\(\int \frac {(e+f x^2)^2}{(a+b x^2)^{3/2} \sqrt {c+d x^2}} \, dx\) [83]

Optimal result

Integrand size = 32, antiderivative size = 278 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {f^2 x \sqrt {c+d x^2}}{b d \sqrt {a+b x^2}}+\frac {\left (b^2 d e^2+2 a^2 d f^2-a b f (2 d e+c f)\right ) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{\sqrt {a} b^{3/2} d (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {\sqrt {a} \left (a c f^2+b e (d e-2 c f)\right ) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{b^{3/2} c (b c-a d) \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

f^2*x*(d*x^2+c)^(1/2)/b/d/(b*x^2+a)^(1/2)+(b^2*d*e^2+2*a^2*d*f^2-a*b*f*(c* 
f+2*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^(1/2)/b^(3/2)/d/(-a*d+b*c)/(b*x^2+a)^(1/2)/(a*(d*x^2+c) 
/c/(b*x^2+a))^(1/2)-a^(1/2)*(a*c*f^2+b*e*(-2*c*f+d*e))*(d*x^2+c)^(1/2)*Inv 
erseJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/b^(3/2)/c/(-a*d+ 
b*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 = 10.77 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.90 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {-\sqrt {\frac {b}{a}} d (b e-a f)^2 x \left (c+d x^2\right )-i c \left (b^2 d e^2+2 a^2 d f^2-a b f (2 d e+c f)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i (-b c+a d) \left (-b d e^2+a c f^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{a^2 \left (\frac {b}{a}\right )^{3/2} d (-b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2}} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(862\) vs. \(2(278)=556\).

Time = 1.04 (sec) , antiderivative size = 862, normalized size of antiderivative = 3.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {433, 2009}

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.

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

\(\Big \downarrow \) 433

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((d*e^2*x*Sqrt[a + b*x^2])/(a*(b*c - a*d)*Sqrt[c + d*x^2])) + (2*d*e*f*x* 
Sqrt[a + b*x^2])/(b*(b*c - a*d)*Sqrt[c + d*x^2]) + ((b*c - 2*a*d)*f^2*x*Sq 
rt[a + b*x^2])/(b^2*(b*c - a*d)*Sqrt[c + d*x^2]) + (b*e^2*x*Sqrt[c + d*x^2 
])/(a*(b*c - a*d)*Sqrt[a + b*x^2]) - (2*e*f*x*Sqrt[c + d*x^2])/((b*c - a*d 
)*Sqrt[a + b*x^2]) + (a*f^2*x*Sqrt[c + d*x^2])/(b*(b*c - a*d)*Sqrt[a + b*x 
^2]) + (Sqrt[c]*Sqrt[d]*e^2*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/S 
qrt[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]) - (2*Sqrt[c]*Sqrt[d]*e*f*Sqrt[a + b*x^2]*EllipticE[ 
ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*(b*c - a*d)*Sqrt[(c*(a + 
 b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (Sqrt[c]*(b*c - 2*a*d)*f^2*Sq 
rt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b^ 
2*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2 
]) - (Sqrt[c]*Sqrt[d]*e^2*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqr 
t[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]) + (2*c^(3/2)*e*f*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sq 
rt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + b 
*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*f^2*Sqrt[a + b*x^2]*El 
lipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[d]*(b*c - a 
*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 433

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 6.42 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.61

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {\left (b d \,x^{2}+b c \right ) x \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{b^{2} a \left (a d -b c \right ) \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {f \left (a f -2 b e \right )}{b^{2}}+\frac {a^{2} f^{2}-2 a b f e +b^{2} e^{2}}{b^{2} a}+\frac {c \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}{b a \left (a d -b c \right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {\left (\frac {f^{2}}{b}+\frac {\left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) d}{b \left (a d -b c \right ) a}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(448\)
default	\(\frac {\left (-\sqrt {-\frac {b}{a}}\, a^{2} d^{2} f^{2} x^{3}+2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} e f \,x^{3}-\sqrt {-\frac {b}{a}}\, b^{2} d^{2} e^{2} x^{3}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,d^{2} e^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} c d \,f^{2}-\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b \,c^{2} f^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d e f +\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c d \,e^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c d \,f^{2} x +2 \sqrt {-\frac {b}{a}}\, a b c d e f x -\sqrt {-\frac {b}{a}}\, b^{2} c d \,e^{2} x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{b a d \sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(650\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.80 \[ \int \frac {\left (e+f x^2\right )^2}{\left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \, dx=\frac {{\left ({\left (b^{3} c^{2} d e^{2} - 2 \, a b^{2} c^{2} d e f - {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d\right )} f^{2}\right )} x^{3} + {\left (a b^{2} c^{2} d e^{2} - 2 \, a^{2} b c^{2} d e f - {\left (a^{2} b c^{3} - 2 \, a^{3} c^{2} d\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left ({\left ({\left (b^{3} c^{2} d + a b^{2} d^{3}\right )} e^{2} - 2 \, {\left (a b^{2} c^{2} d + a b^{2} c d^{2}\right )} e f - {\left (a b^{2} c^{3} - 2 \, a^{2} b c^{2} d - a^{2} b c d^{2}\right )} f^{2}\right )} x^{3} + {\left ({\left (a b^{2} c^{2} d + a^{2} b d^{3}\right )} e^{2} - 2 \, {\left (a^{2} b c^{2} d + a^{2} b c d^{2}\right )} e f - {\left (a^{2} b c^{3} - 2 \, a^{3} c^{2} d - a^{3} c d^{2}\right )} f^{2}\right )} x\right )} \sqrt {b d} \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (a b^{2} c d^{2} e^{2} - 2 \, a^{2} b c d^{2} e f - {\left (a b^{2} c^{2} d - a^{2} b c d^{2}\right )} f^{2} x^{2} - {\left (a^{2} b c^{2} d - 2 \, a^{3} c d^{2}\right )} f^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{{\left (a b^{4} c^{2} d^{2} - a^{2} b^{3} c d^{3}\right )} x^{3} + {\left (a^{2} b^{3} c^{2} d^{2} - a^{3} b^{2} c d^{3}\right )} x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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