3.10.43 \(\int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx\) [943]

3.10.43.1 Optimal result

Integrand size = 26, antiderivative size = 307 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {d (b c-2 a d) x \sqrt {a+b x^2}}{3 a c^2 \sqrt {c+d x^2}}-\frac {\sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {(b c-2 a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 a c^2 x}-\frac {\sqrt {d} (b c-2 a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 a c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}-\frac {b \sqrt {d} \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

output

1/3*d*(-2*a*d+b*c)*x*(b*x^2+a)^(1/2)/a/c^2/(d*x^2+c)^(1/2)-1/3*(-2*a*d+b*c 
)*(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/c^(3/2)/(c*(b*x 
^2+a)/a/(d*x^2+c))^(1/2)/(d*x^2+c)^(1/2)-1/3*b*(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))*d^(1/2)*(b*x^2+a)^(1/2)/a/c^(1/2)/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2)/(d*x 
^2+c)^(1/2)-1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/c/x^3-1/3*(-2*a*d+b*c)*(b* 
x^2+a)^(1/2)*(d*x^2+c)^(1/2)/a/c^2/x
 

3.10.43.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.63 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {-\frac {\left (a+b x^2\right ) \left (c+d x^2\right ) \left (a c+b c x^2-2 a d x^2\right )}{a}+i \sqrt {\frac {b}{a}} c (-b c+2 a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i \sqrt {\frac {b}{a}} c (b c-a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 c^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

input

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

output

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

3.10.43.3 Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 303, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {377, 445, 27, 406, 320, 388, 313}

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.

\(\displaystyle \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx\)

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

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

3.10.43.3.1 Defintions of rubi rules used

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 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 377

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

rule 388

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b   Int[Sqrt[ 
a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]
 

rule 406

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

rule 445

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

3.10.43.4 Maple [A] (verified)

Time = 5.69 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.02

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}+c b \,x^{2}+a c \right )}{3 c^{2} x^{3} a}-\frac {b d \left (\frac {a c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a d -b c \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 a \,c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(314\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {\sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 c \,x^{3}}+\frac {\left (2 a d -b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 a \,c^{2} x}-\frac {b d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{3 c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {b \left (2 a d -b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 c a \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(341\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-\sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}+\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-\sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-2 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) c^{2} x^{3} a \sqrt {-\frac {b}{a}}}\)	\(418\)

input

int((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2),x,method=_RETURNVERBOSE)
 

output

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

3.10.43.5 Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {{\left (b^{2} c - 2 \, a b d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (b^{2} c - {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a^{2} c + {\left (a b c - 2 \, a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, a^{2} c^{2} x^{3}} \]

input

integrate((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

output

1/3*((b^2*c - 2*a*b*d)*sqrt(a*c)*x^3*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt(- 
b/a)), a*d/(b*c)) - (b^2*c - (a^2 + 2*a*b)*d)*sqrt(a*c)*x^3*sqrt(-b/a)*ell 
iptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (a^2*c + (a*b*c - 2*a^2*d)*x^2) 
*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a^2*c^2*x^3)
 

3.10.43.6 Sympy [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {a + b x^{2}}}{x^{4} \sqrt {c + d x^{2}}}\, dx \]

input

integrate((b*x**2+a)**(1/2)/x**4/(d*x**2+c)**(1/2),x)
 

output

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

3.10.43.7 Maxima [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]

input

integrate((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

output

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

3.10.43.8 Giac [F]

\[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {d x^{2} + c} x^{4}} \,d x } \]

input

integrate((b*x^2+a)^(1/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="giac")
 

output

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

3.10.43.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b x^2}}{x^4 \sqrt {c+d x^2}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{x^4\,\sqrt {d\,x^2+c}} \,d x \]

input

int((a + b*x^2)^(1/2)/(x^4*(c + d*x^2)^(1/2)),x)
 

output

int((a + b*x^2)^(1/2)/(x^4*(c + d*x^2)^(1/2)), x)