\(\int \frac {1}{(a+b x^2)^{5/2} \sqrt {c+d x^2}} \, dx\) [160]

Optimal result

Integrand size = 23, antiderivative size = 256 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {b x \sqrt {c+d x^2}}{3 a (b c-a d) \left (a+b x^2\right )^{3/2}}+\frac {2 \sqrt {b} (b c-2 a d) \sqrt {c+d x^2} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{3 a^{3/2} (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}}-\frac {d (b c-3 a d) \sqrt {c+d x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{3 \sqrt {a} \sqrt {b} c (b c-a d)^2 \sqrt {a+b x^2} \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}}} \] Output:

1/3*b*x*(d*x^2+c)^(1/2)/a/(-a*d+b*c)/(b*x^2+a)^(3/2)+2/3*b^(1/2)*(-2*a*d+b 
*c)*(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^(3/2)/(-a*d+b*c)^2/(b*x^2+a)^(1/2)/(a*(d*x^2+c)/c/(b*x^2+a))^ 
(1/2)-1/3*d*(-3*a*d+b*c)*(d*x^2+c)^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/ 
a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(1/2)/c/(-a*d+b*c)^2/(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 = 2.83 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=\frac {b \sqrt {\frac {b}{a}} x \left (c+d x^2\right ) \left (-5 a^2 d+2 b^2 c x^2+a b \left (3 c-4 d x^2\right )\right )-2 i b c (-b c+2 a d) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i \left (2 b^2 c^2-5 a b c d+3 a^2 d^2\right ) \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 a^2 \sqrt {\frac {b}{a}} (b c-a d)^2 \left (a+b x^2\right )^{3/2} \sqrt {c+d x^2}} \] Input:

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

Output:

(b*Sqrt[b/a]*x*(c + d*x^2)*(-5*a^2*d + 2*b^2*c*x^2 + a*b*(3*c - 4*d*x^2)) 
- (2*I)*b*c*(-(b*c) + 2*a*d)*(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)] - I*(2*b^2*c^2 - 5*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]*El 
lipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(3*a^2*Sqrt[b/a]*(b*c - a*d) 
^2*(a + b*x^2)^(3/2)*Sqrt[c + d*x^2])
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {316, 25, 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.

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

\(\Big \downarrow \) 316

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 400

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

\(\Big \downarrow \) 313

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

\(\Big \downarrow \) 320

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 14.50 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.74

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {x \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}{3 b a \left (a d -b c \right ) \left (x^{2}+\frac {a}{b}\right )^{2}}-\frac {2 \left (b d \,x^{2}+b c \right ) x \left (2 a d -b c \right )}{3 a^{2} \left (a d -b c \right )^{2} \sqrt {\left (x^{2}+\frac {a}{b}\right ) \left (b d \,x^{2}+b c \right )}}+\frac {\left (-\frac {d}{3 \left (a d -b c \right ) a}+\frac {\frac {4 a d}{3}-\frac {2 b c}{3}}{a^{2} \left (a d -b c \right )}+\frac {2 b c \left (2 a d -b c \right )}{3 a^{2} \left (a d -b c \right )^{2}}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}-\frac {2 b \left (2 a d -b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 \left (a d -b c \right )^{2} a^{2} \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+x^{2} b c +a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {x^{2} d +c}}\)	\(445\)
default	\(\frac {-4 \sqrt {-\frac {b}{a}}\, a \,b^{2} d^{2} x^{5}+2 \sqrt {-\frac {b}{a}}\, b^{3} c d \,x^{5}+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} b \,d^{2} x^{2}-5 \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^{2} c d \,x^{2}+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^{3} c^{2} x^{2}+4 \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^{2} c d \,x^{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 ) b^{3} c^{2} x^{2}-5 \sqrt {-\frac {b}{a}}\, a^{2} b \,d^{2} x^{3}-\sqrt {-\frac {b}{a}}\, a \,b^{2} c d \,x^{3}+2 \sqrt {-\frac {b}{a}}\, b^{3} c^{2} x^{3}+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^{3} d^{2}-5 \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} b c d +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^{2} c^{2}+4 \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} b c d -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^{2} c^{2}-5 \sqrt {-\frac {b}{a}}\, a^{2} b c d x +3 \sqrt {-\frac {b}{a}}\, a \,b^{2} c^{2} x}{3 \sqrt {x^{2} d +c}\, \left (a d -b c \right )^{2} \sqrt {-\frac {b}{a}}\, a^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\)	\(752\)

Input:

int(1/(b*x^2+a)^(5/2)/(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)*(-1/3/b/a/(a*d 
-b*c)*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/(x^2+a/b)^2-2/3*(b*d*x^2+b*c)/ 
a^2/(a*d-b*c)^2*x*(2*a*d-b*c)/((x^2+a/b)*(b*d*x^2+b*c))^(1/2)+(-1/3*d/(a*d 
-b*c)/a+2/3*(2*a*d-b*c)/(a*d-b*c)/a^2+2/3*b*c/a^2/(a*d-b*c)^2*(2*a*d-b*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)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-2/3*b*(2*a* 
d-b*c)/(a*d-b*c)^2/a^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)*(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 [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.72 \[ \int \frac {1}{\left (a+b x^2\right )^{5/2} \sqrt {c+d x^2}} \, dx=-\frac {2 \, {\left (a^{2} b^{3} c^{2} - 2 \, a^{3} b^{2} c d + {\left (b^{5} c^{2} - 2 \, a b^{4} c d\right )} x^{4} + 2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, a^{2} b^{3} c^{2} - 3 \, a^{5} d^{2} + {\left (2 \, b^{5} c^{2} - 3 \, a^{3} b^{2} d^{2} + {\left (a^{2} b^{3} - 4 \, a b^{4}\right )} c d\right )} x^{4} + {\left (a^{4} b - 4 \, a^{3} b^{2}\right )} c d + 2 \, {\left (2 \, a b^{4} c^{2} - 3 \, a^{4} b d^{2} + {\left (a^{3} b^{2} - 4 \, a^{2} b^{3}\right )} c d\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (2 \, {\left (a b^{4} c^{2} - 2 \, a^{2} b^{3} c d\right )} x^{3} + {\left (3 \, a^{2} b^{3} c^{2} - 5 \, a^{3} b^{2} c d\right )} x\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, {\left (a^{5} b^{3} c^{3} - 2 \, a^{6} b^{2} c^{2} d + a^{7} b c d^{2} + {\left (a^{3} b^{5} c^{3} - 2 \, a^{4} b^{4} c^{2} d + a^{5} b^{3} c d^{2}\right )} x^{4} + 2 \, {\left (a^{4} b^{4} c^{3} - 2 \, a^{5} b^{3} c^{2} d + a^{6} b^{2} c d^{2}\right )} x^{2}\right )}} \] Input:

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

Output:

-1/3*(2*(a^2*b^3*c^2 - 2*a^3*b^2*c*d + (b^5*c^2 - 2*a*b^4*c*d)*x^4 + 2*(a* 
b^4*c^2 - 2*a^2*b^3*c*d)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_e(arcsin(x*sqr 
t(-b/a)), a*d/(b*c)) - (2*a^2*b^3*c^2 - 3*a^5*d^2 + (2*b^5*c^2 - 3*a^3*b^2 
*d^2 + (a^2*b^3 - 4*a*b^4)*c*d)*x^4 + (a^4*b - 4*a^3*b^2)*c*d + 2*(2*a*b^4 
*c^2 - 3*a^4*b*d^2 + (a^3*b^2 - 4*a^2*b^3)*c*d)*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 - 2*a^2*b^3*c* 
d)*x^3 + (3*a^2*b^3*c^2 - 5*a^3*b^2*c*d)*x)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c 
))/(a^5*b^3*c^3 - 2*a^6*b^2*c^2*d + a^7*b*c*d^2 + (a^3*b^5*c^3 - 2*a^4*b^4 
*c^2*d + a^5*b^3*c*d^2)*x^4 + 2*(a^4*b^4*c^3 - 2*a^5*b^3*c^2*d + a^6*b^2*c 
*d^2)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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