\(\int \frac {e+f x^2}{(a+b x^2)^{5/2} \sqrt {c+d x^2}} \, dx\) [36]

Optimal result

Integrand size = 30, antiderivative size = 289 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {(b e-a f) x \sqrt {c+d x^2}}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {\left (2 b^2 c e+a^2 d f-a b (4 d e-c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} \sqrt {b} (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d (b c e-3 a d e+2 a c f) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} \sqrt {b} c (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*(-a*f+b*e)*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(3/2)+1/3*(2*b^2*c 
*e+a^2*d*f-a*b*(-c*f+4*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)/b^(1/2)/(-a*d+b*c)^2/(b*x^2+a) 
^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)-1/3*d*(2*a*c*f-3*a*d*e+b*c*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/(-a*d+b*c)^2/(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 = 10.70 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.04 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (2 a^3 d f+2 b^3 c e x^2+a^2 b d \left (-5 e+f x^2\right )+a b^2 \left (3 c e-4 d e x^2+c f x^2\right )\right )+i c \left (2 b^2 c e+a^2 d f+a b (-4 d e+c f)\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) (2 b c e-3 a d e+a c f) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} (b c-a d)^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

Integrate[(e + f*x^2)/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]
 

Output:

(Sqrt[b/a]*x*(c + d*x^2)*(2*a^3*d*f + 2*b^3*c*e*x^2 + a^2*b*d*(-5*e + f*x^ 
2) + a*b^2*(3*c*e - 4*d*e*x^2 + c*f*x^2)) + I*c*(2*b^2*c*e + a^2*d*f + a*b 
*(-4*d*e + c*f))*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellip 
ticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + I*(-(b*c) + a*d)*(2*b*c*e - 3* 
a*d*e + a*c*f)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Ellipti 
cF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*(b*c - a*d)^2*(a 
 + b*x^2)^(3/2)*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.04, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

Input:

Int[(e + f*x^2)/((a + b*x^2)^(5/2)*Sqrt[c + d*x^2]),x]
 

Output:

((b*e - a*f)*x*Sqrt[c + d*x^2])/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)) + (((2 
*b^2*c*e + a^2*d*f - a*b*(4*d*e - c*f))*Sqrt[c + d*x^2]*EllipticE[ArcTan[( 
Sqrt[b]*x)/Sqrt[a]], 1 - (a*d)/(b*c)])/(Sqrt[a]*Sqrt[b]*(b*c - a*d)*Sqrt[a 
 + b*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]) - (Sqrt[c]*Sqrt[d]*(b*c*e 
 - 3*a*d*e + 2*a*c*f)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c] 
], 1 - (b*c)/(a*d)])/(a*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]* 
Sqrt[c + d*x^2]))/(3*a*(b*c - a*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 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 [A] (verified)

Time = 10.06 (sec) , antiderivative size = 532, normalized size of antiderivative = 1.84

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {x \left (a f -b e \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b^{2} a \left (a d -b c \right ) \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) x \left (a^{2} d f +a b c f -4 a b d e +2 c e \,b^{2}\right )}{3 b \,a^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (\frac {d \left (a f -b e \right )}{3 b \left (a d -b c \right ) a}-\frac {a^{2} d f +a b c f -4 a b d e +2 c e \,b^{2}}{3 \left (a d -b c \right ) b \,a^{2}}-\frac {c \left (a^{2} d f +a b c f -4 a b d e +2 c e \,b^{2}\right )}{3 a^{2} \left (a d -b c \right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {\left (a^{2} d f +a b c f -4 a b d e +2 c e \,b^{2}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 \left (a d -b c \right )^{2} a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(532\)
default	\(\text {Expression too large to display}\)	\(1351\)

Input:

int((f*x^2+e)/(b*x^2+a)^(5/2)/(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)*(1/3/b^2/a/(a* 
d-b*c)*x*(a*f-b*e)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2+1/3*(b* 
d*x^2+b*c)/b/a^2/(a*d-b*c)^2*x*(a^2*d*f+a*b*c*f-4*a*b*d*e+2*b^2*c*e)/((x^2 
+a/b)*(b*d*x^2+b*c))^(1/2)+(1/3*d/b*(a*f-b*e)/(a*d-b*c)/a-1/3/(a*d-b*c)/b* 
(a^2*d*f+a*b*c*f-4*a*b*d*e+2*b^2*c*e)/a^2-1/3*c/a^2/(a*d-b*c)^2*(a^2*d*f+a 
*b*c*f-4*a*b*d*e+2*b^2*c*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))+1/3*(a^2*d*f+a*b*c*f-4*a*b*d*e+2*b^2*c*e)/(a*d-b*c)^2/a^2 
*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))-Elliptic 
E(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. 647 vs. \(2 (266) = 532\).

Time = 0.12 (sec) , antiderivative size = 647, normalized size of antiderivative = 2.24 \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {{\left ({\left (2 \, {\left (b^{5} c^{2} - 2 \, a b^{4} c d\right )} e + {\left (a b^{4} c^{2} + a^{2} b^{3} c d\right )} f\right )} x^{4} + 2 \, {\left (2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d\right )} e + {\left (a^{2} b^{3} c^{2} + a^{3} b^{2} c d\right )} f\right )} x^{2} + 2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d\right )} e + {\left (a^{3} b^{2} c^{2} + a^{4} b c d\right )} f\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left ({\left (2 \, b^{5} c^{2} - 3 \, a^{3} b^{2} d^{2} + {\left (a^{2} b^{3} - 4 \, a b^{4}\right )} c d\right )} e + {\left (a b^{4} c^{2} + {\left (2 \, a^{3} b^{2} + a^{2} b^{3}\right )} c d\right )} f\right )} x^{4} + 2 \, {\left ({\left (2 \, a b^{4} c^{2} - 3 \, a^{4} b d^{2} + {\left (a^{3} b^{2} - 4 \, a^{2} b^{3}\right )} c d\right )} e + {\left (a^{2} b^{3} c^{2} + {\left (2 \, a^{4} b + a^{3} b^{2}\right )} c d\right )} f\right )} x^{2} + {\left (2 \, a^{2} b^{3} c^{2} - 3 \, a^{5} d^{2} + {\left (a^{4} b - 4 \, a^{3} b^{2}\right )} c d\right )} e + {\left (a^{3} b^{2} c^{2} + {\left (2 \, a^{5} + a^{4} b\right )} c d\right )} f\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d\right )} e + {\left (a^{2} b^{3} c^{2} + a^{3} b^{2} c d\right )} f\right )} x^{3} + {\left (2 \, a^{4} b c d f + {\left (3 \, a^{2} b^{3} c^{2} - 5 \, a^{3} b^{2} c d\right )} e\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{5} b^{3} c^{3} - 2 \, a^{6} b^{2} c^{2} d + a^{7} b c d^{2} + {\left (a^{3} b^{5} c^{3} - 2 \, a^{4} b^{4} c^{2} d + a^{5} b^{3} c d^{2}\right )} x^{4} + 2 \, {\left (a^{4} b^{4} c^{3} - 2 \, a^{5} b^{3} c^{2} d + a^{6} b^{2} c d^{2}\right )} x^{2}\right )}} \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

Output:

-1/3*(((2*(b^5*c^2 - 2*a*b^4*c*d)*e + (a*b^4*c^2 + a^2*b^3*c*d)*f)*x^4 + 2 
*(2*(a*b^4*c^2 - 2*a^2*b^3*c*d)*e + (a^2*b^3*c^2 + a^3*b^2*c*d)*f)*x^2 + 2 
*(a^2*b^3*c^2 - 2*a^3*b^2*c*d)*e + (a^3*b^2*c^2 + a^4*b*c*d)*f)*sqrt(a*c)* 
sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (((2*b^5*c^2 - 3* 
a^3*b^2*d^2 + (a^2*b^3 - 4*a*b^4)*c*d)*e + (a*b^4*c^2 + (2*a^3*b^2 + a^2*b 
^3)*c*d)*f)*x^4 + 2*((2*a*b^4*c^2 - 3*a^4*b*d^2 + (a^3*b^2 - 4*a^2*b^3)*c* 
d)*e + (a^2*b^3*c^2 + (2*a^4*b + a^3*b^2)*c*d)*f)*x^2 + (2*a^2*b^3*c^2 - 3 
*a^5*d^2 + (a^4*b - 4*a^3*b^2)*c*d)*e + (a^3*b^2*c^2 + (2*a^5 + a^4*b)*c*d 
)*f)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (( 
2*(a*b^4*c^2 - 2*a^2*b^3*c*d)*e + (a^2*b^3*c^2 + a^3*b^2*c*d)*f)*x^3 + (2* 
a^4*b*c*d*f + (3*a^2*b^3*c^2 - 5*a^3*b^2*c*d)*e)*x)*sqrt(b*x^2 + a)*sqrt(d 
*x^2 + c))/(a^5*b^3*c^3 - 2*a^6*b^2*c^2*d + a^7*b*c*d^2 + (a^3*b^5*c^3 - 2 
*a^4*b^4*c^2*d + a^5*b^3*c*d^2)*x^4 + 2*(a^4*b^4*c^3 - 2*a^5*b^3*c^2*d + a 
^6*b^2*c*d^2)*x^2)
 

Sympy [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {e + f x^{2}}{\left (a + b x^{2}\right )^{\frac {5}{2}} \sqrt {c + d x^{2}}}\, dx \] Input:

integrate((f*x**2+e)/(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)
 

Output:

Integral((e + f*x**2)/((a + b*x**2)**(5/2)*sqrt(c + d*x**2)), x)
 

Maxima [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)), x)
 

Giac [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int { \frac {f x^{2} + e}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} \sqrt {d x^{2} + c}} \,d x } \] Input:

integrate((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x, algorithm="giac")
 

Output:

integrate((f*x^2 + e)/((b*x^2 + a)^(5/2)*sqrt(d*x^2 + c)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\int \frac {f\,x^2+e}{{\left (b\,x^2+a\right )}^{5/2}\,\sqrt {d\,x^2+c}} \,d x \] Input:

int((e + f*x^2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)),x)
 

Output:

int((e + f*x^2)/((a + b*x^2)^(5/2)*(c + d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {e+f x^2}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, x^{2}}{b^{3} d \,x^{8}+3 a \,b^{2} d \,x^{6}+b^{3} c \,x^{6}+3 a^{2} b d \,x^{4}+3 a \,b^{2} c \,x^{4}+a^{3} d \,x^{2}+3 a^{2} b c \,x^{2}+a^{3} c}d x \right ) f +\left (\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d \,x^{8}+3 a \,b^{2} d \,x^{6}+b^{3} c \,x^{6}+3 a^{2} b d \,x^{4}+3 a \,b^{2} c \,x^{4}+a^{3} d \,x^{2}+3 a^{2} b c \,x^{2}+a^{3} c}d x \right ) e \] Input:

int((f*x^2+e)/(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)
 

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a**3*c + a**3*d*x**2 + 3*a** 
2*b*c*x**2 + 3*a**2*b*d*x**4 + 3*a*b**2*c*x**4 + 3*a*b**2*d*x**6 + b**3*c* 
x**6 + b**3*d*x**8),x)*f + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3*c 
 + a**3*d*x**2 + 3*a**2*b*c*x**2 + 3*a**2*b*d*x**4 + 3*a*b**2*c*x**4 + 3*a 
*b**2*d*x**6 + b**3*c*x**6 + b**3*d*x**8),x)*e