\(\int \frac {(c+d \sqrt {a+b x^2})^{3/2}}{x^6} \, dx\) [265]

Optimal result

Integrand size = 23, antiderivative size = 429 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=-\frac {b d \sqrt {a+b x^2} \sqrt {c+d \sqrt {a+b x^2}}}{10 a x^3}-\frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{5 x^5}-\frac {b^2 d \sqrt {c+d \sqrt {a+b x^2}} \left (a c d-\left (4 c^2-3 a d^2\right ) \sqrt {a+b x^2}\right )}{20 a^2 \left (c^2-a d^2\right ) x}+\frac {b^2 d \left (4 c^2-3 a d^2\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {c+d \sqrt {a+b x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{20 a^{3/2} \left (c^2-a d^2\right ) x \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}}}-\frac {b^2 c d \sqrt {-\frac {b x^2}{a}} \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{5 a^{3/2} x \sqrt {c+d \sqrt {a+b x^2}}} \] Output:

-1/10*b*d*(b*x^2+a)^(1/2)*(c+d*(b*x^2+a)^(1/2))^(1/2)/a/x^3-1/5*(c+d*(b*x^ 
2+a)^(1/2))^(3/2)/x^5-1/20*b^2*d*(c+d*(b*x^2+a)^(1/2))^(1/2)*(a*c*d-(-3*a* 
d^2+4*c^2)*(b*x^2+a)^(1/2))/a^2/(-a*d^2+c^2)/x+1/20*b^2*d*(-3*a*d^2+4*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)/x/((c+d*(b*x^2+a)^(1/2))/(c+a^(1/2)*d))^(1/2)-1/5*b^2*c*d* 
(-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)/x/(c+d*(b*x^2+a)^(1/2))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 38.54 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.30 \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\frac {\sqrt {-c-\sqrt {a} d} \left (-c+\sqrt {a} d\right ) \left (4 a^3 d^2-4 b^2 c d x^4 \sqrt {a+b x^2}+a b d x^2 \left (-b d x^2+2 c \sqrt {a+b x^2}\right )+a^2 \left (4 c^2+6 b d^2 x^2+8 c d \sqrt {a+b x^2}\right )\right )-i b^2 \left (-4 c^2+3 a d^2\right ) x^4 \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right )|\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )+i \sqrt {a} b^2 d \left (-4 c+3 \sqrt {a} d\right ) x^4 \sqrt {\frac {d \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \sqrt {\frac {d \left (\sqrt {a}+\sqrt {a+b x^2}\right )}{c+d \sqrt {a+b x^2}}} \left (c+d \sqrt {a+b x^2}\right )^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\sqrt {a} d}}{\sqrt {c+d \sqrt {a+b x^2}}}\right ),\frac {c-\sqrt {a} d}{c+\sqrt {a} d}\right )}{20 a^2 \sqrt {-c-\sqrt {a} d} \left (c-\sqrt {a} d\right ) x^5 \sqrt {c+d \sqrt {a+b x^2}}} \] Input:

Integrate[(c + d*Sqrt[a + b*x^2])^(3/2)/x^6,x]
 

Output:

(Sqrt[-c - Sqrt[a]*d]*(-c + Sqrt[a]*d)*(4*a^3*d^2 - 4*b^2*c*d*x^4*Sqrt[a + 
 b*x^2] + a*b*d*x^2*(-(b*d*x^2) + 2*c*Sqrt[a + b*x^2]) + a^2*(4*c^2 + 6*b* 
d^2*x^2 + 8*c*d*Sqrt[a + b*x^2])) - I*b^2*(-4*c^2 + 3*a*d^2)*x^4*Sqrt[(d*( 
-Sqrt[a] + Sqrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*Sqrt[(d*(Sqrt[a] + S 
qrt[a + b*x^2]))/(c + d*Sqrt[a + b*x^2])]*(c + d*Sqrt[a + b*x^2])^(3/2)*El 
lipticE[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]*b^2*d*(-4*c + 3*Sqrt[a]*d)*x^4*Sqr 
t[(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)])/(20*a^2*Sqrt[-c - Sqrt[a]*d]*(c - Sqrt[ 
a]*d)*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.

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

\(\Big \downarrow \) 7299

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

Input:

Int[(c + d*Sqrt[a + b*x^2])^(3/2)/x^6,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

\[\int \frac {\left (c +d \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{x^{6}}d x\]

Input:

int((c+d*(b*x^2+a)^(1/2))^(3/2)/x^6,x)
 

Output:

int((c+d*(b*x^2+a)^(1/2))^(3/2)/x^6,x)
 

Fricas [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \] Input:

integrate((c+d*(b*x^2+a)^(1/2))^(3/2)/x^6,x, algorithm="fricas")
 

Output:

integral((sqrt(b*x^2 + a)*d + c)^(3/2)/x^6, x)
 

Sympy [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\int \frac {\left (c + d \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{x^{6}}\, dx \] Input:

integrate((c+d*(b*x**2+a)**(1/2))**(3/2)/x**6,x)
 

Output:

Integral((c + d*sqrt(a + b*x**2))**(3/2)/x**6, x)
 

Maxima [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \] Input:

integrate((c+d*(b*x^2+a)^(1/2))^(3/2)/x^6,x, algorithm="maxima")
 

Output:

integrate((sqrt(b*x^2 + a)*d + c)^(3/2)/x^6, x)
 

Giac [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\int { \frac {{\left (\sqrt {b x^{2} + a} d + c\right )}^{\frac {3}{2}}}{x^{6}} \,d x } \] Input:

integrate((c+d*(b*x^2+a)^(1/2))^(3/2)/x^6,x, algorithm="giac")
 

Output:

integrate((sqrt(b*x^2 + a)*d + c)^(3/2)/x^6, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\int \frac {{\left (c+d\,\sqrt {b\,x^2+a}\right )}^{3/2}}{x^6} \,d x \] Input:

int((c + d*(a + b*x^2)^(1/2))^(3/2)/x^6,x)
 

Output:

int((c + d*(a + b*x^2)^(1/2))^(3/2)/x^6, x)
 

Reduce [F]

\[ \int \frac {\left (c+d \sqrt {a+b x^2}\right )^{3/2}}{x^6} \, dx=\text {too large to display} \] Input:

int((c+d*(b*x^2+a)^(1/2))^(3/2)/x^6,x)
 

Output:

( - 2*sqrt(a + b*x**2)*sqrt(sqrt(a + b*x**2)*d + c)*d + 3*int(sqrt(sqrt(a 
+ b*x**2)*d + c)/(17*a**4*d**6*x**4 + 34*a**3*b*d**6*x**6 - 42*a**3*c**2*d 
**4*x**4 + 17*a**2*b**2*d**6*x**8 - 67*a**2*b*c**2*d**4*x**6 + 33*a**2*c** 
4*d**2*x**4 - 25*a*b**2*c**2*d**4*x**8 + 41*a*b*c**4*d**2*x**6 - 8*a*c**6* 
x**4 + 8*b**2*c**4*d**2*x**8 - 8*b*c**6*x**6),x)*a**3*b*c*d**6*x**5 - 5*in 
t(sqrt(sqrt(a + b*x**2)*d + c)/(17*a**4*d**6*x**4 + 34*a**3*b*d**6*x**6 - 
42*a**3*c**2*d**4*x**4 + 17*a**2*b**2*d**6*x**8 - 67*a**2*b*c**2*d**4*x**6 
 + 33*a**2*c**4*d**2*x**4 - 25*a*b**2*c**2*d**4*x**8 + 41*a*b*c**4*d**2*x* 
*6 - 8*a*c**6*x**4 + 8*b**2*c**4*d**2*x**8 - 8*b*c**6*x**6),x)*a**2*b*c**3 
*d**4*x**5 + 2*int(sqrt(sqrt(a + b*x**2)*d + c)/(17*a**4*d**6*x**4 + 34*a* 
*3*b*d**6*x**6 - 42*a**3*c**2*d**4*x**4 + 17*a**2*b**2*d**6*x**8 - 67*a**2 
*b*c**2*d**4*x**6 + 33*a**2*c**4*d**2*x**4 - 25*a*b**2*c**2*d**4*x**8 + 41 
*a*b*c**4*d**2*x**6 - 8*a*c**6*x**4 + 8*b**2*c**4*d**2*x**8 - 8*b*c**6*x** 
6),x)*a*b*c**5*d**2*x**5 + 3*int(sqrt(sqrt(a + b*x**2)*d + c)/(17*a**4*d** 
6*x**2 + 34*a**3*b*d**6*x**4 - 42*a**3*c**2*d**4*x**2 + 17*a**2*b**2*d**6* 
x**6 - 67*a**2*b*c**2*d**4*x**4 + 33*a**2*c**4*d**2*x**2 - 25*a*b**2*c**2* 
d**4*x**6 + 41*a*b*c**4*d**2*x**4 - 8*a*c**6*x**2 + 8*b**2*c**4*d**2*x**6 
- 8*b*c**6*x**4),x)*a**2*b**2*c*d**6*x**5 - 5*int(sqrt(sqrt(a + b*x**2)*d 
+ c)/(17*a**4*d**6*x**2 + 34*a**3*b*d**6*x**4 - 42*a**3*c**2*d**4*x**2 + 1 
7*a**2*b**2*d**6*x**6 - 67*a**2*b*c**2*d**4*x**4 + 33*a**2*c**4*d**2*x*...