\(\int \frac {x^6}{(a-b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [281]

Optimal result

Integrand size = 27, antiderivative size = 319 \[ \int \frac {x^6}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {a x^3 \sqrt {c+d x^2}}{b (b c+a d) \sqrt {a-b x^2}}+\frac {(b c+4 a d) x \sqrt {a-b x^2} \sqrt {c+d x^2}}{3 b^2 d (b c+a d)}+\frac {\sqrt {a} \left (2 b^2 c^2-3 a b c d-8 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 (b c+a d) \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}-\frac {2 \sqrt {a} c (b c-2 a d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{3 b^{5/2} d^2 \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Output:

a*x^3*(d*x^2+c)^(1/2)/b/(a*d+b*c)/(-b*x^2+a)^(1/2)+1/3*(4*a*d+b*c)*x*(-b*x 
^2+a)^(1/2)*(d*x^2+c)^(1/2)/b^2/d/(a*d+b*c)+1/3*a^(1/2)*(-8*a^2*d^2-3*a*b* 
c*d+2*b^2*c^2)*(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^(5/2)/d^2/(a*d+b*c)/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1 
/2)-2/3*a^(1/2)*c*(-2*a*d+b*c)*(1-b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)*Ellipti 
cF(b^(1/2)*x/a^(1/2),(-a*d/b/c)^(1/2))/b^(5/2)/d^2/(-b*x^2+a)^(1/2)/(d*x^2 
+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.28 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.84 \[ \int \frac {x^6}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {-\frac {b}{a}} d x \left (c+d x^2\right ) \left (4 a^2 d-b^2 c x^2+a b \left (c-d x^2\right )\right )+i c \left (-2 b^2 c^2+3 a b c d+8 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-2 i c \left (-b^2 c^2+a b c d+2 a^2 d^2\right ) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{3 b^2 \sqrt {-\frac {b}{a}} d^2 (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[-(b/a)]*d*x*(c + d*x^2)*(4*a^2*d - b^2*c*x^2 + a*b*(c - d*x^2)) + I* 
c*(-2*b^2*c^2 + 3*a*b*c*d + 8*a^2*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2 
)/c]*EllipticE[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))] - (2*I)*c*(-(b^2 
*c^2) + a*b*c*d + 2*a^2*d^2)*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip 
ticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(3*b^2*Sqrt[-(b/a)]*d^2*( 
b*c + a*d)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.03, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {372, 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.

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

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 444

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(a*x^3*Sqrt[c + d*x^2])/(b*(b*c + a*d)*Sqrt[a - b*x^2]) - (-1/3*((b*c + 4* 
a*d)*x*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])/(b*d) + (-((Sqrt[a]*(2*b^2*c^2 - 3 
*a*b*c*d - 8*a^2*d^2)*Sqrt[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])) + (2*Sqrt[a]*c*(b*c - 2*a*d)*(b*c + a*d)*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))/(b*(b*c + a*d))
 

Defintions of rubi rules used

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 372

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

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 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 = 12.18 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.31

method	result	size
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}-b c \right ) a^{2} x}{b^{3} \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {x \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}{3 d \,b^{2}}+\frac {\left (-\frac {c \,a^{2}}{b^{2} \left (a d +b c \right )}-\frac {a c}{3 b^{2} d}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {\left (-\frac {a}{b^{2}}-\frac {d \,a^{2}}{b^{2} \left (a d +b c \right )}-\frac {2 a d -2 b c}{3 d \,b^{2}}\right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(417\)
default	\(\frac {\left (-\sqrt {\frac {b}{a}}\, a b \,d^{3} x^{5}-\sqrt {\frac {b}{a}}\, b^{2} c \,d^{2} x^{5}+4 \sqrt {\frac {b}{a}}\, a^{2} d^{3} x^{3}-\sqrt {\frac {b}{a}}\, b^{2} c^{2} d \,x^{3}+4 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c \,d^{2}+2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} d -2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{3}-8 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a^{2} c \,d^{2}-3 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a b \,c^{2} d +2 \sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b^{2} c^{3}+4 \sqrt {\frac {b}{a}}\, a^{2} c \,d^{2} x +\sqrt {\frac {b}{a}}\, a b \,c^{2} d x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{3 b^{2} \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, d^{2} \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(520\)
risch	\(\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, x}{3 b^{2} d}-\frac {\left (\frac {-\frac {b \left (5 a d -2 b c \right ) c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}+\frac {a b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {3 d \,a^{2} \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}}{b}+\frac {3 a^{3} d \left (\frac {\left (-b d \,x^{2}-b c \right ) x}{a \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {a}{b}\right ) \left (-b d \,x^{2}-b c \right )}}+\frac {\left (-\frac {1}{a}+\frac {b c}{a \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}-\frac {b c \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{a \left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}\right )}{b}\right ) \sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{3 b^{2} d \sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(712\)

Input:

int(x^6/(-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^3*a^2/(a*d+b*c)*x/((x^2-a/b)*(-b*d*x^2-b*c))^(1/2)+1/3/d/b^2*x*(-b* 
d*x^4+a*d*x^2-b*c*x^2+a*c)^(1/2)+(-1/b^2*c*a^2/(a*d+b*c)-1/3/b^2/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^2-1/b^2*d 
*a^2/(a*d+b*c)-1/3/d/b^2*(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*(EllipticF(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 = 0.09 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.22 \[ \int \frac {x^6}{\left (a-b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=-\frac {{\left ({\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d - 8 \, a^{3} b d^{2}\right )} x^{3} - {\left (2 \, a^{2} b^{2} c^{2} - 3 \, a^{3} b c d - 8 \, a^{4} d^{2}\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + {\left ({\left (8 \, a^{3} b d^{2} - {\left (2 \, a b^{3} - b^{4}\right )} c^{2} + {\left (3 \, a^{2} b^{2} + 4 \, a b^{3}\right )} c d\right )} x^{3} - {\left (8 \, a^{4} d^{2} - {\left (2 \, a^{2} b^{2} - a b^{3}\right )} c^{2} + {\left (3 \, a^{3} b + 4 \, a^{2} b^{2}\right )} c d\right )} x\right )} \sqrt {-b d} \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - {\left (2 \, a b^{3} c^{2} - 3 \, a^{2} b^{2} c d - 8 \, a^{3} b d^{2} + {\left (b^{4} c d + a b^{3} d^{2}\right )} x^{4} - 2 \, {\left (b^{4} c^{2} - a b^{3} c d - 2 \, a^{2} b^{2} d^{2}\right )} x^{2}\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left ({\left (b^{6} c d^{2} + a b^{5} d^{3}\right )} x^{3} - {\left (a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x\right )}} \] Input:

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

Output:

-1/3*(((2*a*b^3*c^2 - 3*a^2*b^2*c*d - 8*a^3*b*d^2)*x^3 - (2*a^2*b^2*c^2 - 
3*a^3*b*c*d - 8*a^4*d^2)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/ 
b)/x), -b*c/(a*d)) + ((8*a^3*b*d^2 - (2*a*b^3 - b^4)*c^2 + (3*a^2*b^2 + 4* 
a*b^3)*c*d)*x^3 - (8*a^4*d^2 - (2*a^2*b^2 - a*b^3)*c^2 + (3*a^3*b + 4*a^2* 
b^2)*c*d)*x)*sqrt(-b*d)*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), -b*c/(a* 
d)) - (2*a*b^3*c^2 - 3*a^2*b^2*c*d - 8*a^3*b*d^2 + (b^4*c*d + a*b^3*d^2)*x 
^4 - 2*(b^4*c^2 - a*b^3*c*d - 2*a^2*b^2*d^2)*x^2)*sqrt(-b*x^2 + a)*sqrt(d* 
x^2 + c))/((b^6*c*d^2 + a*b^5*d^3)*x^3 - (a*b^5*c*d^2 + a^2*b^4*d^3)*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(3*sqrt(c + d*x**2)*sqrt(a - b*x**2)*c*x - 2*sqrt(c + d*x**2)*sqrt(a - b*x 
**2)*d*x**3 + 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*d**2 - 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*c*d - 
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*d**2*x**2 + i 
nt((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)*b**2*c*d*x**2 - 3*in 
t((sqrt(c + d*x**2)*sqrt(a - b*x**2))/(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*c**2 + 3*int((sqrt(c 
+ d*x**2)*sqrt(a - b*x**2))/(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*c**2*x**2)/(6*b*d**2*(a - b*x**2 
))