\(\int \sqrt {d+e x^2} (a-c x^4)^{3/2} \, dx\) [444]

Optimal result

Integrand size = 24, antiderivative size = 634 \[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=-\frac {d \left (15 c d^2-68 a e^2\right ) \sqrt {d+e x^2} \sqrt {a-c x^4}}{384 e^3 x}+\frac {5}{192} \left (12 a+\frac {c d^2}{e^2}\right ) x \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {c d x^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}{48 e}-\frac {1}{8} c x^5 \sqrt {d+e x^2} \sqrt {a-c x^4}-\frac {c d \left (d+\frac {\sqrt {a} e}{\sqrt {c}}\right ) \left (15 c d^2-68 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right )|\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}}+\frac {\sqrt {a} \sqrt {c} d \left (5 c d^2+196 a e^2\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{384 e^2 \sqrt {d+e x^2} \sqrt {a-c x^4}}-\frac {\left (5 c^2 d^4-24 a c d^2 e^2-48 a^2 e^4\right ) \sqrt {1-\frac {a}{c x^4}} x^3 \sqrt {\frac {\sqrt {a} \left (d+e x^2\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) x^2}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {a}}{\sqrt {c} x^2}}}{\sqrt {2}}\right ),\frac {2 d}{d+\frac {\sqrt {a} e}{\sqrt {c}}}\right )}{128 e^3 \sqrt {d+e x^2} \sqrt {a-c x^4}} \] Output:

-1/384*d*(-68*a*e^2+15*c*d^2)*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)/e^3/x+5/192 
*(12*a+c*d^2/e^2)*x*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/48*c*d*x^3*(e*x^2+d 
)^(1/2)*(-c*x^4+a)^(1/2)/e-1/8*c*x^5*(e*x^2+d)^(1/2)*(-c*x^4+a)^(1/2)-1/38 
4*c*d*(d+a^(1/2)*e/c^(1/2))*(-68*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)+1/384*a^(1/2)*c^(1/2)*d*(196*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)-1/128*(-48*a^2*e^ 
4-24*a*c*d^2*e^2+5*c^2*d^4)*(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]

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

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

Output:

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

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

\(\Big \downarrow \) 1571

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

\[ \int \sqrt {d+e x^2} \left (a-c x^4\right )^{3/2} \, dx=\frac {60 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, a \,e^{2} x +5 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c \,d^{2} x -4 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c d e \,x^{3}-24 \sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, c \,e^{2} x^{5}-68 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c d \,e^{2}+15 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{4}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) c^{2} d^{3}+72 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} e^{3}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}\, x^{2}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{2} e +132 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a^{2} d \,e^{2}-5 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-c \,x^{4}+a}}{-c e \,x^{6}-c d \,x^{4}+a e \,x^{2}+a d}d x \right ) a c \,d^{3}}{192 e^{2}} \] Input:

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

Output:

(60*sqrt(d + e*x**2)*sqrt(a - c*x**4)*a*e**2*x + 5*sqrt(d + e*x**2)*sqrt(a 
 - c*x**4)*c*d**2*x - 4*sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*d*e*x**3 - 24* 
sqrt(d + e*x**2)*sqrt(a - c*x**4)*c*e**2*x**5 - 68*int((sqrt(d + e*x**2)*s 
qrt(a - c*x**4)*x**4)/(a*d + a*e*x**2 - c*d*x**4 - c*e*x**6),x)*a*c*d*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**3 + 72*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**2*e**3 + 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**2*e + 132*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*d*e**2 - 5*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*c*d**3)/(192*e**2)