\(\int \frac {e+f x^2}{\sqrt {a-b x^2} (c+d x^2)^{3/2}} \, dx\) [44]

Optimal result

Integrand size = 31, antiderivative size = 207 \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {(d e-c f) \sqrt {a-b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1+\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) \sqrt {\frac {c \left (a-b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {\sqrt {c} (b e+a f) \sqrt {a-b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1+\frac {b c}{a d}\right )}{a \sqrt {d} (b c+a d) \sqrt {\frac {c \left (a-b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

(-c*f+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^(1/2)/d^(1/2)/(a*d+b*c)/(c*(-b*x^2+a)/a/(d*x^2+c))^(1 
/2)/(d*x^2+c)^(1/2)+c^(1/2)*(a*f+b*e)*(-b*x^2+a)^(1/2)*InverseJacobiAM(arc 
tan(d^(1/2)*x/c^(1/2)),(1+b*c/a/d)^(1/2))/a/d^(1/2)/(a*d+b*c)/(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 = 8.40 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.06 \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {b}{a}} d (d e-c f) x \left (a-b x^2\right )+i b c (-d e+c f) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i c (b c+a d) f \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} c d (b c+a d) \sqrt {a-b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

(Sqrt[-(b/a)]*d*(d*e - c*f)*x*(a - b*x^2) + I*b*c*(-(d*e) + c*f)*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)*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*Elli 
pticF[I*ArcSinh[Sqrt[-(b/a)]*x], -((a*d)/(b*c))])/(Sqrt[-(b/a)]*c*d*(b*c + 
 a*d)*Sqrt[a - b*x^2]*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.24, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {402, 25, 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.

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

((d*e - c*f)*x*Sqrt[a - b*x^2])/(c*(b*c + a*d)*Sqrt[c + d*x^2]) + ((Sqrt[a 
]*Sqrt[b]*(d*e - c*f)*Sqrt[1 - (b*x^2)/a]*Sqrt[c + d*x^2]*EllipticE[ArcSin 
[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(d*Sqrt[a - b*x^2]*Sqrt[1 + (d*x^2 
)/c]) + (Sqrt[a]*c*(b*c + a*d)*f*Sqrt[1 - (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*E 
llipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]], -((a*d)/(b*c))])/(Sqrt[b]*d*Sqrt[a - 
 b*x^2]*Sqrt[c + d*x^2]))/(c*(b*c + a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 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 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 = 7.95 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.69

method	result	size
default	\(\frac {\left (\sqrt {\frac {b}{a}}\, b c d f \,x^{3}-\sqrt {\frac {b}{a}}\, b \,d^{2} e \,x^{3}+\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) a c d f +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} f -\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b \,c^{2} f +\sqrt {\frac {-b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-\frac {a d}{b c}}\right ) b c d e -\sqrt {\frac {b}{a}}\, a c d f x +\sqrt {\frac {b}{a}}\, a \,d^{2} e x \right ) \sqrt {x^{2} d +c}\, \sqrt {-b \,x^{2}+a}}{\sqrt {\frac {b}{a}}\, d c \left (a d +b c \right ) \left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right )}\)	\(349\)
elliptic	\(\frac {\sqrt {\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (-b d \,x^{2}+a d \right ) x \left (c f -d e \right )}{c d \left (a d +b c \right ) \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (-b d \,x^{2}+a d \right )}}+\frac {\left (\frac {f}{d}-\frac {c f -d e}{d c}+\frac {a \left (c f -d e \right )}{c \left (a d +b c \right )}\right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )}{\sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}}+\frac {b \left (c f -d e \right ) \sqrt {1-\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {b}{a}}, \sqrt {-1-\frac {a d -b c}{c b}}\right )\right )}{\left (a d +b c \right ) \sqrt {\frac {b}{a}}\, \sqrt {-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c}\, d}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(376\)

Input:

int((f*x^2+e)/(-b*x^2+a)^(1/2)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

((b/a)^(1/2)*b*c*d*f*x^3-(b/a)^(1/2)*b*d^2*e*x^3+((-b*x^2+a)/a)^(1/2)*((d* 
x^2+c)/c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*a*c*d*f+((-b*x^2 
+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(x*(b/a)^(1/2),(-a*d/b/c)^(1/2)) 
*b*c^2*f-((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticE(x*(b/a)^(1/2), 
(-a*d/b/c)^(1/2))*b*c^2*f+((-b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipti 
cE(x*(b/a)^(1/2),(-a*d/b/c)^(1/2))*b*c*d*e-(b/a)^(1/2)*a*c*d*f*x+(b/a)^(1/ 
2)*a*d^2*e*x)*(d*x^2+c)^(1/2)*(-b*x^2+a)^(1/2)/(b/a)^(1/2)/d/c/(a*d+b*c)/( 
-b*d*x^4+a*d*x^2-b*c*x^2+a*c)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.24 \[ \int \frac {e+f x^2}{\sqrt {a-b x^2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {{\left (a b d^{2} e - a b c d f\right )} \sqrt {-b x^{2} + a} \sqrt {d x^{2} + c} x + {\left (b^{2} c d e - b^{2} c^{2} f + {\left (b^{2} d^{2} e - b^{2} c d f\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {b}{a}} E(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c}) + {\left ({\left (a b - b^{2}\right )} c d e + {\left ({\left (a b - b^{2}\right )} d^{2} e + {\left (b^{2} c d + a^{2} d^{2}\right )} f\right )} x^{2} + {\left (b^{2} c^{2} + a^{2} c d\right )} f\right )} \sqrt {a c} \sqrt {\frac {b}{a}} F(\arcsin \left (x \sqrt {\frac {b}{a}}\right )\,|\,-\frac {a d}{b c})}{a b^{2} c^{3} d + a^{2} b c^{2} d^{2} + {\left (a b^{2} c^{2} d^{2} + a^{2} b c d^{3}\right )} x^{2}} \] Input:

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

Output:

((a*b*d^2*e - a*b*c*d*f)*sqrt(-b*x^2 + a)*sqrt(d*x^2 + c)*x + (b^2*c*d*e - 
 b^2*c^2*f + (b^2*d^2*e - b^2*c*d*f)*x^2)*sqrt(a*c)*sqrt(b/a)*elliptic_e(a 
rcsin(x*sqrt(b/a)), -a*d/(b*c)) + ((a*b - b^2)*c*d*e + ((a*b - b^2)*d^2*e 
+ (b^2*c*d + a^2*d^2)*f)*x^2 + (b^2*c^2 + a^2*c*d)*f)*sqrt(a*c)*sqrt(b/a)* 
elliptic_f(arcsin(x*sqrt(b/a)), -a*d/(b*c)))/(a*b^2*c^3*d + a^2*b*c^2*d^2 
+ (a*b^2*c^2*d^2 + a^2*b*c*d^3)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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