\(\int \frac {\sqrt {a+b x^2} (e+f x^2)}{(c+d x^2)^{7/2}} \, dx\) [6]

Optimal result

Integrand size = 30, antiderivative size = 378 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {(d e-c f) x \sqrt {a+b x^2}}{5 c d \left (c+d x^2\right )^{5/2}}-\frac {(a d (4 d e+c f)-b c (3 d e+2 c f)) x \sqrt {a+b x^2}}{15 c^2 d (b c-a d) \left (c+d x^2\right )^{3/2}}+\frac {\left (2 a^2 d^2 (4 d e+c f)+b^2 c^2 (3 d e+2 c f)-a b c d (13 d e+2 c f)\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{15 c^{5/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (b c (6 d e-c f)-a d (4 d e+c f)) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{15 c^{3/2} d^{3/2} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/5*(-c*f+d*e)*x*(b*x^2+a)^(1/2)/c/d/(d*x^2+c)^(5/2)-1/15*(a*d*(c*f+4*d*e) 
-b*c*(2*c*f+3*d*e))*x*(b*x^2+a)^(1/2)/c^2/d/(-a*d+b*c)/(d*x^2+c)^(3/2)+1/1 
5*(2*a^2*d^2*(c*f+4*d*e)+b^2*c^2*(2*c*f+3*d*e)-a*b*c*d*(2*c*f+13*d*e))*(b* 
x^2+a)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/ 
2))/c^(5/2)/d^(3/2)/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c) 
^(1/2)+1/15*b*(b*c*(-c*f+6*d*e)-a*d*(c*f+4*d*e))*(b*x^2+a)^(1/2)*InverseJa 
cobiAM(arctan(d^(1/2)*x/c^(1/2)),(1-b*c/a/d)^(1/2))/c^(3/2)/d^(3/2)/(-a*d+ 
b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.61 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (3 c^2 (b c-a d)^2 (d e-c f)+c (b c-a d) (-a d (4 d e+c f)+b c (3 d e+2 c f)) \left (c+d x^2\right )+\left (2 a^2 d^2 (4 d e+c f)+b^2 c^2 (3 d e+2 c f)-a b c d (13 d e+2 c f)\right ) \left (c+d x^2\right )^2\right )+i b c \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right )^2 \sqrt {1+\frac {d x^2}{c}} \left (\left (2 a^2 d^2 (4 d e+c f)+b^2 c^2 (3 d e+2 c f)-a b c d (13 d e+2 c f)\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-(b c-a d) (-a d (4 d e+c f)+b c (3 d e+2 c f)) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{15 \sqrt {\frac {b}{a}} c^3 d^2 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{5/2}} \] Input:

Integrate[(Sqrt[a + b*x^2]*(e + f*x^2))/(c + d*x^2)^(7/2),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.06, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {401, 25, 402, 25, 400, 313, 320}

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 401

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

\(\displaystyle \frac {-\frac {-\frac {a b (b c (6 d e-c f)-a d (c f+4 d e)) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}-\frac {\sqrt {a+b x^2} \left (2 a^2 d^2 (c f+4 d e)-a b c d (2 c f+13 d e)+b^2 c^2 (2 c f+3 d e)\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}-\frac {x \sqrt {a+b x^2} (a d (c f+4 d e)-b c (2 c f+3 d e))}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}}{5 c d}+\frac {x \sqrt {a+b x^2} (d e-c f)}{5 c d \left (c+d x^2\right )^{5/2}}\)

\(\Big \downarrow \) 320

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

Input:

Int[(Sqrt[a + b*x^2]*(e + f*x^2))/(c + d*x^2)^(7/2),x]
 

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 400

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^ 
(3/2)), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(Sqrt[a + b*x^2]* 
Sqrt[c + d*x^2]), x], x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[Sqrt[a + b*x^ 
2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a] & 
& PosQ[d/c]
 

rule 401

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/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1))   Int[(a + b*x^2)^(p + 1)*( 
c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + 
(b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L 
tQ[p, -1] && GtQ[q, 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(753\) vs. \(2(349)=698\).

Time = 6.05 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.99

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (c f -d e \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{5 c \,d^{4} \left (x^{2}+\frac {c}{d}\right )^{3}}+\frac {\left (a c d f +4 a \,d^{2} e -2 b \,c^{2} f -3 b c d e \right ) x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{15 d^{3} c^{2} \left (a d -b c \right ) \left (x^{2}+\frac {c}{d}\right )^{2}}+\frac {\left (b d \,x^{2}+a d \right ) x \left (2 a^{2} c f \,d^{2}+8 a^{2} d^{3} e -2 a b \,c^{2} d f -13 a b c \,d^{2} e +2 b^{2} c^{3} f +3 b^{2} c^{2} d e \right )}{15 c^{3} d^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {b \left (a c d f +4 a \,d^{2} e -2 b \,c^{2} f -3 b c d e \right )}{15 \left (a d -b c \right ) c^{2} d^{2}}+\frac {2 a^{2} c f \,d^{2}+8 a^{2} d^{3} e -2 a b \,c^{2} d f -13 a b c \,d^{2} e +2 b^{2} c^{3} f +3 b^{2} c^{2} d e}{15 d^{2} \left (a d -b c \right ) c^{3}}-\frac {a \left (2 a^{2} c f \,d^{2}+8 a^{2} d^{3} e -2 a b \,c^{2} d f -13 a b c \,d^{2} e +2 b^{2} c^{3} f +3 b^{2} c^{2} d e \right )}{15 d \,c^{3} \left (a d -b c \right )^{2}}\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 \left (2 a^{2} c f \,d^{2}+8 a^{2} d^{3} e -2 a b \,c^{2} d f -13 a b c \,d^{2} e +2 b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \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 )}{15 d^{2} \left (a d -b c \right )^{2} c^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(754\)
default	\(\text {Expression too large to display}\)	\(2882\)

Input:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1175 vs. \(2 (349) = 698\).

Time = 0.14 (sec) , antiderivative size = 1175, normalized size of antiderivative = 3.11 \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(7/2),x, algorithm="fricas")
 

Output:

-1/15*((((3*b^3*c^2*d^4 - 13*a*b^2*c*d^5 + 8*a^2*b*d^6)*e + 2*(b^3*c^3*d^3 
 - a*b^2*c^2*d^4 + a^2*b*c*d^5)*f)*x^6 + 3*((3*b^3*c^3*d^3 - 13*a*b^2*c^2* 
d^4 + 8*a^2*b*c*d^5)*e + 2*(b^3*c^4*d^2 - a*b^2*c^3*d^3 + a^2*b*c^2*d^4)*f 
)*x^4 + 3*((3*b^3*c^4*d^2 - 13*a*b^2*c^3*d^3 + 8*a^2*b*c^2*d^4)*e + 2*(b^3 
*c^5*d - a*b^2*c^4*d^2 + a^2*b*c^3*d^3)*f)*x^2 + (3*b^3*c^5*d - 13*a*b^2*c 
^4*d^2 + 8*a^2*b*c^3*d^3)*e + 2*(b^3*c^6 - a*b^2*c^5*d + a^2*b*c^4*d^2)*f) 
*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (((3*b 
^3*c^2*d^4 - (6*a^2*b + 13*a*b^2)*c*d^5 + 4*(a^3 + 2*a^2*b)*d^6)*e + (2*b^ 
3*c^3*d^3 + (a^2*b - 2*a*b^2)*c^2*d^4 + (a^3 + 2*a^2*b)*c*d^5)*f)*x^6 + 3* 
((3*b^3*c^3*d^3 - (6*a^2*b + 13*a*b^2)*c^2*d^4 + 4*(a^3 + 2*a^2*b)*c*d^5)* 
e + (2*b^3*c^4*d^2 + (a^2*b - 2*a*b^2)*c^3*d^3 + (a^3 + 2*a^2*b)*c^2*d^4)* 
f)*x^4 + 3*((3*b^3*c^4*d^2 - (6*a^2*b + 13*a*b^2)*c^3*d^3 + 4*(a^3 + 2*a^2 
*b)*c^2*d^4)*e + (2*b^3*c^5*d + (a^2*b - 2*a*b^2)*c^4*d^2 + (a^3 + 2*a^2*b 
)*c^3*d^3)*f)*x^2 + (3*b^3*c^5*d - (6*a^2*b + 13*a*b^2)*c^4*d^2 + 4*(a^3 + 
 2*a^2*b)*c^3*d^3)*e + (2*b^3*c^6 + (a^2*b - 2*a*b^2)*c^5*d + (a^3 + 2*a^2 
*b)*c^4*d^2)*f)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/ 
(b*c)) - (((3*a*b^2*c^2*d^4 - 13*a^2*b*c*d^5 + 8*a^3*d^6)*e + 2*(a*b^2*c^3 
*d^3 - a^2*b*c^2*d^4 + a^3*c*d^5)*f)*x^5 + ((9*a*b^2*c^3*d^3 - 33*a^2*b*c^ 
2*d^4 + 20*a^3*c*d^5)*e + (6*a*b^2*c^4*d^2 - 7*a^2*b*c^3*d^3 + 5*a^3*c^2*d 
^4)*f)*x^3 + ((9*a*b^2*c^4*d^2 - 26*a^2*b*c^3*d^3 + 15*a^3*c^2*d^4)*e +...
 

Sympy [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {a + b x^{2}} \left (e + f x^{2}\right )}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((b*x**2+a)**(1/2)*(f*x**2+e)/(d*x**2+c)**(7/2),x)
 

Output:

Integral(sqrt(a + b*x**2)*(e + f*x**2)/(c + d*x**2)**(7/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(7/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int { \frac {\sqrt {b x^{2} + a} {\left (f x^{2} + e\right )}}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(7/2),x, algorithm="giac")
 

Output:

integrate(sqrt(b*x^2 + a)*(f*x^2 + e)/(d*x^2 + c)^(7/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\int \frac {\sqrt {b\,x^2+a}\,\left (f\,x^2+e\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \] Input:

int(((a + b*x^2)^(1/2)*(e + f*x^2))/(c + d*x^2)^(7/2),x)
 

Output:

int(((a + b*x^2)^(1/2)*(e + f*x^2))/(c + d*x^2)^(7/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a+b x^2} \left (e+f x^2\right )}{\left (c+d x^2\right )^{7/2}} \, dx=\text {too large to display} \] Input:

int((b*x^2+a)^(1/2)*(f*x^2+e)/(d*x^2+c)^(7/2),x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*f*x - sqrt(c + d*x**2)*sqrt(a + b* 
x**2)*b*e*x + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a**2*c**4* 
d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a**2*c*d**4*x**6 + 
2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*b*c**3*d**2*x**4 + 8 
*a*b*c**2*d**3*x**6 + 7*a*b*c*d**4*x**8 + 2*a*b*d**5*x**10 - b**2*c**5*x** 
2 - 4*b**2*c**4*d*x**4 - 6*b**2*c**3*d**2*x**6 - 4*b**2*c**2*d**3*x**8 - b 
**2*c*d**4*x**10),x)*a**2*b*c**3*d**2*f + 6*int((sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*x**4)/(2*a**2*c**4*d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d**3* 
x**4 + 8*a**2*c*d**4*x**6 + 2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d*x** 
2 + 2*a*b*c**3*d**2*x**4 + 8*a*b*c**2*d**3*x**6 + 7*a*b*c*d**4*x**8 + 2*a* 
b*d**5*x**10 - b**2*c**5*x**2 - 4*b**2*c**4*d*x**4 - 6*b**2*c**3*d**2*x**6 
 - 4*b**2*c**2*d**3*x**8 - b**2*c*d**4*x**10),x)*a**2*b*c**2*d**3*f*x**2 + 
 6*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**4)/(2*a**2*c**4*d + 8*a**2*c* 
*3*d**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a**2*c*d**4*x**6 + 2*a**2*d**5*x 
**8 - a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*b*c**3*d**2*x**4 + 8*a*b*c**2*d** 
3*x**6 + 7*a*b*c*d**4*x**8 + 2*a*b*d**5*x**10 - b**2*c**5*x**2 - 4*b**2*c* 
*4*d*x**4 - 6*b**2*c**3*d**2*x**6 - 4*b**2*c**2*d**3*x**8 - b**2*c*d**4*x* 
*10),x)*a**2*b*c*d**4*f*x**4 + 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x* 
*4)/(2*a**2*c**4*d + 8*a**2*c**3*d**2*x**2 + 12*a**2*c**2*d**3*x**4 + 8*a* 
*2*c*d**4*x**6 + 2*a**2*d**5*x**8 - a*b*c**5 - 2*a*b*c**4*d*x**2 + 2*a*...