\(\int \frac {\sqrt {a+b x^2} (c+d x^2)^{3/2} (e+f x^2)}{x^6} \, dx\) [94]

Optimal result

Integrand size = 33, antiderivative size = 453 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^6} \, dx=-\frac {b \left (2 b^2 c^2 e-a b c (7 d e+5 c f)-a^2 d (3 d e+35 c f)\right ) x \sqrt {c+d x^2}}{15 a^2 c \sqrt {a+b x^2}}-\frac {d (b c e+3 a d e+20 a c f) \sqrt {a+b x^2} \sqrt {c+d x^2}}{15 a c x}+\frac {(2 b c e-3 a d e-5 a c f) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}{15 a^2 x^3}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}+\frac {\sqrt {b} \left (2 b^2 c^2 e-a b c (7 d e+5 c f)-a^2 d (3 d e+35 c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{15 a^{3/2} c \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d \left (b^2 c e-15 a^2 d f-a b (9 d e+25 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{15 \sqrt {a} \sqrt {b} c \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

-1/15*b*(2*b^2*c^2*e-a*b*c*(5*c*f+7*d*e)-a^2*d*(35*c*f+3*d*e))*x*(d*x^2+c) 
^(1/2)/a^2/c/(b*x^2+a)^(1/2)-1/15*d*(20*a*c*f+3*a*d*e+b*c*e)*(b*x^2+a)^(1/ 
2)*(d*x^2+c)^(1/2)/a/c/x+1/15*(-5*a*c*f-3*a*d*e+2*b*c*e)*(b*x^2+a)^(3/2)*( 
d*x^2+c)^(1/2)/a^2/x^3-1/5*e*(b*x^2+a)^(3/2)*(d*x^2+c)^(3/2)/a/x^5+1/15*b^ 
(1/2)*(2*b^2*c^2*e-a*b*c*(5*c*f+7*d*e)-a^2*d*(35*c*f+3*d*e))*(d*x^2+c)^(1/ 
2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1/2))/a^(3/2 
)/c/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/15*d*(b^2*c*e-15*a^2 
*d*f-a*b*(25*c*f+9*d*e))*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/ 
a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(1/2)/c/(b*x^2+a)^(1/2)/(a*(d*x^2+c) 
/c/(b*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 3.48 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} \left (e+f x^2\right )}{x^6} \, dx=\frac {-\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-2 b^2 c^2 e x^4+a b c x^2 \left (7 d e x^2+c \left (e+5 f x^2\right )\right )+a^2 \left (3 d^2 e x^4+c^2 \left (3 e+5 f x^2\right )+c d \left (6 e x^2+20 f x^4\right )\right )\right )-i b c \left (-2 b^2 c^2 e+a b c (7 d e+5 c f)+a^2 d (3 d e+35 c f)\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i c (-b c+a d) \left (-2 b^2 c e+15 a^2 d f+a b (6 d e+5 c f)\right ) x^5 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{15 a^2 \sqrt {\frac {b}{a}} c x^5 \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(-(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-2*b^2*c^2*e*x^4 + a*b*c*x^2*(7*d*e* 
x^2 + c*(e + 5*f*x^2)) + a^2*(3*d^2*e*x^4 + c^2*(3*e + 5*f*x^2) + c*d*(6*e 
*x^2 + 20*f*x^4)))) - I*b*c*(-2*b^2*c^2*e + a*b*c*(7*d*e + 5*c*f) + a^2*d* 
(3*d*e + 35*c*f))*x^5*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)*(-2*b^2*c*e + 15*a 
^2*d*f + a*b*(6*d*e + 5*c*f))*x^5*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]* 
EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(15*a^2*Sqrt[b/a]*c*x^5*Sq 
rt[a + b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 419, normalized size of antiderivative = 0.92, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {442, 25, 442, 25, 27, 442, 25, 406, 320, 388, 313}

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.

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

\(\Big \downarrow \) 442

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 442

\(\displaystyle -\frac {\frac {\int -\frac {d \sqrt {b x^2+a} \left (a (b c e+3 a d e+20 a c f)-\left (-15 d f a^2-b (6 d e+5 c f) a+2 b^2 c e\right ) x^2\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {\int \frac {d \sqrt {b x^2+a} \left (a (b c e+3 a d e+20 a c f)-\left (-15 d f a^2-b (6 d e+5 c f) a+2 b^2 c e\right ) x^2\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {d \int \frac {\sqrt {b x^2+a} \left (a (b c e+3 a d e+20 a c f)-\left (-15 d f a^2-b (6 d e+5 c f) a+2 b^2 c e\right ) x^2\right )}{x^2 \sqrt {d x^2+c}}dx}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 442

\(\displaystyle -\frac {-\frac {d \left (\frac {\int -\frac {b \left (-d (3 d e+35 c f) a^2-b c (7 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c \left (-15 d f a^2-b (9 d e+25 c f) a+b^2 c e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (20 a c f+3 a d e+b c e)}{c x}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {d \left (-\frac {\int \frac {b \left (-d (3 d e+35 c f) a^2-b c (7 d e+5 c f) a+2 b^2 c^2 e\right ) x^2+a c \left (-15 d f a^2-b (9 d e+25 c f) a+b^2 c e\right )}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (20 a c f+3 a d e+b c e)}{c x}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 406

\(\displaystyle -\frac {-\frac {d \left (-\frac {b \left (a^2 (-d) (35 c f+3 d e)-a b c (5 c f+7 d e)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+a c \left (-15 a^2 d f-a b (25 c f+9 d e)+b^2 c e\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (20 a c f+3 a d e+b c e)}{c x}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 320

\(\displaystyle -\frac {-\frac {d \left (-\frac {b \left (a^2 (-d) (35 c f+3 d e)-a b c (5 c f+7 d e)+2 b^2 c^2 e\right ) \int \frac {x^2}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx+\frac {c^{3/2} \sqrt {a+b x^2} \left (-15 a^2 d f-a b (25 c f+9 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (20 a c f+3 a d e+b c e)}{c x}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 388

\(\displaystyle -\frac {-\frac {d \left (-\frac {b \left (a^2 (-d) (35 c f+3 d e)-a b c (5 c f+7 d e)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b}\right )+\frac {c^{3/2} \sqrt {a+b x^2} \left (-15 a^2 d f-a b (25 c f+9 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (20 a c f+3 a d e+b c e)}{c x}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

\(\Big \downarrow \) 313

\(\displaystyle -\frac {-\frac {d \left (-\frac {\frac {c^{3/2} \sqrt {a+b x^2} \left (-15 a^2 d f-a b (25 c f+9 d e)+b^2 c e\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+b \left (a^2 (-d) (35 c f+3 d e)-a b c (5 c f+7 d e)+2 b^2 c^2 e\right ) \left (\frac {x \sqrt {a+b x^2}}{b \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}\right )}{c}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2} (20 a c f+3 a d e+b c e)}{c x}\right )}{3 a}-\frac {\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (-5 a c f-3 a d e+2 b c e)}{3 a x^3}}{5 a}-\frac {e \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}{5 a x^5}\)

Input:

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

Output:

-1/5*(e*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2))/(a*x^5) - (-1/3*((2*b*c*e - 3 
*a*d*e - 5*a*c*f)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])/(a*x^3) - (d*(-((a*(b 
*c*e + 3*a*d*e + 20*a*c*f)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x)) - (b*(2 
*b^2*c^2*e - a*b*c*(7*d*e + 5*c*f) - a^2*d*(3*d*e + 35*c*f))*((x*Sqrt[a + 
b*x^2])/(b*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[a + b*x^2]*EllipticE[ArcTan[(S 
qrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*( 
c + d*x^2))]*Sqrt[c + d*x^2])) + (c^(3/2)*(b^2*c*e - 15*a^2*d*f - a*b*(9*d 
*e + 25*c*f))*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - ( 
b*c)/(a*d)])/(Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 
]))/c))/(3*a))/(5*a)
 

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 313

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim 
p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c 
+ d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ 
[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
 

rule 320

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( 
c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre 
eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

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

rule 442

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

Maple [A] (verified)

Time = 7.32 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.11

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {c e \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 x^{5}}-\frac {\left (5 a c f +6 a d e +b c e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a \,x^{3}}-\frac {\left (20 a^{2} c f d +3 a^{2} d^{2} e +5 a b \,c^{2} f +7 a b c d e -2 b^{2} c^{2} e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 a^{2} c x}+\frac {\left (a \,d^{2} f +2 b c d f +b \,d^{2} e -\frac {b d \left (5 a c f +6 a d e +b c e \right )}{15 a}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (f b \,d^{2}+\frac {b d \left (20 a^{2} c f d +3 a^{2} d^{2} e +5 a b \,c^{2} f +7 a b c d e -2 b^{2} c^{2} e \right )}{15 a^{2} c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(502\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}\, \left (20 d f \,x^{4} a^{2} c +3 a^{2} d^{2} e \,x^{4}+5 a b \,c^{2} f \,x^{4}+7 a b c d e \,x^{4}-2 b^{2} c^{2} e \,x^{4}+5 a^{2} c^{2} f \,x^{2}+6 a^{2} c d e \,x^{2}+a b \,c^{2} e \,x^{2}+3 a^{2} c^{2} e \right )}{15 x^{5} a^{2} c}+\frac {d \left (-\frac {b \left (35 a^{2} c f d +3 a^{2} d^{2} e +5 a b \,c^{2} f +7 a b c d e -2 b^{2} c^{2} e \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}-\frac {a \,b^{2} c^{2} e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {25 a^{2} b \,c^{2} f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {15 a^{3} c d f \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {9 a^{2} b c d e \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}}{15 a^{2} c \sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(718\)
default	\(\text {Expression too large to display}\)	\(1178\)

Input:

int((b*x^2+a)^(1/2)*(d*x^2+c)^(3/2)*(f*x^2+e)/x^6,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)*(-1/5*c*e*(b*d 
*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^5-1/15*(5*a*c*f+6*a*d*e+b*c*e)/a*(b*d*x^ 
4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x^3-1/15/a^2/c*(20*a^2*c*d*f+3*a^2*d^2*e+5*a* 
b*c^2*f+7*a*b*c*d*e-2*b^2*c^2*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/x+(a* 
d^2*f+2*b*c*d*f+b*d^2*e-1/15*b*d*(5*a*c*f+6*a*d*e+b*c*e)/a)/(-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)*Ell 
ipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-(f*b*d^2+1/15*b*d*(20*a^2* 
c*d*f+3*a^2*d^2*e+5*a*b*c^2*f+7*a*b*c*d*e-2*b^2*c^2*e)/a^2/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 [F]

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

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

Output:

integral((d*f*x^4 + (d*e + c*f)*x^2 + c*e)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c) 
/x^6, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d*f*x**2 + 10*sqrt(c + d*x* 
*2)*sqrt(a + b*x**2)*a**2*d**2*f*x**4 - sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
a*b*c**2*e - 10*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c**2*f*x**2 - 5*sqrt 
(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d*e*x**2 + 20*sqrt(c + d*x**2)*sqrt(a 
+ b*x**2)*a*b*c*d*f*x**4 + 5*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*d**2*e* 
x**4 + 15*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*f*x**4 + 3*sqrt(c + 
d*x**2)*sqrt(a + b*x**2)*b**2*c*d*e*x**4 - 10*int((sqrt(c + d*x**2)*sqrt(a 
 + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + 
a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**3*b*d**4*f*x**5 - 25 
*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + 
 a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x** 
4),x)*a**2*b**2*c*d**3*f*x**5 - 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x 
**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x** 
4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a**2*b**2*d**4*e*x**5 - 30*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 
 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2*c**2*x**2 + b**2*c*d*x**4),x)*a*b 
**3*c**2*d**2*f*x**5 - 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a** 
2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*b*c*d*x**2 + a*b*d**2*x**4 + b**2* 
c**2*x**2 + b**2*c*d*x**4),x)*a*b**3*c*d**3*e*x**5 - 15*int((sqrt(c + d*x* 
*2)*sqrt(a + b*x**2)*x**2)/(a**2*c*d + a**2*d**2*x**2 + a*b*c**2 + 2*a*...