\(\int \frac {1}{(a+b x^2)^{5/2} (c+d x^2)^{3/2}} \, dx\) [213]

Optimal result

Integrand size = 23, antiderivative size = 323 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {d} (b c-9 a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

[Out]

1/3*b*x/a/(-a*d+b*c)/(b*x^2+a)^(3/2)/(d*x^2+c)^(1/2)+2/3*b*(-3*a*d+b*c)*x/a^2/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(d*
x^2+c)^(1/2)+1/3*(-3*a^2*d^2-7*a*b*c*d+2*b^2*c^2)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticE(x*d^(1/2)/
c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*d^(1/2)*(b*x^2+a)^(1/2)/a^2/(-a*d+b*c)^3/c^(1/2)/(c*(b*x^2+a)/a/(
d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*b*(-9*a*d+b*c)*(1/(1+d*x^2/c))^(1/2)*(1+d*x^2/c)^(1/2)*EllipticF(x*d^(1/2)
/c^(1/2)/(1+d*x^2/c)^(1/2),(1-b*c/a/d)^(1/2))*c^(1/2)*d^(1/2)*(b*x^2+a)^(1/2)/a^2/(-a*d+b*c)^3/(c*(b*x^2+a)/a/
(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {425, 541, 539, 429, 422} \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {d} \sqrt {a+b x^2} \left (-3 a^2 d^2-7 a b c d+2 b^2 c^2\right ) E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac {b \sqrt {c} \sqrt {d} \sqrt {a+b x^2} (b c-9 a d) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c+d x^2} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac {2 b x (b c-3 a d)}{3 a^2 \sqrt {a+b x^2} \sqrt {c+d x^2} (b c-a d)^2}+\frac {b x}{3 a \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2} (b c-a d)} \]

[In]

Int[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]

[Out]

(b*x)/(3*a*(b*c - a*d)*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2]) + (2*b*(b*c - 3*a*d)*x)/(3*a^2*(b*c - a*d)^2*Sqrt[a
+ b*x^2]*Sqrt[c + d*x^2]) + (Sqrt[d]*(2*b^2*c^2 - 7*a*b*c*d - 3*a^2*d^2)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqr
t[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*a^2*Sqrt[c]*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c
+ d*x^2]) - (b*Sqrt[c]*Sqrt[d]*(b*c - 9*a*d)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/
(a*d)])/(3*a^2*(b*c - a*d)^3*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[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 425

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

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 539

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)^(3/2)), x_Symbol] :> Dist[(b*e - a*
f)/(b*c - a*d), Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] - Dist[(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 541

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(
-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Dist[1/(a*n*(b*c - a
*d)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*
f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}-\frac {\int \frac {-2 b c+3 a d-3 b d x^2}{\left (a+b x^2\right )^{3/2} \left (c+d x^2\right )^{3/2}} \, dx}{3 a (b c-a d)} \\ & = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\int \frac {a d (b c+3 a d)+2 b d (b c-3 a d) x^2}{\sqrt {a+b x^2} \left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 (b c-a d)^2} \\ & = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}-\frac {(b d (b c-9 a d)) \int \frac {1}{\sqrt {a+b x^2} \sqrt {c+d x^2}} \, dx}{3 a (b c-a d)^3}+\frac {\left (d \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x^2}}{\left (c+d x^2\right )^{3/2}} \, dx}{3 a^2 (b c-a d)^3} \\ & = \frac {b x}{3 a (b c-a d) \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}}+\frac {2 b (b c-3 a d) x}{3 a^2 (b c-a d)^2 \sqrt {a+b x^2} \sqrt {c+d x^2}}+\frac {\sqrt {d} \left (2 b^2 c^2-7 a b c d-3 a^2 d^2\right ) \sqrt {a+b x^2} E\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 \sqrt {c} (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {c} \sqrt {d} (b c-9 a d) \sqrt {a+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a^2 (b c-a d)^3 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.40 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (3 a^4 d^3+6 a^3 b d^3 x^2-2 b^4 c^2 x^2 \left (c+d x^2\right )+a^2 b^2 d \left (8 c^2+8 c d x^2+3 d^2 x^4\right )+a b^3 c \left (-3 c^2+4 c d x^2+7 d^2 x^4\right )\right )+i b c \left (-2 b^2 c^2+7 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+2 i b c \left (b^2 c^2-4 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} c (-b c+a d)^3 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \]

[In]

Integrate[1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x]

[Out]

(Sqrt[b/a]*x*(3*a^4*d^3 + 6*a^3*b*d^3*x^2 - 2*b^4*c^2*x^2*(c + d*x^2) + a^2*b^2*d*(8*c^2 + 8*c*d*x^2 + 3*d^2*x
^4) + a*b^3*c*(-3*c^2 + 4*c*d*x^2 + 7*d^2*x^4)) + I*b*c*(-2*b^2*c^2 + 7*a*b*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[
1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] + (2*I)*b*c*(b^2*c^2 - 4*a*b
*c*d + 3*a^2*d^2)*(a + b*x^2)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/
(b*c)])/(3*a^2*Sqrt[b/a]*c*(-(b*c) + a*d)^3*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])

Maple [A] (verified)

Time = 6.71 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.74

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 a \left (a d -b c \right )^{2} \left (x^{2}+\frac {a}{b}\right )^{2}}+\frac {\left (b d \,x^{2}+b c \right ) b x \left (7 a d -2 b c \right )}{3 a^{2} \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (b d \,x^{2}+a d \right ) d^{2} x}{c \left (a d -b c \right )^{3} \sqrt {\left (x^{2}+\frac {c}{d}\right ) \left (b d \,x^{2}+a d \right )}}+\frac {\left (\frac {b d}{3 \left (a d -b c \right )^{2} a}-\frac {b \left (7 a d -2 b c \right )}{3 \left (a d -b c \right )^{2} a^{2}}-\frac {b^{2} c \left (7 a d -2 b c \right )}{3 a^{2} \left (a d -b c \right )^{3}}+\frac {d^{2}}{\left (a d -b c \right )^{2} c}-\frac {a \,d^{3}}{c \left (a d -b c \right )^{3}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (-\frac {b^{2} d \left (7 a d -2 b c \right )}{3 \left (a d -b c \right )^{3} a^{2}}-\frac {b \,d^{3}}{\left (a d -b c \right )^{3} c}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(561\)
default	\(-\frac {-3 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} d^{3} x^{5}-7 \sqrt {-\frac {b}{a}}\, a \,b^{3} c \,d^{2} x^{5}+2 \sqrt {-\frac {b}{a}}\, b^{4} c^{2} d \,x^{5}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c \,d^{2} x^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{2} d \,x^{2}+2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{3} x^{2}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c \,d^{2} x^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{2} d \,x^{2}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{4} c^{3} x^{2}-6 \sqrt {-\frac {b}{a}}\, a^{3} b \,d^{3} x^{3}-8 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c \,d^{2} x^{3}-4 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{2} d \,x^{3}+2 \sqrt {-\frac {b}{a}}\, b^{4} c^{3} x^{3}+6 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{2}-8 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d +2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3}+3 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{3} b c \,d^{2}+7 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a^{2} b^{2} c^{2} d -2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a \,b^{3} c^{3}-3 \sqrt {-\frac {b}{a}}\, a^{4} d^{3} x -8 \sqrt {-\frac {b}{a}}\, a^{2} b^{2} c^{2} d x +3 \sqrt {-\frac {b}{a}}\, a \,b^{3} c^{3} x}{3 \sqrt {d \,x^{2}+c}\, \left (a d -b c \right )^{3} a^{2} \sqrt {-\frac {b}{a}}\, c \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(964\)

[In]

int(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

((b*x^2+a)*(d*x^2+c))^(1/2)/(b*x^2+a)^(1/2)/(d*x^2+c)^(1/2)*(1/3/a/(a*d-b*c)^2*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)
^(1/2)/(x^2+a/b)^2+1/3*(b*d*x^2+b*c)*b/a^2/(a*d-b*c)^3*x*(7*a*d-2*b*c)/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(b*d*x^
2+a*d)*d^2/c/(a*d-b*c)^3*x/((x^2+c/d)*(b*d*x^2+a*d))^(1/2)+(1/3*b*d/(a*d-b*c)^2/a-1/3/(a*d-b*c)^2*b*(7*a*d-2*b
*c)/a^2-1/3*b^2*c/a^2/(a*d-b*c)^3*(7*a*d-2*b*c)+d^2/(a*d-b*c)^2/c-a*d^3/c/(a*d-b*c)^3)/(-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
))-(-1/3*b^2*d*(7*a*d-2*b*c)/(a*d-b*c)^3/a^2-b*d^3/(a*d-b*c)^3/c)*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. 856 vs. \(2 (305) = 610\).

Time = 0.13 (sec) , antiderivative size = 856, normalized size of antiderivative = 2.65 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=-\frac {{\left (2 \, a^{2} b^{3} c^{3} - 7 \, a^{3} b^{2} c^{2} d - 3 \, a^{4} b c d^{2} + {\left (2 \, b^{5} c^{2} d - 7 \, a b^{4} c d^{2} - 3 \, a^{2} b^{3} d^{3}\right )} x^{6} + {\left (2 \, b^{5} c^{3} - 3 \, a b^{4} c^{2} d - 17 \, a^{2} b^{3} c d^{2} - 6 \, a^{3} b^{2} d^{3}\right )} x^{4} + {\left (4 \, a b^{4} c^{3} - 12 \, a^{2} b^{3} c^{2} d - 13 \, a^{3} b^{2} c d^{2} - 3 \, a^{4} b d^{3}\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 (2 \, a^{2} b^{3} c^{3} + {\left (2 \, b^{5} c^{2} d + {\left (a^{2} b^{3} - 7 \, a b^{4}\right )} c d^{2} - 3 \, {\left (3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d^{3}\right )} x^{6} + {\left (2 \, b^{5} c^{3} + {\left (a^{2} b^{3} - 3 \, a b^{4}\right )} c^{2} d - {\left (7 \, a^{3} b^{2} + 17 \, a^{2} b^{3}\right )} c d^{2} - 6 \, {\left (3 \, a^{4} b + a^{3} b^{2}\right )} d^{3}\right )} x^{4} + {\left (a^{4} b - 7 \, a^{3} b^{2}\right )} c^{2} d - 3 \, {\left (3 \, a^{5} + a^{4} b\right )} c d^{2} + {\left (4 \, a b^{4} c^{3} + 2 \, {\left (a^{3} b^{2} - 6 \, a^{2} b^{3}\right )} c^{2} d - {\left (17 \, a^{4} b + 13 \, a^{3} b^{2}\right )} c d^{2} - 3 \, {\left (3 \, a^{5} + a^{4} b\right )} d^{3}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (2 \, a b^{4} c^{2} d - 7 \, a^{2} b^{3} c d^{2} - 3 \, a^{3} b^{2} d^{3}\right )} x^{5} + 2 \, {\left (a b^{4} c^{3} - 2 \, a^{2} b^{3} c^{2} d - 4 \, a^{3} b^{2} c d^{2} - 3 \, a^{4} b d^{3}\right )} x^{3} + {\left (3 \, a^{2} b^{3} c^{3} - 8 \, a^{3} b^{2} c^{2} d - 3 \, a^{5} d^{3}\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{5} b^{3} c^{5} - 3 \, a^{6} b^{2} c^{4} d + 3 \, a^{7} b c^{3} d^{2} - a^{8} c^{2} d^{3} + {\left (a^{3} b^{5} c^{4} d - 3 \, a^{4} b^{4} c^{3} d^{2} + 3 \, a^{5} b^{3} c^{2} d^{3} - a^{6} b^{2} c d^{4}\right )} x^{6} + {\left (a^{3} b^{5} c^{5} - a^{4} b^{4} c^{4} d - 3 \, a^{5} b^{3} c^{3} d^{2} + 5 \, a^{6} b^{2} c^{2} d^{3} - 2 \, a^{7} b c d^{4}\right )} x^{4} + {\left (2 \, a^{4} b^{4} c^{5} - 5 \, a^{5} b^{3} c^{4} d + 3 \, a^{6} b^{2} c^{3} d^{2} + a^{7} b c^{2} d^{3} - a^{8} c d^{4}\right )} x^{2}\right )}} \]

[In]

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

-1/3*((2*a^2*b^3*c^3 - 7*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + (2*b^5*c^2*d - 7*a*b^4*c*d^2 - 3*a^2*b^3*d^3)*x^6 + (
2*b^5*c^3 - 3*a*b^4*c^2*d - 17*a^2*b^3*c*d^2 - 6*a^3*b^2*d^3)*x^4 + (4*a*b^4*c^3 - 12*a^2*b^3*c^2*d - 13*a^3*b
^2*c*d^2 - 3*a^4*b*d^3)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (2*a^2*b^3*c^3
+ (2*b^5*c^2*d + (a^2*b^3 - 7*a*b^4)*c*d^2 - 3*(3*a^3*b^2 + a^2*b^3)*d^3)*x^6 + (2*b^5*c^3 + (a^2*b^3 - 3*a*b
^4)*c^2*d - (7*a^3*b^2 + 17*a^2*b^3)*c*d^2 - 6*(3*a^4*b + a^3*b^2)*d^3)*x^4 + (a^4*b - 7*a^3*b^2)*c^2*d - 3*(3
*a^5 + a^4*b)*c*d^2 + (4*a*b^4*c^3 + 2*(a^3*b^2 - 6*a^2*b^3)*c^2*d - (17*a^4*b + 13*a^3*b^2)*c*d^2 - 3*(3*a^5
+ a^4*b)*d^3)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - ((2*a*b^4*c^2*d - 7*a^2*
b^3*c*d^2 - 3*a^3*b^2*d^3)*x^5 + 2*(a*b^4*c^3 - 2*a^2*b^3*c^2*d - 4*a^3*b^2*c*d^2 - 3*a^4*b*d^3)*x^3 + (3*a^2*
b^3*c^3 - 8*a^3*b^2*c^2*d - 3*a^5*d^3)*x)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^5*b^3*c^5 - 3*a^6*b^2*c^4*d + 3*
a^7*b*c^3*d^2 - a^8*c^2*d^3 + (a^3*b^5*c^4*d - 3*a^4*b^4*c^3*d^2 + 3*a^5*b^3*c^2*d^3 - a^6*b^2*c*d^4)*x^6 + (a
^3*b^5*c^5 - a^4*b^4*c^4*d - 3*a^5*b^3*c^3*d^2 + 5*a^6*b^2*c^2*d^3 - 2*a^7*b*c*d^4)*x^4 + (2*a^4*b^4*c^5 - 5*a
^5*b^3*c^4*d + 3*a^6*b^2*c^3*d^2 + a^7*b*c^2*d^3 - a^8*c*d^4)*x^2)

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b x^{2}\right )^{\frac {5}{2}} \left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(b*x**2+a)**(5/2)/(d*x**2+c)**(3/2),x)

[Out]

Integral(1/((a + b*x**2)**(5/2)*(c + d*x**2)**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b x^{2} + a\right )}^{\frac {5}{2}} {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(b*x^2+a)^(5/2)/(d*x^2+c)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/2)*(d*x^2 + c)^(3/2)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{5/2}\,{\left (d\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)),x)

[Out]

int(1/((a + b*x^2)^(5/2)*(c + d*x^2)^(3/2)), x)