x^2\right )^{3/2}}\right )}{8 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 a c^2 x^4}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{\sqrt {c x^2} \sqrt {a+b \sqrt {c x^2}}}d\sqrt {c x^2}}{2 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{a \sqrt {c x^2}}\right )}{4 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 a c x^2}\right )}{6 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 a \left (c x^2\right )^{3/2}}\right )}{8 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 a c^2 x^4}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {\int \frac {1}{\frac {c x^2}{b}-\frac {a}{b}}d\sqrt {a+b \sqrt {c x^2}}}{a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{a \sqrt {c x^2}}\right )}{4 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 a c x^2}\right )}{6 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 a \left (c x^2\right )^{3/2}}\right )}{8 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 a c^2 x^4}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (c x^2\right )^{5/2} \left (\frac {1}{10} b \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sqrt {c x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {\sqrt {a+b \sqrt {c x^2}}}{a \sqrt {c x^2}}\right )}{4 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{2 a c x^2}\right )}{6 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{3 a \left (c x^2\right )^{3/2}}\right )}{8 a}-\frac {\sqrt {a+b \sqrt {c x^2}}}{4 a c^2 x^4}\right )-\frac {\sqrt {a+b \sqrt {c x^2}}}{5 \left (c x^2\right )^{5/2}}\right )}{x^5}\)

Input:

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

Output:

((c*x^2)^(5/2)*(-1/5*Sqrt[a + b*Sqrt[c*x^2]]/(c*x^2)^(5/2) + (b*(-1/4*Sqrt 
[a + b*Sqrt[c*x^2]]/(a*c^2*x^4) - (7*b*(-1/3*Sqrt[a + b*Sqrt[c*x^2]]/(a*(c 
*x^2)^(3/2)) - (5*b*(-1/2*Sqrt[a + b*Sqrt[c*x^2]]/(a*c*x^2) - (3*b*(-(Sqrt 
[a + b*Sqrt[c*x^2]]/(a*Sqrt[c*x^2])) + (b*ArcTanh[Sqrt[a + b*Sqrt[c*x^2]]/ 
Sqrt[a]])/a^(3/2)))/(4*a)))/(6*a)))/(8*a)))/10))/x^5
 

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 892

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1))   Subst[Int[x^m*(a + b 
*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x 
] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
 

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.61

method	result	size
default	\(-\frac {105 a^{\frac {17}{2}} \sqrt {a +b \sqrt {c \,x^{2}}}+790 a^{\frac {15}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {3}{2}}-896 a^{\frac {13}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {5}{2}}+490 a^{\frac {11}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {7}{2}}-105 a^{\frac {9}{2}} \left (a +b \sqrt {c \,x^{2}}\right )^{\frac {9}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \sqrt {c \,x^{2}}}}{\sqrt {a}}\right ) a^{4} b^{5} \left (c \,x^{2}\right )^{\frac {5}{2}}}{1920 x^{5} a^{\frac {17}{2}}}\)	\(133\)

Input:

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

Output:

-1/1920*(105*a^(17/2)*(a+b*(c*x^2)^(1/2))^(1/2)+790*a^(15/2)*(a+b*(c*x^2)^ 
(1/2))^(3/2)-896*a^(13/2)*(a+b*(c*x^2)^(1/2))^(5/2)+490*a^(11/2)*(a+b*(c*x 
^2)^(1/2))^(7/2)-105*a^(9/2)*(a+b*(c*x^2)^(1/2))^(9/2)+105*arctanh((a+b*(c 
*x^2)^(1/2))^(1/2)/a^(1/2))*a^4*b^5*(c*x^2)^(5/2))/x^5/a^(17/2)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\left [\frac {105 \, b^{5} c^{2} x^{5} \sqrt {\frac {c}{a}} \log \left (\frac {b c x^{2} - 2 \, \sqrt {\sqrt {c x^{2}} b + a} a x \sqrt {\frac {c}{a}} + 2 \, \sqrt {c x^{2}} a}{x^{2}}\right ) + 2 \, {\left (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \, {\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{3840 \, a^{4} x^{5}}, -\frac {105 \, b^{5} c^{2} x^{5} \sqrt {-\frac {c}{a}} \arctan \left (-\frac {{\left (a b c x^{2} \sqrt {-\frac {c}{a}} - \sqrt {c x^{2}} a^{2} \sqrt {-\frac {c}{a}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{b^{2} c^{2} x^{3} - a^{2} c x}\right ) - {\left (105 \, b^{4} c^{2} x^{4} + 56 \, a^{2} b^{2} c x^{2} - 384 \, a^{4} - 2 \, {\left (35 \, a b^{3} c x^{2} + 24 \, a^{3} b\right )} \sqrt {c x^{2}}\right )} \sqrt {\sqrt {c x^{2}} b + a}}{1920 \, a^{4} x^{5}}\right ] \] Input:

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

Output:

[1/3840*(105*b^5*c^2*x^5*sqrt(c/a)*log((b*c*x^2 - 2*sqrt(sqrt(c*x^2)*b + a 
)*a*x*sqrt(c/a) + 2*sqrt(c*x^2)*a)/x^2) + 2*(105*b^4*c^2*x^4 + 56*a^2*b^2* 
c*x^2 - 384*a^4 - 2*(35*a*b^3*c*x^2 + 24*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x 
^2)*b + a))/(a^4*x^5), -1/1920*(105*b^5*c^2*x^5*sqrt(-c/a)*arctan(-(a*b*c* 
x^2*sqrt(-c/a) - sqrt(c*x^2)*a^2*sqrt(-c/a))*sqrt(sqrt(c*x^2)*b + a)/(b^2* 
c^2*x^3 - a^2*c*x)) - (105*b^4*c^2*x^4 + 56*a^2*b^2*c*x^2 - 384*a^4 - 2*(3 
5*a*b^3*c*x^2 + 24*a^3*b)*sqrt(c*x^2))*sqrt(sqrt(c*x^2)*b + a))/(a^4*x^5)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.55 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\frac {1}{1920} \, b^{5} c^{\frac {5}{2}} {\left (\frac {105 \, \arctan \left (\frac {\sqrt {b \sqrt {c} x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{4}} + \frac {105 \, {\left (b \sqrt {c} x + a\right )}^{\frac {9}{2}} - 490 \, {\left (b \sqrt {c} x + a\right )}^{\frac {7}{2}} a + 896 \, {\left (b \sqrt {c} x + a\right )}^{\frac {5}{2}} a^{2} - 790 \, {\left (b \sqrt {c} x + a\right )}^{\frac {3}{2}} a^{3} - 105 \, \sqrt {b \sqrt {c} x + a} a^{4}}{a^{4} b^{5} c^{\frac {5}{2}} x^{5}}\right )} \] Input:

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

Output:

1/1920*b^5*c^(5/2)*(105*arctan(sqrt(b*sqrt(c)*x + a)/sqrt(-a))/(sqrt(-a)*a 
^4) + (105*(b*sqrt(c)*x + a)^(9/2) - 490*(b*sqrt(c)*x + a)^(7/2)*a + 896*( 
b*sqrt(c)*x + a)^(5/2)*a^2 - 790*(b*sqrt(c)*x + a)^(3/2)*a^3 - 105*sqrt(b* 
sqrt(c)*x + a)*a^4)/(a^4*b^5*c^(5/2)*x^5))
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.72 \[ \int \frac {\sqrt {a+b \sqrt {c x^2}}}{x^6} \, dx=\frac {-96 \sqrt {c}\, \sqrt {\sqrt {c}\, b x +a}\, a^{4} b x -140 \sqrt {c}\, \sqrt {\sqrt {c}\, b x +a}\, a^{2} b^{3} c \,x^{3}-768 \sqrt {\sqrt {c}\, b x +a}\, a^{5}+112 \sqrt {\sqrt {c}\, b x +a}\, a^{3} b^{2} c \,x^{2}+210 \sqrt {\sqrt {c}\, b x +a}\, a \,b^{4} c^{2} x^{4}+105 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b x +a}-\sqrt {a}\right ) b^{5} c^{2} x^{5}-105 \sqrt {c}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, b x +a}+\sqrt {a}\right ) b^{5} c^{2} x^{5}}{3840 a^{5} x^{5}} \] Input:

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

Output:

( - 96*sqrt(c)*sqrt(sqrt(c)*b*x + a)*a**4*b*x - 140*sqrt(c)*sqrt(sqrt(c)*b 
*x + a)*a**2*b**3*c*x**3 - 768*sqrt(sqrt(c)*b*x + a)*a**5 + 112*sqrt(sqrt( 
c)*b*x + a)*a**3*b**2*c*x**2 + 210*sqrt(sqrt(c)*b*x + a)*a*b**4*c**2*x**4 
+ 105*sqrt(c)*sqrt(a)*log(sqrt(sqrt(c)*b*x + a) - sqrt(a))*b**5*c**2*x**5 
- 105*sqrt(c)*sqrt(a)*log(sqrt(sqrt(c)*b*x + a) + sqrt(a))*b**5*c**2*x**5) 
/(3840*a**5*x**5)