\(\int \frac {x^6 (e+f x^2)}{(a-b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [275]

Optimal result

Integrand size = 34, antiderivative size = 483 \[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {(b e+a f) x^5 \sqrt {c+d x^2}}{b (b c+a d) \sqrt {a-b x^2}}+\frac {\left (24 a^2 d^2 f+b^2 c (5 d e-4 c f)+5 a b d (4 d e+c f)\right ) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{15 b^3 d^2 (b c+a d)}+\frac {(5 b d e+b c f+6 a d f) x^3 \sqrt {a-b x^2} \sqrt {c+d x^2}}{5 b^2 d (b c+a d)}-\frac {\sqrt {a} \left (48 a^3 d^3 f-2 b^3 c^2 (5 d e-4 c f)+3 a b^2 c d (5 d e-3 c f)+8 a^2 b d^2 (5 d e+2 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{15 b^{7/2} d^3 (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} c \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{15 b^{7/2} d^3 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

(a*f+b*e)*x^5*(d*x^2+c)^(1/2)/b/(a*d+b*c)/(-b*x^2+a)^(1/2)+1/15*(24*a^2*d^ 
2*f+b^2*c*(-4*c*f+5*d*e)+5*a*b*d*(c*f+4*d*e))*x*(-b*x^2+a)^(1/2)*(d*x^2+c) 
^(1/2)/b^3/d^2/(a*d+b*c)+1/5*(6*a*d*f+b*c*f+5*b*d*e)*x^3*(-b*x^2+a)^(1/2)* 
(d*x^2+c)^(1/2)/b^2/d/(a*d+b*c)-1/15*a^(1/2)*(48*a^3*d^3*f-2*b^3*c^2*(-4*c 
*f+5*d*e)+3*a*b^2*c*d*(-3*c*f+5*d*e)+8*a^2*b*d^2*(2*c*f+5*d*e))*(1-b*x^2/a 
)^(1/2)*(d*x^2+c)^(1/2)*EllipticE(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(7 
/2)/d^3/(a*d+b*c)/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+1/15*a^(1/2)*c*(24*a^ 
2*d^2*f+a*b*d*(-13*c*f+20*d*e)-2*b^2*c*(-4*c*f+5*d*e))*(1-b*x^2/a)^(1/2)*( 
1+d*x^2/c)^(1/2)*EllipticF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(7/2)/d^3 
/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.09 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.83 \[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {-\frac {b}{a}} d x \left (c+d x^2\right ) \left (-24 a^3 d^2 f+b^3 c x^2 \left (5 d e-4 c f+3 d f x^2\right )+a^2 b d \left (-20 d e-5 c f+6 d f x^2\right )+a b^2 \left (4 c^2 f+c d \left (-5 e+2 f x^2\right )+d^2 x^2 \left (5 e+3 f x^2\right )\right )\right )+i c \left (48 a^3 d^3 f+3 a b^2 c d (5 d e-3 c f)+8 a^2 b d^2 (5 d e+2 c f)+2 b^3 c^2 (-5 d e+4 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)+2 b^2 c (-5 d e+4 c f)\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{15 b^3 \sqrt {-\frac {b}{a}} d^3 (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

Integrate[(x^6*(e + f*x^2))/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
 

Output:

(-(Sqrt[-(b/a)]*d*x*(c + d*x^2)*(-24*a^3*d^2*f + b^3*c*x^2*(5*d*e - 4*c*f 
+ 3*d*f*x^2) + a^2*b*d*(-20*d*e - 5*c*f + 6*d*f*x^2) + a*b^2*(4*c^2*f + c* 
d*(-5*e + 2*f*x^2) + d^2*x^2*(5*e + 3*f*x^2)))) + I*c*(48*a^3*d^3*f + 3*a* 
b^2*c*d*(5*d*e - 3*c*f) + 8*a^2*b*d^2*(5*d*e + 2*c*f) + 2*b^3*c^2*(-5*d*e 
+ 4*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt 
[-(b/a)]*x], -((a*d)/(b*c))] - I*c*(b*c + a*d)*(24*a^2*d^2*f + a*b*d*(20*d 
*e - 13*c*f) + 2*b^2*c*(-5*d*e + 4*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x 
^2)/c]*EllipticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(15*b^3*Sqrt[ 
-(b/a)]*d^3*(b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.324, Rules used = {440, 25, 444, 444, 399, 323, 323, 321, 331, 330, 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.

\(\displaystyle \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 440

\(\displaystyle \frac {\int -\frac {x^4 \left ((5 b d e+b c f+6 a d f) x^2+5 c (b e+a f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{b (a d+b c)}+\frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\int \frac {x^4 \left ((5 b d e+b c f+6 a d f) x^2+5 c (b e+a f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{b (a d+b c)}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\int \frac {x^2 \left (\left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right ) x^2+3 a c (5 b d e+b c f+6 a d f)\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 444

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\int \frac {\left (-2 c^2 (5 d e-4 c f) b^3+3 a c d (5 d e-3 c f) b^2+8 a^2 d^2 (5 d e+2 c f) b+48 a^3 d^3 f\right ) x^2+a c \left (c (5 d e-4 c f) b^2+5 a d (4 d e+c f) b+24 a^2 d^2 f\right )}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {d x^2+c}}dx}{d}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 323

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \int \frac {1}{\sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1}}dx}{d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {a-b x^2}}dx}{d}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 331

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {d x^2+c}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) \int \frac {\sqrt {\frac {d x^2}{c}+1}}{\sqrt {1-\frac {b x^2}{a}}}dx}{d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x^5 \sqrt {c+d x^2} (a f+b e)}{b \sqrt {a-b x^2} (a d+b c)}-\frac {\frac {\frac {\frac {\sqrt {a} \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} \left (48 a^3 d^3 f+8 a^2 b d^2 (2 c f+5 d e)+3 a b^2 c d (5 d e-3 c f)-2 b^3 c^2 (5 d e-4 c f)\right ) E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {\frac {d x^2}{c}+1}}-\frac {\sqrt {a} c \sqrt {1-\frac {b x^2}{a}} \sqrt {\frac {d x^2}{c}+1} (a d+b c) \left (24 a^2 d^2 f+a b d (20 d e-13 c f)-2 b^2 c (5 d e-4 c f)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {c+d x^2}}}{3 b d}-\frac {1}{3} x \sqrt {a-b x^2} \sqrt {c+d x^2} \left (\frac {24 a^2 d f}{b}+5 a (c f+4 d e)+\frac {b c (5 d e-4 c f)}{d}\right )}{5 b d}-\frac {x^3 \sqrt {a-b x^2} \sqrt {c+d x^2} (6 a d f+b c f+5 b d e)}{5 b d}}{b (a d+b c)}\)

Input:

Int[(x^6*(e + f*x^2))/((a - b*x^2)^(3/2)*Sqrt[c + d*x^2]),x]
 

Output:

((b*e + a*f)*x^5*Sqrt[c + d*x^2])/(b*(b*c + a*d)*Sqrt[a - b*x^2]) - (-1/5* 
((5*b*d*e + b*c*f + 6*a*d*f)*x^3*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + 
(-1/3*(((24*a^2*d*f)/b + (b*c*(5*d*e - 4*c*f))/d + 5*a*(4*d*e + c*f))*x*Sq 
rt[a - b*x^2]*Sqrt[c + d*x^2]) + ((Sqrt[a]*(48*a^3*d^3*f - 2*b^3*c^2*(5*d* 
e - 4*c*f) + 3*a*b^2*c*d*(5*d*e - 3*c*f) + 8*a^2*b*d^2*(5*d*e + 2*c*f))*Sq 
rt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]], - 
((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2)/c]) - (Sqrt[a] 
*c*(b*c + a*d)*(24*a^2*d^2*f + a*b*d*(20*d*e - 13*c*f) - 2*b^2*c*(5*d*e - 
4*c*f))*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[(Sqrt[b]* 
x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])) 
/(3*b*d))/(5*b*d))/(b*(b*c + a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( 
d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] &&  !GtQ[c, 0]
 

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 330

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

rule 331

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

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 440

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

rule 444

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

Maple [A] (verified)

Time = 20.69 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.22

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) a^{2} x \left (a f +b e \right )}{b^{4} \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {f \,x^{3} \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{5 b^{2} d}-\frac {\left (-\frac {a f +b e}{b^{2}}-\frac {f \left (4 a d -4 b c \right )}{5 b^{2} d}\right ) x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 b d}+\frac {\left (-\frac {c \,a^{2} \left (a f +b e \right )}{b^{3} \left (a d +b c \right )}+\frac {\left (-\frac {a f +b e}{b^{2}}-\frac {f \left (4 a d -4 b c \right )}{5 b^{2} d}\right ) a c}{3 b d}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (-\frac {a \left (a f +b e \right )}{b^{3}}-\frac {d \,a^{2} \left (a f +b e \right )}{b^{3} \left (a d +b c \right )}-\frac {3 f a c}{5 b^{2} d}+\frac {\left (-\frac {a f +b e}{b^{2}}-\frac {f \left (4 a d -4 b c \right )}{5 b^{2} d}\right ) \left (2 a d -2 b c \right )}{3 b d}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(587\)
risch	\(\frac {x \left (3 b d f \,x^{2}+9 a d f -4 b c f +5 b d e \right ) \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}{15 b^{3} d^{2}}-\frac {\left (-\frac {\left (33 f \,d^{2} a^{2}-17 f d c b a +25 a b \,d^{2} e +8 f \,c^{2} b^{2}-10 d \,b^{2} c e \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {a \left (15 f \,d^{2} a^{2}+9 f d c b a +15 a b \,d^{2} e -4 f \,c^{2} b^{2}+5 d \,b^{2} c e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{b \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {15 a^{3} d^{2} \left (a f +b e \right ) \left (\frac {\left (-b d \,x^{2}-b c \right ) x}{a \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (-\frac {1}{a}+\frac {b c}{a \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{a \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 b^{3} d^{2} \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(721\)
default	\(\text {Expression too large to display}\)	\(1276\)

Input:

int(x^6*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE 
)
 

Output:

((-b*x^2+a)*(d*x^2+c))^(1/2)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(-(-b*d*x^2- 
b*c)/b^4*a^2/(a*d+b*c)*x*(a*f+b*e)/((x^2-a/b)*(-b*d*x^2-b*c))^(1/2)+1/5*f/ 
b^2/d*x^3*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)-1/3*(-1/b^2*(a*f+b*e)-1/5*f 
/b^2/d*(4*a*d-4*b*c))/b/d*x*(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)+(-1/b^3*c 
*a^2/(a*d+b*c)*(a*f+b*e)+1/3*(-1/b^2*(a*f+b*e)-1/5*f/b^2/d*(4*a*d-4*b*c))/ 
b/d*a*c)/(b/a)^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2 
-b*c*x^2+a*c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-(-a/ 
b^3*(a*f+b*e)-1/b^3*d*a^2*(a*f+b*e)/(a*d+b*c)-3/5*f/b^2/d*a*c+1/3*(-1/b^2* 
(a*f+b*e)-1/5*f/b^2/d*(4*a*d-4*b*c))/b/d*(2*a*d-2*b*c))*c/(b/a)^(1/2)*(1-b 
*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(-b*d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)/d*(El 
lipticF(x*(b/a)^(1/2),(-1-(a*d-b*c)/c/b)^(1/2))-EllipticE(x*(b/a)^(1/2),(- 
1-(a*d-b*c)/c/b)^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 837, normalized size of antiderivative = 1.73 \[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

integrate(x^6*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="fri 
cas")
 

Output:

-1/15*(((5*(2*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 - 8*a^3*b^2*d^3)*e - (8*a*b^4* 
c^3 - 9*a^2*b^3*c^2*d + 16*a^3*b^2*c*d^2 + 48*a^4*b*d^3)*f)*x^3 - (5*(2*a^ 
2*b^3*c^2*d - 3*a^3*b^2*c*d^2 - 8*a^4*b*d^3)*e - (8*a^2*b^3*c^3 - 9*a^3*b^ 
2*c^2*d + 16*a^4*b*c*d^2 + 48*a^5*d^3)*f)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic 
_e(arcsin(sqrt(a/b)/x), -b*c/(a*d)) + ((5*(8*a^3*b^2*d^3 - (2*a*b^4 - b^5) 
*c^2*d + (3*a^2*b^3 + 4*a*b^4)*c*d^2)*e + (48*a^4*b*d^3 + 4*(2*a*b^4 - b^5 
)*c^3 - (9*a^2*b^3 - 5*a*b^4)*c^2*d + 8*(2*a^3*b^2 + 3*a^2*b^3)*c*d^2)*f)* 
x^3 - (5*(8*a^4*b*d^3 - (2*a^2*b^3 - a*b^4)*c^2*d + (3*a^3*b^2 + 4*a^2*b^3 
)*c*d^2)*e + (48*a^5*d^3 + 4*(2*a^2*b^3 - a*b^4)*c^3 - (9*a^3*b^2 - 5*a^2* 
b^3)*c^2*d + 8*(2*a^4*b + 3*a^3*b^2)*c*d^2)*f)*x)*sqrt(-b*d)*sqrt(a/b)*ell 
iptic_f(arcsin(sqrt(a/b)/x), -b*c/(a*d)) - (3*(b^5*c*d^2 + a*b^4*d^3)*f*x^ 
6 + (5*(b^5*c*d^2 + a*b^4*d^3)*e - 2*(2*b^5*c^2*d - a*b^4*c*d^2 - 3*a^2*b^ 
3*d^3)*f)*x^4 - (10*(b^5*c^2*d - a*b^4*c*d^2 - 2*a^2*b^3*d^3)*e - (8*b^5*c 
^3 - 5*a*b^4*c^2*d + 11*a^2*b^3*c*d^2 + 24*a^3*b^2*d^3)*f)*x^2 + 5*(2*a*b^ 
4*c^2*d - 3*a^2*b^3*c*d^2 - 8*a^3*b^2*d^3)*e - (8*a*b^4*c^3 - 9*a^2*b^3*c^ 
2*d + 16*a^3*b^2*c*d^2 + 48*a^4*b*d^3)*f)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c) 
)/((b^7*c*d^3 + a*b^6*d^4)*x^3 - (a*b^6*c*d^3 + a^2*b^5*d^4)*x)
 

Sympy [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^{6} \left (e + f x^{2}\right )}{\left (a - b x^{2}\right )^{\frac {3}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate(x**6*(f*x**2+e)/(-b*x**2+a)**(3/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral(x**6*(e + f*x**2)/((a - b*x**2)**(3/2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )} x^{6}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate(x^6*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="max 
ima")
 

Output:

integrate((f*x^2 + e)*x^6/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int { \frac {{\left (f x^{2} + e\right )} x^{6}}{{\left (-b x^{2} + a\right )}^{\frac {3}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate(x^6*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x, algorithm="gia 
c")
 

Output:

integrate((f*x^2 + e)*x^6/((-b*x^2 + a)^(3/2)*sqrt(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\int \frac {x^6\,\left (f\,x^2+e\right )}{{\left (a-b\,x^2\right )}^{3/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((x^6*(e + f*x^2))/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((x^6*(e + f*x^2))/((a - b*x^2)^(3/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {x^6 \left (e+f x^2\right )}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx =\text {Too large to display} \] Input:

int(x^6*(f*x^2+e)/(-b*x^2+a)^(3/2)/(d*x^2+c)^(1/2),x)
 

Output:

(18*sqrt(c + d*x**2)*sqrt(a - b*x**2)*a*c*d*f*x - 12*sqrt(c + d*x**2)*sqrt 
(a - b*x**2)*a*d**2*f*x**3 - 12*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c**2*f 
*x + 15*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*c*d*e*x + 8*sqrt(c + d*x**2)*s 
qrt(a - b*x**2)*b*c*d*f*x**3 - 10*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*d**2 
*e*x**3 - 6*sqrt(c + d*x**2)*sqrt(a - b*x**2)*b*d**2*f*x**5 + 48*int((sqrt 
(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 
 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**3*d**3*f - 8*int((sqrt(c 
+ d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 2* 
a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*c*d**2*f + 40*int((sqrt( 
c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 
2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**3*e - 48*int((sqrt( 
c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x**2 - 
2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a**2*b*d**3*f*x**2 + 4*int((s 
qrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x** 
2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c**2*d*f - 5*int(( 
sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x* 
*2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d**2*e + 8*int( 
(sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 - 2*a*b*c*x 
**2 - 2*a*b*d*x**4 + b**2*c*x**4 + b**2*d*x**6),x)*a*b**2*c*d**2*f*x**2 - 
40*int((sqrt(c + d*x**2)*sqrt(a - b*x**2)*x**4)/(a**2*c + a**2*d*x**2 -...