\(\int \frac {d+e x^2}{(a-c x^4)^{3/2}} \, dx\) [415]

Optimal result

Integrand size = 20, antiderivative size = 156 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x^2\right )}{2 a \sqrt {a-c x^4}}-\frac {e \sqrt {1-\frac {c x^4}{a}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{a} c^{3/4} \sqrt {a-c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{2 a^{3/4} c^{3/4} \sqrt {a-c x^4}} \] Output:

1/2*x*(e*x^2+d)/a/(-c*x^4+a)^(1/2)-1/2*e*(1-c*x^4/a)^(1/2)*EllipticE(c^(1/ 
4)*x/a^(1/4),I)/a^(1/4)/c^(3/4)/(-c*x^4+a)^(1/2)+1/2*(c^(1/2)*d+a^(1/2)*e) 
*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/a^(3/4)/c^(3/4)/(-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 = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.64 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {3 d x+3 d x \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {c x^4}{a}\right )+2 e x^3 \sqrt {1-\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {c x^4}{a}\right )}{6 a \sqrt {a-c x^4}} \] Input:

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

Output:

(3*d*x + 3*d*x*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[1/4, 1/2, 5/4, (c*x^4 
)/a] + 2*e*x^3*Sqrt[1 - (c*x^4)/a]*Hypergeometric2F1[3/4, 3/2, 7/4, (c*x^4 
)/a])/(6*a*Sqrt[a - c*x^4])
 

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.01, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {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.

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

\(\Big \downarrow \) 1493

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1513

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

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]
 

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28

method	result	size
elliptic	\(\frac {2 c \left (\frac {e \,x^{3}}{4 a c}+\frac {d x}{4 a c}\right )}{\sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {d \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {e \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\)	\(199\)
default	\(d \left (\frac {x}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\right )+e \left (\frac {x^{3}}{2 a \sqrt {-\left (x^{4}-\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}\, \sqrt {c}}\right )\)	\(206\)

Input:

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

Output:

2*c*(1/4/a*e/c*x^3+1/4*d/a/c*x)/(-(x^4-1/c*a)*c)^(1/2)+1/2*d/a/(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)+1/2/a^(1/2) 
*e/(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 = 120, normalized size of antiderivative = 0.77 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{3/2}} \, dx=-\frac {{\left (c e x^{4} - a e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (c d + c e\right )} x^{4} - a d - a e\right )} \sqrt {a} \left (\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (c e x^{3} + c d x\right )} \sqrt {-c x^{4} + a}}{2 \, {\left (a c^{2} x^{4} - a^{2} c\right )}} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 3.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.53 \[ \int \frac {d+e x^2}{\left (a-c x^4\right )^{3/2}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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