\(\int \frac {(d+e x^2)^2}{\sqrt {a-c x^4}} \, dx\) [406]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=-\frac {e^2 x \sqrt {a-c x^4}}{3 c}+\frac {2 a^{3/4} d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{c^{3/4} \sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \left (3 c d^2-6 \sqrt {a} \sqrt {c} d e+a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{3 c^{5/4} \sqrt {a-c x^4}} \] Output:

-1/3*e^2*x*(-c*x^4+a)^(1/2)/c+2*a^(3/4)*d*e*(1-c*x^4/a)^(1/2)*EllipticE(c^ 
(1/4)*x/a^(1/4),I)/c^(3/4)/(-c*x^4+a)^(1/2)+1/3*a^(1/4)*(3*c*d^2-6*a^(1/2) 
*c^(1/2)*d*e+a*e^2)*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(5/ 
4)/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

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

Time = 10.07 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.75 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=\frac {\left (3 c d^2+a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+e x \left (-a e+c e x^4+2 c d x^2 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {c x^4}{a}\right )\right )}{3 c \sqrt {a-c x^4}} \] Input:

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

Output:

((3*c*d^2 + a*e^2)*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, 
(c*x^4)/a] + e*x*(-(a*e) + c*e*x^4 + 2*c*d*x^2*Sqrt[1 - (c*x^4)/a]*Hyperge 
ometric2F1[1/2, 3/4, 7/4, (c*x^4)/a]))/(3*c*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {1519, 25, 1513, 27, 765, 762, 1390, 1389, 327}

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.

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

\(\Big \downarrow \) 1519

\(\displaystyle -\frac {\int -\frac {3 c d^2+6 c e x^2 d+a e^2}{\sqrt {a-c x^4}}dx}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {3 c d^2+6 c e x^2 d+a e^2}{\sqrt {a-c x^4}}dx}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 1513

\(\displaystyle \frac {\left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+6 \sqrt {a} \sqrt {c} d e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \sqrt {a-c x^4}}dx}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {1}{\sqrt {a-c x^4}}dx+6 \sqrt {c} d e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\frac {\sqrt {1-\frac {c x^4}{a}} \left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \int \frac {1}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+6 \sqrt {c} d e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 762

\(\displaystyle \frac {6 \sqrt {c} d e \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a-c x^4}}dx+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 1390

\(\displaystyle \frac {\frac {6 \sqrt {c} d e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {1-\frac {c x^4}{a}}}dx}{\sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 1389

\(\displaystyle \frac {\frac {6 \sqrt {a} \sqrt {c} d e \sqrt {1-\frac {c x^4}{a}} \int \frac {\sqrt {\frac {\sqrt {c} x^2}{\sqrt {a}}+1}}{\sqrt {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}}dx}{\sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\frac {6 a^{3/4} \sqrt [4]{c} d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt {a-c x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} \left (-6 \sqrt {a} \sqrt {c} d e+a e^2+3 c d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} \sqrt {a-c x^4}}}{3 c}-\frac {e^2 x \sqrt {a-c x^4}}{3 c}\)

Input:

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

Output:

-1/3*(e^2*x*Sqrt[a - c*x^4])/c + ((6*a^(3/4)*c^(1/4)*d*e*Sqrt[1 - (c*x^4)/ 
a]*EllipticE[ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^4] + (a^(1/4)* 
(3*c*d^2 - 6*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[Ar 
cSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]))/(3*c)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1389

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[d/Sq 
rt[a]   Int[Sqrt[1 + e*(x^2/d)]/Sqrt[1 - e*(x^2/d)], x], x] /; FreeQ[{a, c, 
 d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] && GtQ[a, 0]
 

rule 1390

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Simp[Sqrt 
[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], 
x] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 + a*e^2, 0] && NegQ[c/a] &&  !GtQ 
[a, 0] &&  !(LtQ[a, 0] && GtQ[c, 0])
 

rule 1513

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[-c/a, 2]}, Simp[(d*q - e)/q   Int[1/Sqrt[a + c*x^4], x], x] + Simp[e/q 
  Int[(1 + q*x^2)/Sqrt[a + c*x^4], x], x]] /; FreeQ[{a, c, d, e}, x] && Neg 
Q[c/a] && NeQ[c*d^2 + a*e^2, 0]
 

rule 1519

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Sim 
p[e^q*x^(2*q - 3)*((a + c*x^4)^(p + 1)/(c*(4*p + 2*q + 1))), x] + Simp[1/(c 
*(4*p + 2*q + 1))   Int[(a + c*x^4)^p*ExpandToSum[c*(4*p + 2*q + 1)*(d + e* 
x^2)^q - a*(2*q - 3)*e^q*x^(2*q - 4) - c*(4*p + 2*q + 1)*e^q*x^(2*q), x], x 
], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[q, 1]
 

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.15

method	result	size
elliptic	\(-\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\left (d^{2}+\frac {a \,e^{2}}{3 c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {2 d e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\)	\(186\)
default	\(\frac {d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+e^{2} \left (-\frac {x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {a \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{3 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )-\frac {2 d e \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\)	\(246\)
risch	\(-\frac {e^{2} x \sqrt {-c \,x^{4}+a}}{3 c}+\frac {\frac {a \,e^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {3 c \,d^{2} \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {6 e \sqrt {c}\, d \sqrt {a}\, \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}}{3 c}\)	\(251\)

Input:

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

Output:

-1/3*e^2*x*(-c*x^4+a)^(1/2)/c+(d^2+1/3*a*e^2/c)/(1/a^(1/2)*c^(1/2))^(1/2)* 
(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1/2)/(-c*x^4+a) 
^(1/2)*EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-2*d*e*a^(1/2)/(1/a^(1/2)*c 
^(1/2))^(1/2)*(1-1/a^(1/2)*c^(1/2)*x^2)^(1/2)*(1+1/a^(1/2)*c^(1/2)*x^2)^(1 
/2)/(-c*x^4+a)^(1/2)/c^(1/2)*(EllipticF(x*(1/a^(1/2)*c^(1/2))^(1/2),I)-Ell 
ipticE(x*(1/a^(1/2)*c^(1/2))^(1/2),I))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=-\frac {6 \, a \sqrt {-c} d e x \left (\frac {a}{c}\right )^{\frac {3}{4}} E(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) - {\left (3 \, c d^{2} + 6 \, a d e + a e^{2}\right )} \sqrt {-c} x \left (\frac {a}{c}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (\frac {a}{c}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (a e^{2} x^{2} + 6 \, a d e\right )} \sqrt {-c x^{4} + a}}{3 \, a c x} \] Input:

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

Output:

-1/3*(6*a*sqrt(-c)*d*e*x*(a/c)^(3/4)*elliptic_e(arcsin((a/c)^(1/4)/x), -1) 
 - (3*c*d^2 + 6*a*d*e + a*e^2)*sqrt(-c)*x*(a/c)^(3/4)*elliptic_f(arcsin((a 
/c)^(1/4)/x), -1) + (a*e^2*x^2 + 6*a*d*e)*sqrt(-c*x^4 + a))/(a*c*x)
 

Sympy [A] (verification not implemented)

Time = 1.40 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.80 \[ \int \frac {\left (d+e x^2\right )^2}{\sqrt {a-c x^4}} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} + \frac {d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{2 \sqrt {a} \Gamma \left (\frac {7}{4}\right )} + \frac {e^{2} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {9}{4}\right )} \] Input:

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

Output:

d**2*x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), c*x**4*exp_polar(2*I*pi)/a)/(4 
*sqrt(a)*gamma(5/4)) + d*e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), c*x** 
4*exp_polar(2*I*pi)/a)/(2*sqrt(a)*gamma(7/4)) + e**2*x**5*gamma(5/4)*hyper 
((1/2, 5/4), (9/4,), c*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(9/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(a - c*x**4)*e**2*x + int(sqrt(a - c*x**4)/(a - c*x**4),x)*a*e**2 
+ 3*int(sqrt(a - c*x**4)/(a - c*x**4),x)*c*d**2 + 6*int((sqrt(a - c*x**4)* 
x**2)/(a - c*x**4),x)*c*d*e)/(3*c)