\(\int \frac {x^4}{\sqrt {c+d \sqrt {a+b x^2}}} \, dx\) [275]

Optimal result

Integrand size = 23, antiderivative size = 458 \[ \int \frac {x^4}{\sqrt {c+d \sqrt {a+b x^2}}} \, dx=-\frac {4 c \left (32 c^2-33 a d^2\right ) x \sqrt {c+d \sqrt {a+b x^2}}}{315 b^2 d^4}+\frac {4 \left (8 c^2-7 a d^2\right ) x \sqrt {a+b x^2} \sqrt {c+d \sqrt {a+b x^2}}}{105 b^2 d^3}-\frac {2 x^3 \left (8 c-7 d \sqrt {a+b x^2}\right ) \sqrt {c+d \sqrt {a+b x^2}}}{63 b d^2}-\frac {8 \sqrt {a} \left (32 c^4-57 a c^2 d^2+21 a^2 d^4\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {c+d \sqrt {a+b x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{315 b^3 d^5 x \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}}}+\frac {8 \sqrt {a} c \left (32 c^4-65 a c^2 d^2+33 a^2 d^4\right ) \sqrt {-\frac {b x^2}{a}} \sqrt {\frac {c+d \sqrt {a+b x^2}}{c+\sqrt {a} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} d}{c+\sqrt {a} d}\right )}{315 b^3 d^5 x \sqrt {c+d \sqrt {a+b x^2}}} \] Output:

-4/315*c*(-33*a*d^2+32*c^2)*x*(c+d*(b*x^2+a)^(1/2))^(1/2)/b^2/d^4+4/105*(- 
7*a*d^2+8*c^2)*x*(b*x^2+a)^(1/2)*(c+d*(b*x^2+a)^(1/2))^(1/2)/b^2/d^3-2/63* 
x^3*(8*c-7*d*(b*x^2+a)^(1/2))*(c+d*(b*x^2+a)^(1/2))^(1/2)/b/d^2-8/315*a^(1 
/2)*(21*a^2*d^4-57*a*c^2*d^2+32*c^4)*(-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))/b^3/d^5/x/((c+d*(b*x^2+a)^(1/2))/(c+a^(1/2) 
*d))^(1/2)+8/315*a^(1/2)*c*(33*a^2*d^4-65*a*c^2*d^2+32*c^4)*(-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))/b^3/ 
d^5/x/(c+d*(b*x^2+a)^(1/2))^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 80.65 (sec) , antiderivative size = 27114, normalized size of antiderivative = 59.20 \[ \int \frac {x^4}{\sqrt {c+d \sqrt {a+b x^2}}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

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.

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

\(\Big \downarrow \) 7299

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 7299

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^4}{\sqrt {c+d \sqrt {a+b x^2}}} \, dx=\text {too large to display} \] Input:

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

Output:

(2*( - 6*sqrt(a + b*x**2)*sqrt(sqrt(a + b*x**2)*d + c)*a**2*d**2*x + 7*sqr 
t(a + b*x**2)*sqrt(sqrt(a + b*x**2)*d + c)*a*b*d**2*x**3 + 6*sqrt(a + b*x* 
*2)*sqrt(sqrt(a + b*x**2)*d + c)*a*c**2*x - 4*sqrt(a + b*x**2)*sqrt(sqrt(a 
 + b*x**2)*d + c)*b*c**2*x**3 - 196*int((sqrt(sqrt(a + b*x**2)*d + c)*x**6 
)/(7*a**3*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x 
**4 - 15*a*b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4* 
x**2),x)*a**2*b**3*c*d**5 + 224*int((sqrt(sqrt(a + b*x**2)*d + c)*x**6)/(7 
*a**3*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 
- 15*a*b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2 
),x)*a*b**3*c**3*d**3 - 64*int((sqrt(sqrt(a + b*x**2)*d + c)*x**6)/(7*a**3 
*d**4 + 14*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15* 
a*b*c**2*d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2),x)* 
b**3*c**5*d - 217*int((sqrt(sqrt(a + b*x**2)*d + c)*x**4)/(7*a**3*d**4 + 1 
4*a**2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15*a*b*c**2* 
d**2*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2),x)*a**3*b**2 
*c*d**5 + 257*int((sqrt(sqrt(a + b*x**2)*d + c)*x**4)/(7*a**3*d**4 + 14*a* 
*2*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15*a*b*c**2*d**2 
*x**2 + 4*a*c**4 - 4*b**2*c**2*d**2*x**4 + 4*b*c**4*x**2),x)*a**2*b**2*c** 
3*d**3 - 76*int((sqrt(sqrt(a + b*x**2)*d + c)*x**4)/(7*a**3*d**4 + 14*a**2 
*b*d**4*x**2 - 11*a**2*c**2*d**2 + 7*a*b**2*d**4*x**4 - 15*a*b*c**2*d**...