\(\int \frac {e+f x^2}{\sqrt {a+b x^2} (c-d x^2)^{3/2}} \, dx\) [45]

Optimal result

Integrand size = 31, antiderivative size = 237 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {(d e+c f) x \sqrt {a+b x^2}}{c (b c+a d) \sqrt {c-d x^2}}-\frac {(d e+c f) \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {e \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.24 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.90 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} d (d e+c f) x \left (a+b x^2\right )-i b c (d e+c f) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) f \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{\sqrt {\frac {b}{a}} c d (b c+a d) \sqrt {a+b x^2} \sqrt {c-d x^2}} \] 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]*EllipticF[I*A 
rcSinh[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 = 0.39 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {402, 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

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

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 
[c]*(d*e + c*f)*Sqrt[a + b*x^2]*Sqrt[1 - (d*x^2)/c]*EllipticE[ArcSin[(Sqrt 
[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[1 + (b*x^2)/a]*Sqrt[c - d* 
x^2])) + (Sqrt[c]*(b*c + a*d)*e*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*El 
lipticF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(Sqrt[d]*Sqrt[a + b* 
x^2]*Sqrt[c - d*x^2]))/(c*(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 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 = 7.91 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.41

method	result	size
default	\(\frac {\left (\sqrt {\frac {d}{c}}\, b c f \,x^{3}+\sqrt {\frac {d}{c}}\, b d e \,x^{3}+\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b c e -\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a c f -\sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) a d e +\sqrt {\frac {d}{c}}\, a c f x +\sqrt {\frac {d}{c}}\, a d e x \right ) \sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\sqrt {\frac {d}{c}}\, c \left (a d +b c \right ) \left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right )}\)	\(333\)
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {\left (-b d \,x^{2}-a d \right ) x \left (c f +d e \right )}{c d \left (a d +b c \right ) \sqrt {\left (x^{2}-\frac {c}{d}\right ) \left (-b d \,x^{2}-a d \right )}}+\frac {\left (-\frac {f}{d}+\frac {c f +d e}{d c}-\frac {a \left (c f +d e \right )}{c \left (a d +b c \right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}+\frac {\left (c f +d e \right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a d +b c \right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {-x^{2} d +c}\, \sqrt {b \,x^{2}+a}}\)	\(376\)

Input:

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

Output:

((1/c*d)^(1/2)*b*c*f*x^3+(1/c*d)^(1/2)*b*d*e*x^3+((-d*x^2+c)/c)^(1/2)*((b* 
x^2+a)/a)^(1/2)*EllipticF(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*d*e+((-d*x^2 
+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticF(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2 
))*b*c*e-((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*EllipticE(x*(1/c*d)^(1/2 
),(-b*c/a/d)^(1/2))*a*c*f-((-d*x^2+c)/c)^(1/2)*((b*x^2+a)/a)^(1/2)*Ellipti 
cE(x*(1/c*d)^(1/2),(-b*c/a/d)^(1/2))*a*d*e+(1/c*d)^(1/2)*a*c*f*x+(1/c*d)^( 
1/2)*a*d*e*x)*(-d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/(1/c*d)^(1/2)/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 = 0.12 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.07 \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\frac {{\left (a c d^{2} e + a c^{2} d f\right )} \sqrt {b x^{2} + a} \sqrt {-d x^{2} + c} x - {\left (a c d^{2} e + a c^{2} d f - {\left (a d^{3} e + a c d^{2} f\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {d}{c}} E(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d}) - {\left ({\left ({\left (b c^{2} d + a d^{3}\right )} e - {\left (a c^{2} d - a c d^{2}\right )} f\right )} x^{2} - {\left (b c^{3} + a c d^{2}\right )} e + {\left (a c^{3} - a c^{2} d\right )} f\right )} \sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d})}{a b c^{4} d + a^{2} c^{3} d^{2} - {\left (a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} x^{2}} \] Input:

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

Output:

((a*c*d^2*e + a*c^2*d*f)*sqrt(b*x^2 + a)*sqrt(-d*x^2 + c)*x - (a*c*d^2*e + 
 a*c^2*d*f - (a*d^3*e + a*c*d^2*f)*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic_e(arc 
sin(x*sqrt(d/c)), -b*c/(a*d)) - (((b*c^2*d + a*d^3)*e - (a*c^2*d - a*c*d^2 
)*f)*x^2 - (b*c^3 + a*c*d^2)*e + (a*c^3 - a*c^2*d)*f)*sqrt(a*c)*sqrt(d/c)* 
elliptic_f(arcsin(x*sqrt(d/c)), -b*c/(a*d)))/(a*b*c^4*d + a^2*c^3*d^2 - (a 
*b*c^3*d^2 + a^2*c^2*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. \[ \int \frac {e+f x^2}{\sqrt {a+b x^2} \left (c-d x^2\right )^{3/2}} \, dx=\int \frac {f\,x^2+e}{\sqrt {b\,x^2+a}\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \] 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