\(\int \frac {a c+2 b c x^2+b d x^4}{(a+b x^2)^{3/2} (c+d x^2)^{3/2}} \, dx\) [22]

Optimal result

Integrand size = 40, antiderivative size = 220 \[ \int \frac {a c+2 b c x^2+b d x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {x}{\sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{(b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {2 b c^{3/2} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

-x/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)-2*c^(1/2)*d^(1/2)*(b*x^2+a)^(1/2)*Ellip 
ticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/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)+2*b*c^(3/2)*(b*x^2+a)^(1/2)*I 
nverseJacobiAM(arctan(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 [A] (verified)

Time = 10.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.66 \[ \int \frac {a c+2 b c x^2+b d x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {-\frac {b}{a}} x \sqrt {1+\frac {d x^2}{c}} \left (a d+b \left (c+2 d x^2\right )\right )-2 b \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) E\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right )|\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} (-b c+a d) \sqrt {a+b x^2} \sqrt {c+d x^2} \sqrt {1+\frac {d x^2}{c}}} \] Input:

Integrate[(a*c + 2*b*c*x^2 + b*d*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2) 
),x]
 

Output:

(Sqrt[-(b/a)]*x*Sqrt[1 + (d*x^2)/c]*(a*d + b*(c + 2*d*x^2)) - 2*b*Sqrt[1 + 
 (b*x^2)/a]*(c + d*x^2)*EllipticE[ArcSin[Sqrt[-(b/a)]*x], (a*d)/(b*c)])/(S 
qrt[-(b/a)]*(-(b*c) + a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*Sqrt[1 + (d*x^2 
)/c])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(474\) vs. \(2(220)=440\).

Time = 1.08 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[(a*c + 2*b*c*x^2 + b*d*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x]
 

Output:

-((b*c*x)/((b*c - a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) + (a*d*x)/((b*c - 
 a*d)*Sqrt[a + b*x^2]*Sqrt[c + d*x^2]) - (4*b*c^(3/2)*Sqrt[d]*Sqrt[a + b*x 
^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/((b*c - a*d)^ 
2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (2*Sqrt[c]*Sqrt 
[d]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - 
 (b*c)/(a*d)])/((b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c 
 + d*x^2]) - (4*b*c^(3/2)*Sqrt[d]*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d 
]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/((b*c - a*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c 
 + d*x^2))]*Sqrt[c + d*x^2]) + (2*b*c^(3/2)*(b*c + a*d)*Sqrt[a + b*x^2]*El 
lipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*(b*c - a 
*d)^2*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])
 

Defintions of rubi rules used

rule 2009

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

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 8.12 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.70

method	result	size
default	\(\frac {\left (2 \sqrt {-\frac {b}{a}}\, b d \,x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {x^{2} d +c}{c}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b c +\sqrt {-\frac {b}{a}}\, a d x +\sqrt {-\frac {b}{a}}\, b c x \right ) \sqrt {x^{2} d +c}\, \sqrt {b \,x^{2}+a}}{\sqrt {-\frac {b}{a}}\, \left (a d -b c \right ) \left (b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c \right )}\)	\(154\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {2 b d \left (-\frac {x^{3}}{a d -b c}-\frac {\left (a d +b c \right ) x}{2 \left (a d -b c \right ) b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{d b}+\frac {a c}{d b}\right ) b d}}+\frac {\left (1-\frac {a d +b c}{a d -b c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}+\frac {2 b c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\left (a d -b c \right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(357\)

Input:

int((b*d*x^4+2*b*c*x^2+a*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x,method=_RETU 
RNVERBOSE)
 

Output:

(2*(-b/a)^(1/2)*b*d*x^3-2*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Elliptic 
E(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c+(-b/a)^(1/2)*a*d*x+(-b/a)^(1/2)*b*c* 
x)*(d*x^2+c)^(1/2)*(b*x^2+a)^(1/2)/(-b/a)^(1/2)/(a*d-b*c)/(b*d*x^4+a*d*x^2 
+b*c*x^2+a*c)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.12 \[ \int \frac {a c+2 b c x^2+b d x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - 2 \, {\left ({\left (a b + b^{2}\right )} d x^{4} + {\left ({\left (a b + b^{2}\right )} c + {\left (a^{2} + a b\right )} d\right )} x^{2} + {\left (a^{2} + a b\right )} c\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, a b d x^{3} + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{a^{2} b c^{2} - a^{3} c d + {\left (a b^{2} c d - a^{2} b d^{2}\right )} x^{4} + {\left (a b^{2} c^{2} - a^{3} d^{2}\right )} x^{2}} \] Input:

integrate((b*d*x^4+2*b*c*x^2+a*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algor 
ithm="fricas")
 

Output:

(2*(b^2*d*x^4 + a*b*c + (b^2*c + a*b*d)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic 
_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - 2*((a*b + b^2)*d*x^4 + ((a*b + b^2)* 
c + (a^2 + a*b)*d)*x^2 + (a^2 + a*b)*c)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(ar 
csin(x*sqrt(-b/a)), a*d/(b*c)) - (2*a*b*d*x^3 + (a*b*c + a^2*d)*x)*sqrt(b* 
x^2 + a)*sqrt(d*x^2 + c))/(a^2*b*c^2 - a^3*c*d + (a*b^2*c*d - a^2*b*d^2)*x 
^4 + (a*b^2*c^2 - a^3*d^2)*x^2)
 

Sympy [F]

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

integrate((b*d*x**4+2*b*c*x**2+a*c)/(b*x**2+a)**(3/2)/(d*x**2+c)**(3/2),x)
 

Output:

Integral((a*c + 2*b*c*x**2 + b*d*x**4)/((a + b*x**2)**(3/2)*(c + d*x**2)** 
(3/2)), x)
 

Maxima [F]

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

integrate((b*d*x^4+2*b*c*x^2+a*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algor 
ithm="maxima")
 

Output:

integrate((b*d*x^4 + 2*b*c*x^2 + a*c)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2) 
), x)
 

Giac [F]

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

integrate((b*d*x^4+2*b*c*x^2+a*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x, algor 
ithm="giac")
 

Output:

integrate((b*d*x^4 + 2*b*c*x^2 + a*c)/((b*x^2 + a)^(3/2)*(d*x^2 + c)^(3/2) 
), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a c+2 b c x^2+b d x^4}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {b\,d\,x^4+2\,b\,c\,x^2+a\,c}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \] Input:

int((a*c + 2*b*c*x^2 + b*d*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)),x)
 

Output:

int((a*c + 2*b*c*x^2 + b*d*x^4)/((a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)), x)
 

Reduce [F]

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

int((b*d*x^4+2*b*c*x^2+a*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2),x)
 

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*x + 2*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)*a*c**2 + 2*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)*a*c*d*x**2 + 2*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)*b* 
c**2*x**2 + 2*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)*b*c*d*x**4 
)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4)