\(\int \frac {e+f x^2}{(a+b x^2)^{5/4} (c+d x^2)} \, dx\) [562]

Optimal result

Integrand size = 28, antiderivative size = 257 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\frac {2 (b e-a f) \sqrt [4]{1+\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt {b} (b c-a d) \sqrt [4]{a+b x^2}}+\frac {\sqrt [4]{a} (d e-c f) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {d} (-b c+a d)^{3/2} x}-\frac {\sqrt [4]{a} (d e-c f) \sqrt {-\frac {b x^2}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {-b c+a d}},\arcsin \left (\frac {\sqrt [4]{a+b x^2}}{\sqrt [4]{a}}\right ),-1\right )}{\sqrt {d} (-b c+a d)^{3/2} x} \] Output:

2*(-a*f+b*e)*(1+b*x^2/a)^(1/4)*EllipticE(sin(1/2*arctan(b^(1/2)*x/a^(1/2)) 
),2^(1/2))/a^(1/2)/b^(1/2)/(-a*d+b*c)/(b*x^2+a)^(1/4)+a^(1/4)*(-c*f+d*e)*( 
-b*x^2/a)^(1/2)*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),-a^(1/2)*d^(1/2)/(a*d-b 
*c)^(1/2),I)/d^(1/2)/(a*d-b*c)^(3/2)/x-a^(1/4)*(-c*f+d*e)*(-b*x^2/a)^(1/2) 
*EllipticPi((b*x^2+a)^(1/4)/a^(1/4),a^(1/2)*d^(1/2)/(a*d-b*c)^(1/2),I)/d^( 
1/2)/(a*d-b*c)^(3/2)/x
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 10.42 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.37 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\frac {x \left (-\frac {d (-b e+a f) x^2 \sqrt [4]{1+\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{c}+\frac {6 \left (3 a c \left (-b e \left (c+2 d x^2\right )+a d \left (e+2 f x^2\right )\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+(b e-a f) x^2 \left (c+d x^2\right ) \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}{\left (c+d x^2\right ) \left (6 a c \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{4},1,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )-x^2 \left (4 a d \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{4},2,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+b c \operatorname {AppellF1}\left (\frac {3}{2},\frac {5}{4},1,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right )}\right )}{3 a (-b c+a d) \sqrt [4]{a+b x^2}} \] Input:

Integrate[(e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)),x]
 

Output:

(x*(-((d*(-(b*e) + a*f)*x^2*(1 + (b*x^2)/a)^(1/4)*AppellF1[3/2, 1/4, 1, 5/ 
2, -((b*x^2)/a), -((d*x^2)/c)])/c) + (6*(3*a*c*(-(b*e*(c + 2*d*x^2)) + a*d 
*(e + 2*f*x^2))*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2)/c)] + ( 
b*e - a*f)*x^2*(c + d*x^2)*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), 
 -((d*x^2)/c)] + b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c) 
])))/((c + d*x^2)*(6*a*c*AppellF1[1/2, 1/4, 1, 3/2, -((b*x^2)/a), -((d*x^2 
)/c)] - x^2*(4*a*d*AppellF1[3/2, 1/4, 2, 5/2, -((b*x^2)/a), -((d*x^2)/c)] 
+ b*c*AppellF1[3/2, 5/4, 1, 5/2, -((b*x^2)/a), -((d*x^2)/c)])))))/(3*a*(-( 
b*c) + a*d)*(a + b*x^2)^(1/4))
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {402, 27, 405, 227, 225, 212, 310, 993, 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.

\(\displaystyle \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx\)

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 405

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

\(\Big \downarrow \) 227

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

\(\Big \downarrow \) 225

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {a (d e-c f) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+\frac {\sqrt [4]{\frac {b x^2}{a}+1} (b e-a f) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\int \frac {1}{\left (\frac {b x^2}{a}+1\right )^{5/4}}dx\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}\)

\(\Big \downarrow \) 212

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {a (d e-c f) \int \frac {1}{\sqrt [4]{b x^2+a} \left (d x^2+c\right )}dx+\frac {\sqrt [4]{\frac {b x^2}{a}+1} (b e-a f) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}\)

\(\Big \downarrow \) 310

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {2 a \sqrt {-\frac {b x^2}{a}} (d e-c f) \int \frac {\sqrt {b x^2+a}}{\sqrt {1-\frac {b x^2+a}{a}} \left (b c-a d+d \left (b x^2+a\right )\right )}d\sqrt [4]{b x^2+a}}{x}+\frac {\sqrt [4]{\frac {b x^2}{a}+1} (b e-a f) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}\)

\(\Big \downarrow \) 993

\(\displaystyle \frac {2 x (b e-a f)}{a \sqrt [4]{a+b x^2} (b c-a d)}-\frac {\frac {2 a \sqrt {-\frac {b x^2}{a}} (d e-c f) \left (\frac {\int \frac {1}{\left (\sqrt {a d-b c}+\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}-\frac {\int \frac {1}{\left (\sqrt {a d-b c}-\sqrt {d} \sqrt {b x^2+a}\right ) \sqrt {1-\frac {b x^2+a}{a}}}d\sqrt [4]{b x^2+a}}{2 \sqrt {d}}\right )}{x}+\frac {\sqrt [4]{\frac {b x^2}{a}+1} (b e-a f) \left (\frac {2 x}{\sqrt [4]{\frac {b x^2}{a}+1}}-\frac {2 \sqrt {a} E\left (\left .\frac {1}{2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {b}}\right )}{\sqrt [4]{a+b x^2}}}{a (b c-a d)}\)

\(\Big \downarrow \) 1542

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

Input:

Int[(e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)),x]
 

Output:

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

Defintions of rubi rules used

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 212

Int[((a_) + (b_.)*(x_)^2)^(-5/4), x_Symbol] :> Simp[(2/(a^(5/4)*Rt[b/a, 2]) 
)*EllipticE[(1/2)*ArcTan[Rt[b/a, 2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a 
, 0] && PosQ[b/a]
 

rule 225

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[2*(x/(a + b*x^2)^(1/4)) 
, x] - Simp[a   Int[1/(a + b*x^2)^(5/4), x], x] /; FreeQ[{a, b}, x] && GtQ[ 
a, 0] && PosQ[b/a]
 

rule 227

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(1 + b*(x^2/a))^(1/4)/( 
a + b*x^2)^(1/4)   Int[1/(1 + b*(x^2/a))^(1/4), x], x] /; FreeQ[{a, b}, x] 
&& PosQ[a]
 

rule 310

Int[1/(((a_) + (b_.)*(x_)^2)^(1/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Sim 
p[2*(Sqrt[(-b)*(x^2/a)]/x)   Subst[Int[x^2/(Sqrt[1 - x^4/a]*(b*c - a*d + d* 
x^4)), x], x, (a + b*x^2)^(1/4)], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0]
 

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -1]
 

rule 405

Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 
), x_Symbol] :> Simp[f/d   Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d 
 Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, x]
 

rule 993

Int[(x_)^2/(((a_) + (b_.)*(x_)^4)*Sqrt[(c_) + (d_.)*(x_)^4]), x_Symbol] :> 
With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2* 
b)   Int[1/((r + s*x^2)*Sqrt[c + d*x^4]), x], x] - Simp[s/(2*b)   Int[1/((r 
 - s*x^2)*Sqrt[c + d*x^4]), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
a*d, 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]
 

Maple [F]

\[\int \frac {f \,x^{2}+e}{\left (b \,x^{2}+a \right )^{\frac {5}{4}} \left (x^{2} d +c \right )}d x\]

Input:

int((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c),x)
 

Output:

int((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\text {Timed out} \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\int \frac {e + f x^{2}}{\left (a + b x^{2}\right )^{\frac {5}{4}} \left (c + d x^{2}\right )}\, dx \] Input:

integrate((f*x**2+e)/(b*x**2+a)**(5/4)/(d*x**2+c),x)
 

Output:

Integral((e + f*x**2)/((a + b*x**2)**(5/4)*(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/4)*(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{4}} {\left (d x^{2} + c\right )}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c),x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/4)*(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\int \frac {f\,x^2+e}{{\left (b\,x^2+a\right )}^{5/4}\,\left (d\,x^2+c\right )} \,d x \] Input:

int((e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)),x)
 

Output:

int((e + f*x^2)/((a + b*x^2)^(5/4)*(c + d*x^2)), x)
 

Reduce [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/4} \left (c+d x^2\right )} \, dx=\left (\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a c +\left (b \,x^{2}+a \right )^{\frac {1}{4}} a d \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b c \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b d \,x^{4}}d x \right ) f +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} a c +\left (b \,x^{2}+a \right )^{\frac {1}{4}} a d \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b c \,x^{2}+\left (b \,x^{2}+a \right )^{\frac {1}{4}} b d \,x^{4}}d x \right ) e \] Input:

int((f*x^2+e)/(b*x^2+a)^(5/4)/(d*x^2+c),x)
 

Output:

int(x**2/((a + b*x**2)**(1/4)*a*c + (a + b*x**2)**(1/4)*a*d*x**2 + (a + b* 
x**2)**(1/4)*b*c*x**2 + (a + b*x**2)**(1/4)*b*d*x**4),x)*f + int(1/((a + b 
*x**2)**(1/4)*a*c + (a + b*x**2)**(1/4)*a*d*x**2 + (a + b*x**2)**(1/4)*b*c 
*x**2 + (a + b*x**2)**(1/4)*b*d*x**4),x)*e