\(\int \frac {(d+e x^2)^2}{(a-c x^4)^{3/2}} \, dx\) [414]

Optimal result

Integrand size = 22, antiderivative size = 178 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (d^2+\frac {a e^2}{c}+2 d e x^2\right )}{2 a \sqrt {a-c x^4}}-\frac {d e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}+\frac {\left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{5/4} \sqrt {a-c x^4}} \] Output:

1/2*x*(d^2+a*e^2/c+2*d*e*x^2)/a/(-c*x^4+a)^(1/2)-d*e*(1-c*x^4/a)^(1/2)*Ell 
ipticE(c^(1/4)*x/a^(1/4),I)/a^(1/4)/c^(3/4)/(-c*x^4+a)^(1/2)+1/2*(c*d^2+2* 
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 
)/a^(3/4)/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 = 10.08 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.71 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 \left (c d^2+a e^2\right ) x+3 \left (c d^2-a e^2\right ) x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+4 c d e x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{6 a c \sqrt {a-c x^4}} \] Input:

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

Output:

(3*(c*d^2 + a*e^2)*x + 3*(c*d^2 - a*e^2)*x*Sqrt[1 - (c*x^4)/a]*Hypergeomet 
ric2F1[1/4, 1/2, 5/4, (c*x^4)/a] + 4*c*d*e*x^3*Sqrt[1 - (c*x^4)/a]*Hyperge 
ometric2F1[3/4, 3/2, 7/4, (c*x^4)/a])/(6*a*c*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.17, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {1519, 1493, 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.

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

\(\Big \downarrow \) 1519

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

\(\Big \downarrow \) 1493

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(e^2*x)/(c*Sqrt[a - c*x^4]) + ((x*(c*d^2 - a*e^2 + 2*c*d*e*x^2))/(2*a*Sqrt 
[a - c*x^4]) + ((-2*a^(3/4)*c^(1/4)*d*e*Sqrt[1 - (c*x^4)/a]*EllipticE[ArcS 
in[(c^(1/4)*x)/a^(1/4)], -1])/Sqrt[a - c*x^4] + (a^(1/4)*(c*d^2 + 2*Sqrt[a 
]*Sqrt[c]*d*e - a*e^2)*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^ 
(1/4)], -1])/(c^(1/4)*Sqrt[a - c*x^4]))/(2*a))/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 1493

Int[((d_) + (e_.)*(x_)^2)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x 
)*(d + e*x^2)*((a + c*x^4)^(p + 1)/(4*a*(p + 1))), x] + Simp[1/(4*a*(p + 1) 
)   Int[Simp[d*(4*p + 5) + e*(4*p + 7)*x^2, x]*(a + c*x^4)^(p + 1), x], x] 
/; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && Integer 
Q[2*p]
 

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 = 1.08 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.31

method	result	size
elliptic	\(\frac {2 c \left (\frac {e d \,x^{3}}{2 a c}+\frac {\left (a \,e^{2}+c \,d^{2}\right ) x}{4 a \,c^{2}}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\left (-\frac {e^{2}}{c}+\frac {a \,e^{2}+c \,d^{2}}{2 a c}\right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\)	\(233\)
default	\(d^{2} \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e^{2} \left (\frac {x}{2 c \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+2 d e \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\)	\(303\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^2}{\left (a-c x^4\right )^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} d e x^{4} - a c d e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (c^{2} d^{2} + 2 \, c^{2} d e - a c e^{2}\right )} x^{4} - a c d^{2} - 2 \, a c d e + a^{2} e^{2}\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (2 \, c^{2} d e x^{3} + {\left (c^{2} d^{2} + a c e^{2}\right )} x\right )} \sqrt {-c x^{4} + a}}{2 \, {\left (a c^{3} x^{4} - a^{2} c^{2}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(a - c*x**4)*e**2*x - int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2* 
x**8),x)*a**2*e**2 + int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8), 
x)*a*c*d**2 + int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*a*c* 
e**2*x**4 - int(sqrt(a - c*x**4)/(a**2 - 2*a*c*x**4 + c**2*x**8),x)*c**2*d 
**2*x**4 + 2*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8),x 
)*a*c*d*e - 2*int((sqrt(a - c*x**4)*x**2)/(a**2 - 2*a*c*x**4 + c**2*x**8), 
x)*c**2*d*e*x**4)/(c*(a - c*x**4))