\(\int (d+e x^2)^{3/2} \sqrt {a-c x^4} \, dx\) [436]

Optimal result

Integrand size = 24, antiderivative size = 567 \[ \int \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\frac {\left (\frac {3 d^2}{e}-\frac {8 a e}{c}\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 x}+\frac {7}{24} d x \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {1}{6} e x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}+\frac {\left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (3 c d^2-8 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 e \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \left (31 c d^2+8 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{48 \sqrt {c} \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {d \left (c d^2+12 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{16 e \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

1/48*(3*d^2/e-8*a*e/c)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/x+7/24*d*x*(e*x^2+ 
d)^(1/2)*(-c*x^4+a)^(1/2)+1/6*e*x^3*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)+1/48* 
(d+a^(1/2)*e/c^(1/2))*(-8*a*e^2+3*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/(e*x^2+d)^(1 
/2)/(-c*x^4+a)^(1/2)+1/48*a^(1/2)*(8*a*e^2+31*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))/c^ 
(1/2)/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)+1/16*d*(12*a*e^2+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/(e*x^2+d)^(1/2)/(-c*x^4+a)^(1/2)
 

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 1571

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

Input:

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

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d+e x^2\right )^{3/2} \sqrt {a-c x^4} \, dx=\frac {7 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, d x}{24}+\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, e \,x^{3}}{6}+\frac {\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,e^{2}}{3}-\frac {\left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c \,d^{2}}{8}+\frac {11 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a d e}{12}+\frac {17 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a \,d^{2}}{24} \] Input:

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

Output:

(7*sqrt(d + e*x**2)*sqrt(a - c*x**4)*d*x + 4*sqrt(d + e*x**2)*sqrt(a - c*x 
**4)*e*x**3 + 8*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*e**2 - 3*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*d**2 + 22*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*d*e + 17*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*d**2)/24