\(\int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^6} \, dx\) [254]

Optimal result

Integrand size = 23, antiderivative size = 478 \[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^6} \, dx=-\frac {\sqrt {c+d \sqrt {a+b x^2}}}{5 x^5}+\frac {b d \left (a d-c \sqrt {a+b x^2}\right ) \sqrt {c+d \sqrt {a+b x^2}}}{30 a \left (c^2-a d^2\right ) x^3}-\frac {b^2 d \sqrt {c+d \sqrt {a+b x^2}} \left (a d \left (c^2-5 a d^2\right )-4 c \left (c^2-2 a d^2\right ) \sqrt {a+b x^2}\right )}{60 a^2 \left (c^2-a d^2\right )^2 x}+\frac {b^2 c d \left (c^2-2 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {c+d \sqrt {a+b x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{15 a^{3/2} \left (c^2-a d^2\right )^2 x \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}}}-\frac {b^2 d \left (4 c^2-5 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{60 a^{3/2} \left (c^2-a d^2\right ) x \sqrt {c+d \sqrt {a+b x^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.09 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {c+d \sqrt {a+b x^2}}}{x^6} \, dx=\frac {-12 a^2 \left (c^2-a d^2\right )^2 \left (c+d \sqrt {a+b x^2}\right )-2 a b d \left (-c^2+a d^2\right ) x^2 \left (a d-c \sqrt {a+b x^2}\right ) \left (c+d \sqrt {a+b x^2}\right )+b^2 d x^4 \left (c+d \sqrt {a+b x^2}\right ) \left (5 a^2 d^3+4 c^3 \sqrt {a+b x^2}-a c d \left (c+8 d \sqrt {a+b x^2}\right )\right )-\frac {b^2 x^4 \left (4 b c d^2 \sqrt {-c-\sqrt {a} d} \left (c^2-2 a d^2\right ) x^2-4 i c \left (c^3+\sqrt {a} c^2 d-2 a c d^2-2 a^{3/2} d^3\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right )|\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )+i \sqrt {a} d \left (4 c^3+\sqrt {a} c^2 d-8 a c d^2-5 a^{3/2} d^3\right ) \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )\right )}{\sqrt {-c-\sqrt {a} d}}}{60 a^2 \left (c^2-a d^2\right )^2 x^5 \sqrt {c+d \sqrt {a+b x^2}}} \] Input:

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

Output:

(-12*a^2*(c^2 - a*d^2)^2*(c + d*Sqrt[a + b*x^2]) - 2*a*b*d*(-c^2 + a*d^2)* 
x^2*(a*d - c*Sqrt[a + b*x^2])*(c + d*Sqrt[a + b*x^2]) + b^2*d*x^4*(c + d*S 
qrt[a + b*x^2])*(5*a^2*d^3 + 4*c^3*Sqrt[a + b*x^2] - a*c*d*(c + 8*d*Sqrt[a 
 + b*x^2])) - (b^2*x^4*(4*b*c*d^2*Sqrt[-c - Sqrt[a]*d]*(c^2 - 2*a*d^2)*x^2 
 - (4*I)*c*(c^3 + Sqrt[a]*c^2*d - 2*a*c*d^2 - 2*a^(3/2)*d^3)*Sqrt[(d*(-Sqr 
t[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + Sqrt[ 
a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2)*Ellipt 
icE[I*ArcSinh[Sqrt[-c - Sqrt[a]*d]/Sqrt[c + d*Sqrt[a + b*x^2]]], (c - Sqrt 
[a]*d)/(c + Sqrt[a]*d)] + I*Sqrt[a]*d*(4*c^3 + Sqrt[a]*c^2*d - 8*a*c*d^2 - 
 5*a^(3/2)*d^3)*Sqrt[(d*(-Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^ 
2])]*Sqrt[(d*(Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c + d* 
Sqrt[a + b*x^2])^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c - Sqrt[a]*d]/Sqrt[c + d 
*Sqrt[a + b*x^2]]], (c - Sqrt[a]*d)/(c + Sqrt[a]*d)]))/Sqrt[-c - Sqrt[a]*d 
])/(60*a^2*(c^2 - a*d^2)^2*x^5*Sqrt[c + d*Sqrt[a + b*x^2]])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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