\(\int \frac {d+e x^2}{(a+c x^4)^{3/2}} \, dx\) [245]

Optimal result

Integrand size = 19, antiderivative size = 262 \[ \int \frac {d+e x^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x^2\right )}{2 a \sqrt {a+c x^4}}-\frac {e x \sqrt {a+c x^4}}{2 a \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {e \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{2 a^{3/4} c^{3/4} \sqrt {a+c x^4}}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{2}\right )}{4 a^{5/4} c^{3/4} \sqrt {a+c x^4}} \] Output:

1/2*x*(e*x^2+d)/a/(c*x^4+a)^(1/2)-1/2*e*x*(c*x^4+a)^(1/2)/a/c^(1/2)/(a^(1/ 
2)+c^(1/2)*x^2)+1/2*e*(a^(1/2)+c^(1/2)*x^2)*((c*x^4+a)/(a^(1/2)+c^(1/2)*x^ 
2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*2^(1/2))/a^(3/4 
)/c^(3/4)/(c*x^4+a)^(1/2)+1/4*(c^(1/2)*d-a^(1/2)*e)*(a^(1/2)+c^(1/2)*x^2)* 
((c*x^4+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4) 
*x/a^(1/4)),1/2*2^(1/2))/a^(5/4)/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 = 10.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.38 \[ \int \frac {d+e x^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {3 d x+3 d x \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {c x^4}{a}\right )+2 e x^3 \sqrt {1+\frac {c x^4}{a}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {c x^4}{a}\right )}{6 a \sqrt {a+c x^4}} \] 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 = 0.57 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {1493, 25, 1512, 27, 761, 1510}

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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1512

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 761

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

\(\Big \downarrow \) 1510

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

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]) + ((e*(-((x*Sqrt[a + c*x^4])/(Sqrt[a 
] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqr 
t[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], 1/2])/(c^( 
1/4)*Sqrt[a + c*x^4])))/Sqrt[c] + ((d - (Sqrt[a]*e)/Sqrt[c])*(Sqrt[a] + Sq 
rt[c]*x^2)*Sqrt[(a + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*ArcTan[ 
(c^(1/4)*x)/a^(1/4)], 1/2])/(2*a^(1/4)*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 761

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 
1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* 
EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

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 1510

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

rule 1512

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.74 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.81

method	result	size
elliptic	\(-\frac {2 c \left (-\frac {e \,x^{3}}{4 a c}-\frac {d x}{4 a c}\right )}{\sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {d \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}-\frac {i e \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\)	\(212\)
default	\(d \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}+\frac {\sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}}\right )+e \left (\frac {x^{3}}{2 a \sqrt {\left (x^{4}+\frac {a}{c}\right ) c}}-\frac {i \sqrt {1-\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {c}\, x^{2}}{\sqrt {a}}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}, i\right )\right )}{2 \sqrt {a}\, \sqrt {\frac {i \sqrt {c}}{\sqrt {a}}}\, \sqrt {c \,x^{4}+a}\, \sqrt {c}}\right )\)	\(217\)

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

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.45 \[ \int \frac {d+e x^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {{\left (c e x^{4} + a e\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left ({\left (c d + c e\right )} x^{4} + a d + a e\right )} \sqrt {a} \left (-\frac {c}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {c}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + {\left (c e x^{3} + c d x\right )} \sqrt {c x^{4} + a}}{2 \, {\left (a c^{2} x^{4} + a^{2} c\right )}} \] 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(arc 
sin(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 [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.38 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.30 \[ \int \frac {d+e x^2}{\left (a+c x^4\right )^{3/2}} \, dx=\frac {d x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {c x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \] 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(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*exp_ 
polar(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. \[ \int \frac {d+e x^2}{\left (a+c x^4\right )^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (c\,x^4+a\right )}^{3/2}} \,d x \] 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