\(\int (d+e x^2) (d^2-e^2 x^4)^{3/2} \, dx\) [45]

Optimal result

Integrand size = 24, antiderivative size = 190 \[ \int \left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {2}{105} d^2 x \left (15 d+7 e x^2\right ) \sqrt {d^2-e^2 x^4}+\frac {1}{63} x \left (9 d+7 e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2}+\frac {4 d^{11/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{15 \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {32 d^{11/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{105 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

2/105*d^2*x*(7*e*x^2+15*d)*(-e^2*x^4+d^2)^(1/2)+1/63*x*(7*e*x^2+9*d)*(-e^2 
*x^4+d^2)^(3/2)+4/15*d^(11/2)*(1-e^2*x^4/d^2)^(1/2)*EllipticE(e^(1/2)*x/d^ 
(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+32/105*d^(11/2)*(1-e^2*x^4/d^2)^(1/2 
)*EllipticF(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 8.74 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.47 \[ \int \left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {d^2 x \sqrt {d^2-e^2 x^4} \left (3 d \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+e x^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{3 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:

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

Output:

(d^2*x*Sqrt[d^2 - e^2*x^4]*(3*d*Hypergeometric2F1[-3/2, 1/4, 5/4, (e^2*x^4 
)/d^2] + e*x^2*Hypergeometric2F1[-3/2, 3/4, 7/4, (e^2*x^4)/d^2]))/(3*Sqrt[ 
1 - (e^2*x^4)/d^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.

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

\(\Big \downarrow \) 1571

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

Input:

Int[(d + e*x^2)*(d^2 - e^2*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 [A] (verified)

Time = 3.76 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.02

method	result	size
risch	\(\frac {x \left (-35 e^{3} x^{6}-45 d \,e^{2} x^{4}+77 d^{2} e \,x^{2}+135 d^{3}\right ) \sqrt {-e^{2} x^{4}+d^{2}}}{315}+\frac {4 d^{4} \left (\frac {15 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {7 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{105}\)	\(194\)
elliptic	\(-\frac {e^{3} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}-\frac {d \,e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {11 d^{2} e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{45}+\frac {3 d^{3} x \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {4 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{15 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(231\)
default	\(d \left (-\frac {e^{2} x^{5} \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {3 d^{2} x \sqrt {-e^{2} x^{4}+d^{2}}}{7}+\frac {4 d^{4} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{7 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e \left (-\frac {e^{2} x^{7} \sqrt {-e^{2} x^{4}+d^{2}}}{9}+\frac {11 d^{2} x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{45}-\frac {4 d^{5} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{15 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\)	\(238\)

Input:

int((e*x^2+d)*(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/315*x*(-35*e^3*x^6-45*d*e^2*x^4+77*d^2*e*x^2+135*d^3)*(-e^2*x^4+d^2)^(1/ 
2)+4/105*d^4*(15*d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x 
^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2),I)-7*d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2) 
*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*(EllipticF(x*(e/d)^(1/2),I)-Ellipt 
icE(x*(e/d)^(1/2),I)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.77 \[ \int \left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2} \, dx=-\frac {84 \, \sqrt {-e^{2}} d^{5} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 12 \, {\left (7 \, d^{5} + 15 \, d^{4} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + {\left (35 \, e^{5} x^{8} + 45 \, d e^{4} x^{6} - 77 \, d^{2} e^{3} x^{4} - 135 \, d^{3} e^{2} x^{2} + 84 \, d^{4} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{315 \, e^{2} x} \] Input:

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

Output:

-1/315*(84*sqrt(-e^2)*d^5*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) 
- 12*(7*d^5 + 15*d^4*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e) 
/x), -1) + (35*e^5*x^8 + 45*d*e^4*x^6 - 77*d^2*e^3*x^4 - 135*d^3*e^2*x^2 + 
 84*d^4*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
 

Sympy [A] (verification not implemented)

Time = 1.76 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.95 \[ \int \left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {d^{4} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {d^{3} e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} - \frac {d^{2} e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {d e^{3} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {11}{4}\right )} \] Input:

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

Output:

d**4*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d 
**2)/(4*gamma(5/4)) + d**3*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), e* 
*2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(7/4)) - d**2*e**2*x**5*gamma(5/4) 
*hyper((-1/2, 5/4), (9/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(9/4 
)) - d*e**3*x**7*gamma(7/4)*hyper((-1/2, 7/4), (11/4,), e**2*x**4*exp_pola 
r(2*I*pi)/d**2)/(4*gamma(11/4))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (d+e x^2\right ) \left (d^2-e^2 x^4\right )^{3/2} \, dx=\frac {3 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{3} x}{7}+\frac {11 \sqrt {-e^{2} x^{4}+d^{2}}\, d^{2} e \,x^{3}}{45}-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, d \,e^{2} x^{5}}{7}-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{3} x^{7}}{9}+\frac {4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{5}}{7}+\frac {4 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{4} e}{15} \] Input:

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

Output:

(135*sqrt(d**2 - e**2*x**4)*d**3*x + 77*sqrt(d**2 - e**2*x**4)*d**2*e*x**3 
 - 45*sqrt(d**2 - e**2*x**4)*d*e**2*x**5 - 35*sqrt(d**2 - e**2*x**4)*e**3* 
x**7 + 180*int(sqrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**5 + 84*int( 
(sqrt(d**2 - e**2*x**4)*x**2)/(d**2 - e**2*x**4),x)*d**4*e)/315