\(\int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx\) [722]

Optimal result

Integrand size = 34, antiderivative size = 384 \[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=-\frac {4 b x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{15 c^2}+\frac {x^3 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{5 c}+\frac {d \sqrt {c d-b e} \left (9 c^2 d^2-9 b c d e+8 b^2 e^2\right ) \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )|-1+\frac {b e}{c d}\right )}{15 c^{5/2} e^{7/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}-\frac {d \sqrt {c d-b e} \left (9 c^2 d^2-5 b c d e+4 b^2 e^2\right ) \sqrt {1+\frac {e x^2}{d}} \sqrt {1-\frac {c e x^2}{c d-b e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),-1+\frac {b e}{c d}\right )}{15 c^{5/2} e^{7/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}} \] Output:

-4/15*b*x*(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)/c^2+1/5*x^3*(-d*(-b*e+c*d) 
/e^2+b*x^2+c*x^4)^(1/2)/c+1/15*d*(-b*e+c*d)^(1/2)*(8*b^2*e^2-9*b*c*d*e+9*c 
^2*d^2)*(1+e*x^2/d)^(1/2)*(1-c*e*x^2/(-b*e+c*d))^(1/2)*EllipticE(c^(1/2)*e 
^(1/2)*x/(-b*e+c*d)^(1/2),(-1+b*e/c/d)^(1/2))/c^(5/2)/e^(7/2)/(-d*(-b*e+c* 
d)/e^2+b*x^2+c*x^4)^(1/2)-1/15*d*(-b*e+c*d)^(1/2)*(4*b^2*e^2-5*b*c*d*e+9*c 
^2*d^2)*(1+e*x^2/d)^(1/2)*(1-c*e*x^2/(-b*e+c*d))^(1/2)*EllipticF(c^(1/2)*e 
^(1/2)*x/(-b*e+c*d)^(1/2),(-1+b*e/c/d)^(1/2))/c^(5/2)/e^(7/2)/(-d*(-b*e+c* 
d)/e^2+b*x^2+c*x^4)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.82 (sec) , antiderivative size = 321, normalized size of antiderivative = 0.84 \[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {c e \sqrt {\frac {e}{d}} x \left (-4 b+3 c x^2\right ) \left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )+i \left (9 c^3 d^3-18 b c^2 d^2 e+17 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|\frac {c d}{-c d+b e}\right )-i \left (9 c^3 d^3-22 b c^2 d^2 e+21 b^2 c d e^2-8 b^3 e^3\right ) \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),\frac {c d}{-c d+b e}\right )}{15 c^3 e^3 \sqrt {\frac {e}{d}} \sqrt {\frac {\left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )}{e^2}}} \] Input:

Integrate[x^6/Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4],x]
 

Output:

(c*e*Sqrt[e/d]*x*(-4*b + 3*c*x^2)*(d + e*x^2)*(-(c*d) + b*e + c*e*x^2) + I 
*(9*c^3*d^3 - 18*b*c^2*d^2*e + 17*b^2*c*d*e^2 - 8*b^3*e^3)*Sqrt[(-(c*d) + 
b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*ArcSinh[Sqr 
t[e/d]*x], (c*d)/(-(c*d) + b*e)] - I*(9*c^3*d^3 - 22*b*c^2*d^2*e + 21*b^2* 
c*d*e^2 - 8*b^3*e^3)*Sqrt[(-(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 
+ (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)])/(15* 
c^3*e^3*Sqrt[e/d]*Sqrt[((d + e*x^2)*(-(c*d) + b*e + c*e*x^2))/e^2])
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.87, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1442, 25, 27, 1602, 1514, 399, 321, 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.

\(\displaystyle \int \frac {x^6}{\sqrt {\frac {b d e-c d^2}{e^2}+b x^2+c x^4}} \, dx\)

\(\Big \downarrow \) 1442

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1602

\(\displaystyle \frac {-\frac {\int \frac {4 b d (c d-b e)-\left (8 b^2 e^2+9 c d (c d-b e)\right ) x^2}{\sqrt {c x^4+b x^2-\frac {d (c d-b e)}{e^2}}}dx}{3 c}-\frac {4 b e^2 x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}}{5 c e^2}+\frac {x^3 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{5 c}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {-\frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \int \frac {4 b d (c d-b e)-\left (8 b^2 e^2+9 c d (c d-b e)\right ) x^2}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{3 c \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}-\frac {4 b e^2 x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}}{5 c e^2}+\frac {x^3 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{5 c}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {-\frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \left (4 b^2 e^2-5 b c d e+9 c^2 d^2\right ) \int \frac {1}{\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{e}-\frac {d \left (8 b^2 e^2+9 c d (c d-b e)\right ) \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{e}\right )}{3 c \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}-\frac {4 b e^2 x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}}{5 c e^2}+\frac {x^3 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{5 c}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {-\frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \sqrt {c d-b e} \left (4 b^2 e^2-5 b c d e+9 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} e^{3/2}}-\frac {d \left (8 b^2 e^2+9 c d (c d-b e)\right ) \int \frac {\sqrt {\frac {e x^2}{d}+1}}{\sqrt {1-\frac {c e x^2}{c d-b e}}}dx}{e}\right )}{3 c \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}-\frac {4 b e^2 x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}}{5 c e^2}+\frac {x^3 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{5 c}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {-\frac {\sqrt {\frac {e x^2}{d}+1} \sqrt {1-\frac {c e x^2}{c d-b e}} \left (\frac {d \sqrt {c d-b e} \left (4 b^2 e^2-5 b c d e+9 c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right ),\frac {b e}{c d}-1\right )}{\sqrt {c} e^{3/2}}-\frac {d \sqrt {c d-b e} \left (8 b^2 e^2+9 c d (c d-b e)\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {e} x}{\sqrt {c d-b e}}\right )|\frac {b e}{c d}-1\right )}{\sqrt {c} e^{3/2}}\right )}{3 c \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}-\frac {4 b e^2 x \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{3 c}}{5 c e^2}+\frac {x^3 \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}}{5 c}\)

Input:

Int[x^6/Sqrt[(-(c*d^2) + b*d*e)/e^2 + b*x^2 + c*x^4],x]
 

Output:

(x^3*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*x^4])/(5*c) + ((-4*b*e^2*x*Sq 
rt[-((d*(c*d - b*e))/e^2) + b*x^2 + c*x^4])/(3*c) - (Sqrt[1 + (e*x^2)/d]*S 
qrt[1 - (c*e*x^2)/(c*d - b*e)]*(-((d*Sqrt[c*d - b*e]*(8*b^2*e^2 + 9*c*d*(c 
*d - b*e))*EllipticE[ArcSin[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]], -1 + (b* 
e)/(c*d)])/(Sqrt[c]*e^(3/2))) + (d*Sqrt[c*d - b*e]*(9*c^2*d^2 - 5*b*c*d*e 
+ 4*b^2*e^2)*EllipticF[ArcSin[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]], -1 + ( 
b*e)/(c*d)])/(Sqrt[c]*e^(3/2))))/(3*c*Sqrt[-((d*(c*d - b*e))/e^2) + b*x^2 
+ c*x^4]))/(5*c*e^2)
 

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 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 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 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 1442

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] 
:> Simp[d^3*(d*x)^(m - 3)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 1))), 
x] - Simp[d^4/(c*(m + 4*p + 1))   Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b*(m + 
 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x 
] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2* 
p] && (IntegerQ[p] || IntegerQ[m])
 

rule 1514

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

rule 1602

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*( 
x_)^4)^(p_), x_Symbol] :> Simp[e*f*(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 
1)/(c*(m + 4*p + 3))), x] - Simp[f^2/(c*(m + 4*p + 3))   Int[(f*x)^(m - 2)* 
(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p 
+ 3))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c 
, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (IntegerQ[p] | 
| IntegerQ[m])
 

Maple [A] (verified)

Time = 20.37 (sec) , antiderivative size = 411, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{5 c}-\frac {4 b x \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{15 c^{2}}+\frac {4 b \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{15 c^{2} \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}-\frac {2 \left (-\frac {\frac {3 b d}{e}-\frac {3 c \,d^{2}}{e^{2}}}{5 c}+\frac {8 b^{2}}{15 c^{2}}\right ) \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(411\)
elliptic	\(\frac {x^{3} \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{5 c}-\frac {4 b x \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}{15 c^{2}}+\frac {4 b \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{15 c^{2} \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}-\frac {2 \left (-\frac {\frac {3 b d}{e}-\frac {3 c \,d^{2}}{e^{2}}}{5 c}+\frac {8 b^{2}}{15 c^{2}}\right ) \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(411\)
risch	\(-\frac {x \left (-3 c \,x^{2}+4 b \right ) \left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{15 c^{2} e^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}+\frac {\left (-\frac {2 \left (8 b^{2} e^{2}-9 b c d e +9 c^{2} d^{2}\right ) \left (b d e -c \,d^{2}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}\, \left (b \,e^{2}+e \left (b e -2 c d \right )\right )}+\frac {4 b^{2} d e \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}}-\frac {4 d^{2} c b \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4} e^{2}+b \,x^{2} e^{2}+b d e -c \,d^{2}}}\right ) \sqrt {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}}{15 c^{2} e^{2} \sqrt {\frac {\left (e \,x^{2}+d \right ) \left (c e \,x^{2}+b e -c d \right )}{e^{2}}}}\)	\(559\)

Input:

int(x^6/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/5/c*x^3*(c*x^4+b*x^2+b*d/e-c*d^2/e^2)^(1/2)-4/15*b/c^2*x*(c*x^4+b*x^2+b* 
d/e-c*d^2/e^2)^(1/2)+4/15*b/c^2*(b*d/e-c*d^2/e^2)/(-c*e/(b*e-c*d))^(1/2)*( 
1+c*e/(b*e-c*d)*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(c*x^4+b*x^2+b*d/e-c*d^2/e^2) 
^(1/2)*EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d)^(1/2))-2*(-1/5/c*(3 
*b*d/e-3*c*d^2/e^2)+8/15/c^2*b^2)*(b*d/e-c*d^2/e^2)/(-c*e/(b*e-c*d))^(1/2) 
*(1+c*e/(b*e-c*d)*x^2)^(1/2)*(1+e*x^2/d)^(1/2)/(c*x^4+b*x^2+b*d/e-c*d^2/e^ 
2)^(1/2)/(b+(b*e-2*c*d)/e)*(EllipticF(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d 
)^(1/2))-EllipticE(x*(-c*e/(b*e-c*d))^(1/2),(-1+b*e/c/d)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.67 \[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=-\frac {{\left (9 \, c^{2} d^{3} - 9 \, b c d^{2} e + 8 \, b^{2} d e^{2}\right )} \sqrt {c e^{2}} x \sqrt {-\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,-\frac {c d - b e}{c d}) - {\left (9 \, c^{2} d^{3} - 9 \, b c d^{2} e + 4 \, b^{2} e^{3} + 4 \, {\left (2 \, b^{2} - b c\right )} d e^{2}\right )} \sqrt {c e^{2}} x \sqrt {-\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {-\frac {d}{e}}}{x}\right )\,|\,-\frac {c d - b e}{c d}) - {\left (3 \, c^{2} e^{4} x^{4} - 4 \, b c e^{4} x^{2} + 9 \, c^{2} d^{2} e^{2} - 9 \, b c d e^{3} + 8 \, b^{2} e^{4}\right )} \sqrt {\frac {c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}{e^{2}}}}{15 \, c^{3} e^{4} x} \] Input:

integrate(x^6/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="fricas")
 

Output:

-1/15*((9*c^2*d^3 - 9*b*c*d^2*e + 8*b^2*d*e^2)*sqrt(c*e^2)*x*sqrt(-d/e)*el 
liptic_e(arcsin(sqrt(-d/e)/x), -(c*d - b*e)/(c*d)) - (9*c^2*d^3 - 9*b*c*d^ 
2*e + 4*b^2*e^3 + 4*(2*b^2 - b*c)*d*e^2)*sqrt(c*e^2)*x*sqrt(-d/e)*elliptic 
_f(arcsin(sqrt(-d/e)/x), -(c*d - b*e)/(c*d)) - (3*c^2*e^4*x^4 - 4*b*c*e^4* 
x^2 + 9*c^2*d^2*e^2 - 9*b*c*d*e^3 + 8*b^2*e^4)*sqrt((c*e^2*x^4 + b*e^2*x^2 
 - c*d^2 + b*d*e)/e^2))/(c^3*e^4*x)
 

Sympy [F]

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

integrate(x**6/((b*d*e-c*d**2)/e**2+b*x**2+c*x**4)**(1/2),x)
 

Output:

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

Maxima [F]

\[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {x^{6}}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}}} \,d x } \] Input:

integrate(x^6/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="maxima")
 

Output:

integrate(x^6/sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2), x)
 

Giac [F]

\[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int { \frac {x^{6}}{\sqrt {c x^{4} + b x^{2} - \frac {c d^{2} - b d e}{e^{2}}}} \,d x } \] Input:

integrate(x^6/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x, algorithm="giac")
 

Output:

integrate(x^6/sqrt(c*x^4 + b*x^2 - (c*d^2 - b*d*e)/e^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\int \frac {x^6}{\sqrt {b\,x^2-\frac {c\,d^2-b\,d\,e}{e^2}+c\,x^4}} \,d x \] Input:

int(x^6/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2),x)
                                                                                    
                                                                                    
 

Output:

int(x^6/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^6}{\sqrt {\frac {-c d^2+b d e}{e^2}+b x^2+c x^4}} \, dx=\frac {-4 \sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, b x +3 \sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, c \,x^{3}+8 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, x^{2}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) b^{2} e^{2}-9 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, x^{2}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) b c d e +9 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}\, x^{2}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) c^{2} d^{2}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) b^{2} d e -4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {c e \,x^{2}+b e -c d}}{c \,e^{2} x^{4}+b \,e^{2} x^{2}+b d e -c \,d^{2}}d x \right ) b c \,d^{2}}{15 c^{2} e} \] Input:

int(x^6/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(1/2),x)
 

Output:

( - 4*sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2)*b*x + 3*sqrt(d + e*x**2) 
*sqrt(b*e - c*d + c*e*x**2)*c*x**3 + 8*int((sqrt(d + e*x**2)*sqrt(b*e - c* 
d + c*e*x**2)*x**2)/(b*d*e + b*e**2*x**2 - c*d**2 + c*e**2*x**4),x)*b**2*e 
**2 - 9*int((sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2)*x**2)/(b*d*e + b* 
e**2*x**2 - c*d**2 + c*e**2*x**4),x)*b*c*d*e + 9*int((sqrt(d + e*x**2)*sqr 
t(b*e - c*d + c*e*x**2)*x**2)/(b*d*e + b*e**2*x**2 - c*d**2 + c*e**2*x**4) 
,x)*c**2*d**2 + 4*int((sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2))/(b*d*e 
 + b*e**2*x**2 - c*d**2 + c*e**2*x**4),x)*b**2*d*e - 4*int((sqrt(d + e*x** 
2)*sqrt(b*e - c*d + c*e*x**2))/(b*d*e + b*e**2*x**2 - c*d**2 + c*e**2*x**4 
),x)*b*c*d**2)/(15*c**2*e)