\(\int \frac {1}{\sqrt {x} (a x+b x^3+c x^5)^{3/2}} \, dx\) [66]

Optimal result

Integrand size = 24, antiderivative size = 468 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {b^2-2 a c+b c x^2}{a \left (b^2-4 a c\right ) \sqrt {x} \sqrt {a x+b x^3+c x^5}}+\frac {2 \sqrt {c} \left (b^2-3 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 \left (b^2-4 a c\right ) x^{3/2}}-\frac {2 \sqrt [4]{c} \left (b^2-3 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{c} \left (2 b^2+\sqrt {a} b \sqrt {c}-6 a c\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \] Output:

(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/x^(1/2)/(c*x^5+b*x^3+a*x)^(1/2)+2*c^(1/ 
2)*(-3*a*c+b^2)*x^(3/2)*(c*x^4+b*x^2+a)/a^2/(-4*a*c+b^2)/(a^(1/2)+c^(1/2)* 
x^2)/(c*x^5+b*x^3+a*x)^(1/2)-2*(-3*a*c+b^2)*(c*x^5+b*x^3+a*x)^(1/2)/a^2/(- 
4*a*c+b^2)/x^(3/2)-2*c^(1/4)*(-3*a*c+b^2)*x^(1/2)*(a^(1/2)+c^(1/2)*x^2)*(( 
c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/ 
4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(7/4)/(-4*a*c+b^2)/(c*x^ 
5+b*x^3+a*x)^(1/2)+1/2*c^(1/4)*(2*b^2+a^(1/2)*b*c^(1/2)-6*a*c)*x^(1/2)*(a^ 
(1/2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*Inverse 
JacobiAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(7 
/4)/(-4*a*c+b^2)/(c*x^5+b*x^3+a*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.79 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx=-\frac {2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-4 a^2 c+2 b^2 x^2 \left (b+c x^2\right )+a \left (b^2-7 b c x^2-6 c^2 x^4\right )\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x} \sqrt {x \left (a+b x^2+c x^4\right )}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 448, normalized size of antiderivative = 0.96, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1971, 25, 1998, 25, 27, 2000, 1511, 27, 1416, 1509}

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.

\(\displaystyle \int \frac {1}{\sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1971

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1998

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2000

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

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \sqrt {a} \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a x^{3/2}}}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\sqrt {a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a x^{3/2}}}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 \left (b^2-3 a c\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a x^{3/2}}}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {c \sqrt {x} \sqrt {a+b x^2+c x^4} \left (\frac {\sqrt [4]{a} \left (\frac {2 \left (b^2-3 a c\right )}{\sqrt {c}}+\sqrt {a} b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {2 \left (b^2-3 a c\right ) \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right )}{\sqrt {c}}\right )}{a \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a x^{3/2}}}{a \left (b^2-4 a c\right )}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}\)

Input:

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

Output:

(b^2 - 2*a*c + b*c*x^2)/(a*(b^2 - 4*a*c)*Sqrt[x]*Sqrt[a*x + b*x^3 + c*x^5] 
) + ((-2*(b^2 - 3*a*c)*Sqrt[a*x + b*x^3 + c*x^5])/(a*x^(3/2)) + (c*Sqrt[x] 
*Sqrt[a + b*x^2 + c*x^4]*((-2*(b^2 - 3*a*c)*(-((x*Sqrt[a + b*x^2 + c*x^4]) 
/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x 
^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1 
/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sq 
rt[c] + (a^(1/4)*(Sqrt[a]*b + (2*(b^2 - 3*a*c))/Sqrt[c])*(Sqrt[a] + Sqrt[c 
]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*Arc 
Tan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(1/4)*Sqrt[a 
+ b*x^2 + c*x^4])))/(a*Sqrt[a*x + b*x^3 + c*x^5]))/(a*(b^2 - 4*a*c))
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1971

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[(-x^(m - q + 1))*(b^2 - 2*a*c + b*c*x^(n - q))*((a*x^q 
 + b*x^n + c*x^(2*n - q))^(p + 1)/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))), x] + 
Simp[1/(a*(n - q)*(p + 1)*(b^2 - 4*a*c))   Int[x^(m - q)*(b^2*(m + p*q + (n 
 - q)*(p + 1) + 1) - 2*a*c*(m + p*q + 2*(n - q)*(p + 1) + 1) + b*c*(m + p*q 
 + (n - q)*(2*p + 3) + 1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p + 1 
), x], x] /; FreeQ[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !Int 
egerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && LtQ[p, -1] && RationalQ[m, 
 q] && LtQ[m + p*q + 1, n - q]
 

rule 1998

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_ 
.)*((A_) + (B_.)*(x_)^(r_.)), x_Symbol] :> Simp[A*x^(m - q + 1)*((a*x^q + b 
*x^n + c*x^(2*n - q))^(p + 1)/(a*(m + p*q + 1))), x] + Simp[1/(a*(m + p*q + 
 1))   Int[x^(m + n - q)*Simp[a*B*(m + p*q + 1) - A*b*(m + p*q + (n - q)*(p 
 + 1) + 1) - A*c*(m + p*q + 2*(n - q)*(p + 1) + 1)*x^(n - q), x]*(a*x^q + b 
*x^n + c*x^(2*n - q))^p, x], x] /; FreeQ[{a, b, c, A, B}, x] && EqQ[r, n - 
q] && EqQ[j, 2*n - q] &&  !IntegerQ[p] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] 
 && RationalQ[m, p, q] && ((GeQ[p, -1] && LtQ[p, 0]) || EqQ[m + p*q + (n - 
q)*(2*p + 1) + 1, 0]) && LeQ[m + p*q, -(n - q)] && NeQ[m + p*q + 1, 0]
 

rule 2000

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(j_.)))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x 
_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - 
q) + c*x^(2*(n - q))]/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)])   Int[x^(m - q/2 
)*((A + B*x^(n - q))/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; Fr 
eeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && Pos 
Q[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1135\) vs. \(2(393)=786\).

Time = 5.70 (sec) , antiderivative size = 1136, normalized size of antiderivative = 2.43

method	result	size
default	\(\text {Expression too large to display}\)	\(1136\)
risch	\(\text {Expression too large to display}\)	\(1497\)

Input:

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

Output:

-1/2/x^(3/2)*(x*(c*x^4+b*x^2+a))^(1/2)*(12*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/ 
2)*(-4*a*c+b^2)^(1/2)*a*c^2*x^4-4*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(-4*a* 
c+b^2)^(1/2)*b^2*c*x^4+12*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*a*b*c^2*x^4-4* 
((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*b^3*c*x^4+a*b*c*(-2*((-4*a*c+b^2)^(1/2)* 
x^2-b*x^2-2*a)/a)^(1/2)*(((-4*a*c+b^2)^(1/2)*x^2+b*x^2+2*a)/a)^(1/2)*Ellip 
ticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)*((b*(-4*a 
*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*x*(-4*a*c+b^2)^(1/2)+12*(-2*((-4*a*c+ 
b^2)^(1/2)*x^2-b*x^2-2*a)/a)^(1/2)*(((-4*a*c+b^2)^(1/2)*x^2+b*x^2+2*a)/a)^ 
(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2 
)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a^2*c^2*x-3*a*b^2*c*(-2*(( 
-4*a*c+b^2)^(1/2)*x^2-b*x^2-2*a)/a)^(1/2)*(((-4*a*c+b^2)^(1/2)*x^2+b*x^2+2 
*a)/a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2 
*2^(1/2)*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*x-12*(-2*((-4*a*c+b 
^2)^(1/2)*x^2-b*x^2-2*a)/a)^(1/2)*(((-4*a*c+b^2)^(1/2)*x^2+b*x^2+2*a)/a)^( 
1/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2) 
*((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a^2*c^2*x+4*(-2*((-4*a*c+b^ 
2)^(1/2)*x^2-b*x^2-2*a)/a)^(1/2)*(((-4*a*c+b^2)^(1/2)*x^2+b*x^2+2*a)/a)^(1 
/2)*EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*2^(1/2)* 
((b*(-4*a*c+b^2)^(1/2)-2*a*c+b^2)/a/c)^(1/2))*a*b^2*c*x+14*((-b+(-4*a*c+b^ 
2)^(1/2))/a)^(1/2)*(-4*a*c+b^2)^(1/2)*a*b*c*x^2-4*((-b+(-4*a*c+b^2)^(1/...
 

Fricas [F]

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

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

Output:

integral(sqrt(c*x^5 + b*x^3 + a*x)*sqrt(x)/(c^2*x^11 + 2*b*c*x^9 + (b^2 + 
2*a*c)*x^7 + 2*a*b*x^5 + a^2*x^3), x)
 

Sympy [F]

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

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

Output:

Integral(1/(sqrt(x)*(x*(a + b*x**2 + c*x**4))**(3/2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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