\(\int \frac {A+B x^2}{(c+d x^2) \sqrt {a-c x^4}} \, dx\) [10]

Optimal result

Integrand size = 29, antiderivative size = 136 \[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=\frac {\sqrt [4]{a} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} (B c-A d) \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} d \sqrt {a-c x^4}} \] Output:

a^(1/4)*B*(1-c*x^4/a)^(1/2)*EllipticF(c^(1/4)*x/a^(1/4),I)/c^(1/4)/d/(-c*x 
^4+a)^(1/2)-a^(1/4)*(-A*d+B*c)*(1-c*x^4/a)^(1/2)*EllipticPi(c^(1/4)*x/a^(1 
/4),-a^(1/2)*d/c^(3/2),I)/c^(5/4)/d/(-c*x^4+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.38 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.95 \[ \int \frac {A+B x^2}{\left (c+d x^2\right ) \sqrt {a-c x^4}} \, dx=-\frac {i \sqrt {1-\frac {c x^4}{a}} \left (B c \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )+(-B c+A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},i \text {arcsinh}\left (\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} x\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}} c d \sqrt {a-c x^4}} \] Input:

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

Output:

((-I)*Sqrt[1 - (c*x^4)/a]*(B*c*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[c]/Sqrt[a]) 
]*x], -1] + (-(B*c) + A*d)*EllipticPi[-((Sqrt[a]*d)/c^(3/2)), I*ArcSinh[Sq 
rt[-(Sqrt[c]/Sqrt[a])]*x], -1]))/(Sqrt[-(Sqrt[c]/Sqrt[a])]*c*d*Sqrt[a - c* 
x^4])
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2229, 765, 762, 1543, 1542}

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.

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

\(\Big \downarrow \) 2229

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

\(\Big \downarrow \) 1543

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

\(\Big \downarrow \) 1542

\(\displaystyle \frac {\sqrt [4]{a} B \sqrt {1-\frac {c x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt [4]{c} d \sqrt {a-c x^4}}-\frac {\sqrt [4]{a} \sqrt {1-\frac {c x^4}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{c^{3/2}},\arcsin \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),-1\right )}{c^{5/4} d \sqrt {a-c x^4}}\)

Input:

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

Output:

(a^(1/4)*B*Sqrt[1 - (c*x^4)/a]*EllipticF[ArcSin[(c^(1/4)*x)/a^(1/4)], -1]) 
/(c^(1/4)*d*Sqrt[a - c*x^4]) - (a^(1/4)*(B*c - A*d)*Sqrt[1 - (c*x^4)/a]*El 
lipticPi[-((Sqrt[a]*d)/c^(3/2)), ArcSin[(c^(1/4)*x)/a^(1/4)], -1])/(c^(5/4 
)*d*Sqrt[a - c*x^4])
 

Defintions of rubi rules used

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 1542

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

rule 1543

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

rule 2229

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

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.27

method	result	size
default	\(\frac {B \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\left (A d -B c \right ) \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right )}{d c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\)	\(173\)
elliptic	\(\frac {B \sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}+\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) A}{c \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}-\frac {\sqrt {1-\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {\sqrt {c}\, x^{2}}{\sqrt {a}}}\, \operatorname {EllipticPi}\left (x \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}, -\frac {\sqrt {a}\, d}{c^{\frac {3}{2}}}, \frac {\sqrt {-\frac {\sqrt {c}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {c}}{\sqrt {a}}}}\right ) B}{d \sqrt {\frac {\sqrt {c}}{\sqrt {a}}}\, \sqrt {-c \,x^{4}+a}}\)	\(258\)

Input:

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

Output:

B/d/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2)*x^2/a 
^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticF(x*(c^(1/2)/a^(1/2))^(1/2),I)+(A*d 
-B*c)/d/c/(c^(1/2)/a^(1/2))^(1/2)*(1-c^(1/2)*x^2/a^(1/2))^(1/2)*(1+c^(1/2) 
*x^2/a^(1/2))^(1/2)/(-c*x^4+a)^(1/2)*EllipticPi(x*(c^(1/2)/a^(1/2))^(1/2), 
-a^(1/2)*d/c^(3/2),(-c^(1/2)/a^(1/2))^(1/2)/(c^(1/2)/a^(1/2))^(1/2))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

Integral((A + B*x**2)/(sqrt(a - c*x**4)*(c + d*x**2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a - c*x**4)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*a + int((s 
qrt(a - c*x**4)*x**2)/(a*c + a*d*x**2 - c**2*x**4 - c*d*x**6),x)*b