3.10.53 \(\int \frac {(a+b x^2)^{3/2}}{x^4 \sqrt {c+d x^2}} \, dx\) [953]

3.10.53.1 Optimal result

Integrand size = 26, antiderivative size = 311 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {2 d (2 b c-a d) x \sqrt {a+b x^2}}{3 c^2 \sqrt {c+d x^2}}-\frac {a \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c x^3}-\frac {2 (2 b c-a d) \sqrt {a+b x^2} \sqrt {c+d x^2}}{3 c^2 x}-\frac {2 \sqrt {d} (2 b c-a d) \sqrt {a+b x^2} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{3 c^{3/2} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}}+\frac {b (3 b c-a d) \sqrt {a+b x^2} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),1-\frac {b c}{a d}\right )}{3 a \sqrt {c} \sqrt {d} \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt {c+d x^2}} \]

output

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

3.10.53.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.46 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.73 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right ) \left (-a c-4 b c x^2+2 a d x^2\right )+2 i b c (-2 b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )-i b c (-b c+a d) x^3 \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{3 \sqrt {\frac {b}{a}} c^2 x^3 \sqrt {a+b x^2} \sqrt {c+d x^2}} \]

input

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

output

(Sqrt[b/a]*(a + b*x^2)*(c + d*x^2)*(-(a*c) - 4*b*c*x^2 + 2*a*d*x^2) + (2*I 
)*b*c*(-2*b*c + 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*b*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*Sqrt[b/a]*c^2*x^3*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])
 

3.10.53.3 Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {376, 445, 25, 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.

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

\(\Big \downarrow \) 376

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 406

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

\(\Big \downarrow \) 320

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

\(\Big \downarrow \) 388

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

\(\Big \downarrow \) 313

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

input

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

output

-1/3*(a*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(c*x^3) + ((-2*(2*b*c - a*d)*Sqrt 
[a + b*x^2]*Sqrt[c + d*x^2])/(c*x) + (b*(2*d*(2*b*c - 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)*(3*b*c - a*d)*Sqrt[a + b*x^2]*Elli 
pticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[(c*(a 
 + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2])))/c)/(3*c)
 

3.10.53.3.1 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 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 376

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

3.10.53.4 Maple [A] (verified)

Time = 5.75 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.09

method	result	size
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (-\frac {a \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 c \,x^{3}}+\frac {2 \left (a d -2 b c \right ) \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}{3 c^{2} x}+\frac {\left (b^{2}-\frac {a b d}{3 c}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}+\frac {2 b \left (a d -2 b c \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{3 c \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(340\)
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-2 a d \,x^{2}+4 c b \,x^{2}+a c \right )}{3 c^{2} x^{3}}-\frac {b \left (\frac {a c d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {3 b \,c^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}}-\frac {\left (2 a \,d^{2}-4 b c d \right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (F\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c}\, d}\right ) \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}{3 c^{2} \sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}}\)	\(403\)
default	\(\frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (2 \sqrt {-\frac {b}{a}}\, a b \,d^{2} x^{6}-4 \sqrt {-\frac {b}{a}}\, b^{2} c d \,x^{6}+b d \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) x^{3} a c -\sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, F\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}-2 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) a b c d \,x^{3}+4 \sqrt {\frac {b \,x^{2}+a}{a}}\, \sqrt {\frac {d \,x^{2}+c}{c}}\, E\left (x \sqrt {-\frac {b}{a}}, \sqrt {\frac {a d}{b c}}\right ) b^{2} c^{2} x^{3}+2 \sqrt {-\frac {b}{a}}\, a^{2} d^{2} x^{4}-3 \sqrt {-\frac {b}{a}}\, a b c d \,x^{4}-4 \sqrt {-\frac {b}{a}}\, b^{2} c^{2} x^{4}+\sqrt {-\frac {b}{a}}\, a^{2} c d \,x^{2}-5 \sqrt {-\frac {b}{a}}\, a b \,c^{2} x^{2}-\sqrt {-\frac {b}{a}}\, a^{2} c^{2}\right )}{3 \left (b d \,x^{4}+a d \,x^{2}+c b \,x^{2}+a c \right ) c^{2} x^{3} \sqrt {-\frac {b}{a}}}\)	\(433\)

input

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

3.10.53.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.54 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{x^4 \sqrt {c+d x^2}} \, dx=\frac {2 \, {\left (2 \, b^{2} c - a b d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left ({\left (3 \, a b + 4 \, b^{2}\right )} c - {\left (a^{2} + 2 \, a b\right )} d\right )} \sqrt {a c} x^{3} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a^{2} c + 2 \, {\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c}}{3 \, a c^{2} x^{3}} \]

input

integrate((b*x^2+a)^(3/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="fricas")
 

output

1/3*(2*(2*b^2*c - a*b*d)*sqrt(a*c)*x^3*sqrt(-b/a)*elliptic_e(arcsin(x*sqrt 
(-b/a)), a*d/(b*c)) - ((3*a*b + 4*b^2)*c - (a^2 + 2*a*b)*d)*sqrt(a*c)*x^3* 
sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (a^2*c + 2*(2*a*b 
*c - a^2*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c))/(a*c^2*x^3)
 

3.10.53.6 Sympy [F]

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

input

integrate((b*x**2+a)**(3/2)/x**4/(d*x**2+c)**(1/2),x)
 

output

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

3.10.53.7 Maxima [F]

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

input

integrate((b*x^2+a)^(3/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="maxima")
 

output

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

3.10.53.8 Giac [F]

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

input

integrate((b*x^2+a)^(3/2)/x^4/(d*x^2+c)^(1/2),x, algorithm="giac")
 

output

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

3.10.53.9 Mupad [F(-1)]

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

input

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

output

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