\(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^{5/2}} \, dx\) [167]

Optimal result

Integrand size = 23, antiderivative size = 408 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}}+\frac {2 b (b c-4 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}}+\frac {d \left (2 b^2 c^2-9 a b c d-a^2 d^2\right ) x \sqrt {a+b x^2}}{3 a^2 c (b c-a d)^3 \left (c+d x^2\right )^{3/2}}+\frac {2 \sqrt {d} (b c+a d) \left (b^2 c^2-6 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 c^{3/2} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {d} \left (b^2 c^2-18 a b c d+a^2 d^2\right ) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^4 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \] Output:

1/3*b*x/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(3/2)+2/3*b*(-4*a*d+b*c)*x/ 
a^2/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(d*x^2+c)^(3/2)+1/3*d*(-a^2*d^2-9*a*b*c*d 
+2*b^2*c^2)*x*(b*x^2+a)^(1/2)/a^2/c/(-a*d+b*c)^3/(d*x^2+c)^(3/2)+2/3*d^(1/ 
2)*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2*c^2)*(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))/a^2/c^(3/2)/(-a*d+b*c)^4/( 
c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*b*d^(1/2)*(a^2*d^2-18*a 
*b*c*d+b^2*c^2)*(b*x^2+a)^(1/2)*InverseJacobiAM(arctan(d^(1/2)*x/c^(1/2)), 
(1-b*c/a/d)^(1/2))/a^2/c^(1/2)/(-a*d+b*c)^4/(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 = 5.66 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.87 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\frac {-\sqrt {\frac {b}{a}} x \left (a^2 c d^3 (b c-a d) \left (a+b x^2\right )^2-2 a^2 d^3 (-5 b c+a d) \left (a+b x^2\right )^2 \left (c+d x^2\right )+a b^3 c^2 (-b c+a d) \left (c+d x^2\right )^2-2 b^3 c^2 (b c-5 a d) \left (a+b x^2\right ) \left (c+d x^2\right )^2\right )+i b c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}} \left (2 \left (b^3 c^3-5 a b^2 c^2 d-5 a^2 b c d^2+a^3 d^3\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-\left (2 b^3 c^3-11 a b^2 c^2 d+8 a^2 b c d^2+a^3 d^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c-a d)^4 \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \] Input:

Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)),x]
 

Output:

(-(Sqrt[b/a]*x*(a^2*c*d^3*(b*c - a*d)*(a + b*x^2)^2 - 2*a^2*d^3*(-5*b*c + 
a*d)*(a + b*x^2)^2*(c + d*x^2) + a*b^3*c^2*(-(b*c) + a*d)*(c + d*x^2)^2 - 
2*b^3*c^2*(b*c - 5*a*d)*(a + b*x^2)*(c + d*x^2)^2)) + I*b*c*(a + b*x^2)*Sq 
rt[1 + (b*x^2)/a]*(c + d*x^2)*Sqrt[1 + (d*x^2)/c]*(2*(b^3*c^3 - 5*a*b^2*c^ 
2*d - 5*a^2*b*c*d^2 + a^3*d^3)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b* 
c)] - (2*b^3*c^3 - 11*a*b^2*c^2*d + 8*a^2*b*c*d^2 + a^3*d^3)*EllipticF[I*A 
rcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(3*a^2*Sqrt[b/a]*c^2*(b*c - a*d)^4*(a 
+ b*x^2)^(3/2)*(c + d*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {316, 25, 402, 27, 402, 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.

\(\displaystyle \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 316

\(\displaystyle \frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {5 b d x^2+2 b c-3 a d}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{3 a (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 b d x^2+2 b c-3 a d}{\left (b x^2+a\right )^{3/2} \left (d x^2+c\right )^{5/2}}dx}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {2 b x (b c-4 a d)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}-\frac {\int -\frac {3 d \left (2 b (b c-4 a d) x^2+a (b c+a d)\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{a (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 d \int \frac {2 b (b c-4 a d) x^2+a (b c+a d)}{\sqrt {b x^2+a} \left (d x^2+c\right )^{5/2}}dx}{a (b c-a d)}+\frac {2 b x (b c-4 a d)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {3 d \left (\frac {\int \frac {b \left (2 b^2 c^2-9 a b d c-a^2 d^2\right ) x^2+a \left (b^2 c^2+9 a b d c-2 a^2 d^2\right )}{\sqrt {b x^2+a} \left (d x^2+c\right )^{3/2}}dx}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (-a^2 d^2-9 a b c d+2 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {2 b x (b c-4 a d)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 400

\(\displaystyle \frac {\frac {3 d \left (\frac {\frac {2 (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) \int \frac {\sqrt {b x^2+a}}{\left (d x^2+c\right )^{3/2}}dx}{b c-a d}-\frac {a b \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (-a^2 d^2-9 a b c d+2 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {2 b x (b c-4 a d)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 313

\(\displaystyle \frac {\frac {3 d \left (\frac {\frac {2 \sqrt {a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {a b \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {b x^2+a} \sqrt {d x^2+c}}dx}{b c-a d}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (-a^2 d^2-9 a b c d+2 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {2 b x (b c-4 a d)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

\(\Big \downarrow \) 320

\(\displaystyle \frac {\frac {3 d \left (\frac {\frac {2 \sqrt {a+b x^2} (a d+b c) \left (a^2 d^2-6 a b c d+b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c} \sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {a+b x^2} \left (a^2 d^2-18 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{\sqrt {d} \sqrt {c+d x^2} (b c-a d) \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}}{3 c (b c-a d)}+\frac {x \sqrt {a+b x^2} \left (-a^2 d^2-9 a b c d+2 b^2 c^2\right )}{3 c \left (c+d x^2\right )^{3/2} (b c-a d)}\right )}{a (b c-a d)}+\frac {2 b x (b c-4 a d)}{a \sqrt {a+b x^2} \left (c+d x^2\right )^{3/2} (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2} (b c-a d)}\)

Input:

Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(5/2)),x]
 

Output:

(b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*(c + d*x^2)^(3/2)) + ((2*b*(b*c - 
 4*a*d)*x)/(a*(b*c - a*d)*Sqrt[a + b*x^2]*(c + d*x^2)^(3/2)) + (3*d*(((2*b 
^2*c^2 - 9*a*b*c*d - a^2*d^2)*x*Sqrt[a + b*x^2])/(3*c*(b*c - a*d)*(c + d*x 
^2)^(3/2)) + ((2*(b*c + a*d)*(b^2*c^2 - 6*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^ 
2]*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[c]*Sqrt[ 
d]*(b*c - a*d)*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (b 
*Sqrt[c]*(b^2*c^2 - 18*a*b*c*d + a^2*d^2)*Sqrt[a + b*x^2]*EllipticF[ArcTan 
[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(Sqrt[d]*(b*c - a*d)*Sqrt[(c*(a + 
 b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]))/(3*c*(b*c - a*d))))/(a*(b*c - 
a*d)))/(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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 316

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) 
), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d))   Int[(a + b*x^2)^(p + 1)*(c + d*x 
^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x 
], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  ! 
( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, 
 p, q, x]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(860\) vs. \(2(373)=746\).

Time = 19.54 (sec) , antiderivative size = 861, normalized size of antiderivative = 2.11

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (\frac {\left (\frac {\left (a d +b c \right ) x^{3}}{3 d b a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{3 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} d^{2}}\right ) \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{d b}+\frac {a c}{d b}\right )^{2}}-\frac {2 b d \left (-\frac {\left (a d +b c \right ) \left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right ) x^{3}}{3 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2}}-\frac {\left (2 d^{4} a^{4}-9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}-9 a \,b^{3} c^{3} d +2 c^{4} b^{4}\right ) x}{6 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2} 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 (\frac {\frac {2}{3} a^{2} d^{2}-2 a b c d +\frac {2}{3} b^{2} c^{2}}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2} c^{2}}-\frac {2 d^{4} a^{4}-9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}-9 a \,b^{3} c^{3} d +2 c^{4} b^{4}}{3 a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\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 (\frac {4 b d \left (a^{3} d^{3}-5 a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d +b^{3} c^{3}\right )}{3 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{2} a^{2} c^{2}}-\frac {2 b d \left (a d +b c \right ) \left (a^{2} d^{2}-6 a b c d +b^{2} c^{2}\right )}{a^{2} c^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )^{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 )}{\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}}\)	\(861\)
default	\(\text {Expression too large to display}\)	\(2057\)

Input:

int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(5/2),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/d/b*(a*d 
+b*c)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^3+1/3/a/c*(a^2*d^2+b^2*c^2)/(a^2*d 
^2-2*a*b*c*d+b^2*c^2)/b^2/d^2*x)*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^4+ 
(a*d+b*c)/d/b*x^2+a*c/d/b)^2-2*b*d*(-1/3*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2* 
c^2)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^2*x^3-1/6*(2*a^4*d^4-9*a^3*b*c*d^ 
3-2*a^2*b^2*c^2*d^2-9*a*b^3*c^3*d+2*b^4*c^4)/a^2/c^2/(a^2*d^2-2*a*b*c*d+b^ 
2*c^2)^2/b/d*x)/((x^4+(a*d+b*c)/d/b*x^2+a*c/d/b)*b*d)^(1/2)+(2/3*(a^2*d^2- 
3*a*b*c*d+b^2*c^2)/(a^2*d^2-2*a*b*c*d+b^2*c^2)/a^2/c^2-1/3*(2*a^4*d^4-9*a^ 
3*b*c*d^3-2*a^2*b^2*c^2*d^2-9*a*b^3*c^3*d+2*b^4*c^4)/a^2/c^2/(a^2*d^2-2*a* 
b*c*d+b^2*c^2)^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))-(4/3*b*d*(a^3*d^3-5*a^2*b*c*d^2-5*a*b^2*c^2*d+b^3*c^3)/(a^2*d^2-2*a* 
b*c*d+b^2*c^2)^2/a^2/c^2-2*b*d*(a*d+b*c)*(a^2*d^2-6*a*b*c*d+b^2*c^2)/a^2/c 
^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)^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)/d*(EllipticF(x*(-b/a)^(1/2) 
,(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(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. 1456 vs. \(2 (373) = 746\).

Time = 0.17 (sec) , antiderivative size = 1456, normalized size of antiderivative = 3.57 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*(2*(a^2*b^4*c^5 - 5*a^3*b^3*c^4*d - 5*a^4*b^2*c^3*d^2 + a^5*b*c^2*d^3 
 + (b^6*c^3*d^2 - 5*a*b^5*c^2*d^3 - 5*a^2*b^4*c*d^4 + a^3*b^3*d^5)*x^8 + 2 
*(b^6*c^4*d - 4*a*b^5*c^3*d^2 - 10*a^2*b^4*c^2*d^3 - 4*a^3*b^3*c*d^4 + a^4 
*b^2*d^5)*x^6 + (b^6*c^5 - a*b^5*c^4*d - 24*a^2*b^4*c^3*d^2 - 24*a^3*b^3*c 
^2*d^3 - a^4*b^2*c*d^4 + a^5*b*d^5)*x^4 + 2*(a*b^5*c^5 - 4*a^2*b^4*c^4*d - 
 10*a^3*b^3*c^3*d^2 - 4*a^4*b^2*c^2*d^3 + a^5*b*c*d^4)*x^2)*sqrt(a*c)*sqrt 
(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*a^2*b^4*c^5 + (2*b 
^6*c^3*d^2 + (a^2*b^4 - 10*a*b^5)*c^2*d^3 - 2*(9*a^3*b^3 + 5*a^2*b^4)*c*d^ 
4 + (a^4*b^2 + 2*a^3*b^3)*d^5)*x^8 + 2*(2*b^6*c^4*d + (a^2*b^4 - 8*a*b^5)* 
c^3*d^2 - (17*a^3*b^3 + 20*a^2*b^4)*c^2*d^3 - (17*a^4*b^2 + 8*a^3*b^3)*c*d 
^4 + (a^5*b + 2*a^4*b^2)*d^5)*x^6 + (a^4*b^2 - 10*a^3*b^3)*c^4*d - 2*(9*a^ 
5*b + 5*a^4*b^2)*c^3*d^2 + (a^6 + 2*a^5*b)*c^2*d^3 + (2*b^6*c^5 + (a^2*b^4 
 - 2*a*b^5)*c^4*d - 2*(7*a^3*b^3 + 24*a^2*b^4)*c^3*d^2 - 2*(35*a^4*b^2 + 2 
4*a^3*b^3)*c^2*d^3 - 2*(7*a^5*b + a^4*b^2)*c*d^4 + (a^6 + 2*a^5*b)*d^5)*x^ 
4 + 2*(2*a*b^5*c^5 + (a^3*b^3 - 8*a^2*b^4)*c^4*d - (17*a^4*b^2 + 20*a^3*b^ 
3)*c^3*d^2 - (17*a^5*b + 8*a^4*b^2)*c^2*d^3 + (a^6 + 2*a^5*b)*c*d^4)*x^2)* 
sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*(a*b 
^5*c^3*d^2 - 5*a^2*b^4*c^2*d^3 - 5*a^3*b^3*c*d^4 + a^4*b^2*d^5)*x^7 + (4*a 
*b^5*c^4*d - 17*a^2*b^4*c^3*d^2 - 22*a^3*b^3*c^2*d^3 - 17*a^4*b^2*c*d^4 + 
4*a^5*b*d^5)*x^5 + 2*(a*b^5*c^5 - 2*a^2*b^4*c^4*d - 11*a^3*b^3*c^3*d^2 ...
 

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{3} d^{3} x^{12}+3 a \,b^{2} d^{3} x^{10}+3 b^{3} c \,d^{2} x^{10}+3 a^{2} b \,d^{3} x^{8}+9 a \,b^{2} c \,d^{2} x^{8}+3 b^{3} c^{2} d \,x^{8}+a^{3} d^{3} x^{6}+9 a^{2} b c \,d^{2} x^{6}+9 a \,b^{2} c^{2} d \,x^{6}+b^{3} c^{3} x^{6}+3 a^{3} c \,d^{2} x^{4}+9 a^{2} b \,c^{2} d \,x^{4}+3 a \,b^{2} c^{3} x^{4}+3 a^{3} c^{2} d \,x^{2}+3 a^{2} b \,c^{3} x^{2}+a^{3} c^{3}}d x \] Input:

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

Output:

int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**3*c**3 + 3*a**3*c**2*d*x**2 + 
3*a**3*c*d**2*x**4 + a**3*d**3*x**6 + 3*a**2*b*c**3*x**2 + 9*a**2*b*c**2*d 
*x**4 + 9*a**2*b*c*d**2*x**6 + 3*a**2*b*d**3*x**8 + 3*a*b**2*c**3*x**4 + 9 
*a*b**2*c**2*d*x**6 + 9*a*b**2*c*d**2*x**8 + 3*a*b**2*d**3*x**10 + b**3*c* 
*3*x**6 + 3*b**3*c**2*d*x**8 + 3*b**3*c*d**2*x**10 + b**3*d**3*x**12),x)