\(\int \frac {(a-c x^4)^{3/2}}{\sqrt {d+e x^2}} \, dx\) [445]

Optimal result

Integrand size = 24, antiderivative size = 580 \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=-\frac {\left (15 c d^2-32 a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 e^3 x}+\frac {5 c d x \sqrt {d+e x^2} \sqrt {a-c x^4}}{24 e^2}-\frac {c x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{6 e}-\frac {c \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (15 c d^2-32 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} \left (5 c d^2+16 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {c d \left (5 c d^2-12 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/48*(-32*a*e^2+15*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e^3/x+5/24*c*d 
*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e^2-1/6*c*x^3*(e*x^2+d)^(1/2)*(-c*x^4+ 
a)^(1/2)/e-1/48*c*(d+a^(1/2)*e/c^(1/2))*(-32*a*e^2+15*c*d^2)*(1-a/c/x^4)^( 
1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^2)^(1/2)*EllipticE(1/2 
*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^( 
1/2))/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/48*a^(1/2)*c^(1/2)*(16*a*e^2+ 
5*c*d^2)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^(1/2)*e)/x^ 
2)^(1/2)*EllipticF(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2^(1/2)*(d/(d 
+a^(1/2)*e/c^(1/2)))^(1/2))/e^2/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)-1/16*c*d* 
(-12*a*e^2+5*c*d^2)*(1-a/c/x^4)^(1/2)*x^3*(a^(1/2)*(e*x^2+d)/(c^(1/2)*d+a^ 
(1/2)*e)/x^2)^(1/2)*EllipticPi(1/2*(1-a^(1/2)/c^(1/2)/x^2)^(1/2)*2^(1/2),2 
,2^(1/2)*(d/(d+a^(1/2)*e/c^(1/2)))^(1/2))/e^3/(e*x^2+d)^(1/2)/(-c*x^4+a)^( 
1/2)
 

Mathematica [F]

\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx \] Input:

Integrate[(a - c*x^4)^(3/2)/Sqrt[d + e*x^2],x]
 

Output:

Integrate[(a - c*x^4)^(3/2)/Sqrt[d + e*x^2], x]
 

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.

\(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx\)

\(\Big \downarrow \) 1571

\(\displaystyle \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}}dx\)

Input:

Int[(a - c*x^4)^(3/2)/Sqrt[d + e*x^2],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U 
nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
 

Maple [F]

\[\int \frac {\left (-c \,x^{4}+a \right )^{\frac {3}{2}}}{\sqrt {e \,x^{2}+d}}d x\]

Input:

int((-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x)
 

Output:

int((-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\text {Timed out} \] Input:

integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {\left (a - c x^{4}\right )^{\frac {3}{2}}}{\sqrt {d + e x^{2}}}\, dx \] Input:

integrate((-c*x**4+a)**(3/2)/(e*x**2+d)**(1/2),x)
 

Output:

Integral((a - c*x**4)**(3/2)/sqrt(d + e*x**2), x)
 

Maxima [F]

\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{\sqrt {e x^{2} + d}} \,d x } \] Input:

integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="maxima")
 

Output:

integrate((-c*x^4 + a)^(3/2)/sqrt(e*x^2 + d), x)
 

Giac [F]

\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int { \frac {{\left (-c x^{4} + a\right )}^{\frac {3}{2}}}{\sqrt {e x^{2} + d}} \,d x } \] Input:

integrate((-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x, algorithm="giac")
 

Output:

integrate((-c*x^4 + a)^(3/2)/sqrt(e*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\int \frac {{\left (a-c\,x^4\right )}^{3/2}}{\sqrt {e\,x^2+d}} \,d x \] Input:

int((a - c*x^4)^(3/2)/(d + e*x^2)^(1/2),x)
 

Output:

int((a - c*x^4)^(3/2)/(d + e*x^2)^(1/2), x)
 

Reduce [F]

\[ \int \frac {\left (a-c x^4\right )^{3/2}}{\sqrt {d+e x^2}} \, dx=\frac {5 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c d x -4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c e \,x^{3}-32 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,e^{2}+15 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c^{2} d^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d e +24 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e^{2}-5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{2}}{24 e^{2}} \] Input:

int((-c*x^4+a)^(3/2)/(e*x^2+d)^(1/2),x)
 

Output:

(5*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d*x - 4*sqrt(d + e*x**2)*sqrt(a - c 
*x**4)*c*e*x**3 - 32*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**4)/(a*d + a 
*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*e**2 + 15*int((sqrt(d + e*x**2)*sqrt 
(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*c**2*d**2 + 2 
*int((sqrt(d + e*x**2)*sqrt(a - c*x**4)*x**2)/(a*d + a*e*x**2 - c*d*x**4 - 
 c*e*x**6),x)*a*c*d*e + 24*int((sqrt(d + e*x**2)*sqrt(a - c*x**4))/(a*d + 
a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a**2*e**2 - 5*int((sqrt(d + e*x**2)*sqr 
t(a - c*x**4))/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d**2)/(24*e** 
2)