\(\int \frac {x^{3/2}}{(a x+b x^3+c x^5)^{3/2}} \, dx\) [117]

Optimal result

Integrand size = 24, antiderivative size = 391 \[ \int \frac {x^{3/2}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {b \sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {b \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a x+b x^3+c x^5}} \]

[Out]

x^(3/2)*(b*c*x^2-2*a*c+b^2)/a/(-4*a*c+b^2)/(c*x^5+b*x^3+a*x)^(1/2)-b*x^(3/2)*(c*x^4+b*x^2+a)*c^(1/2)/a/(-4*a*c
+b^2)/(a^(1/2)+x^2*c^(1/2))/(c*x^5+b*x^3+a*x)^(1/2)+b*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2
*arctan(c^(1/4)*x/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/
2)+x^2*c^(1/2))*x^(1/2)*((c*x^4+b*x^2+a)/(a^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/(-4*a*c+b^2)/(c*x^5+b*x^3+a*x)
^(1/2)-1/2*c^(1/4)*(cos(2*arctan(c^(1/4)*x/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x/a^(1/4)))*EllipticF(sin(2
*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x^2*c^(1/2))*x^(1/2)*((c*x^4+b*x^2+a)/(a
^(1/2)+x^2*c^(1/2))^2)^(1/2)/a^(3/4)/(b-2*a^(1/2)*c^(1/2))/(c*x^5+b*x^3+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {1938, 1967, 1211, 1117, 1209} \[ \int \frac {x^{3/2}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {b \sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a x+b x^3+c x^5}}+\frac {x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {b \sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}} \]

[In]

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

[Out]

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

Rule 1117

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 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> 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 1211

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[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] && PosQ[c/a]

Rule 1938

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] + Dist[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] &&  !IntegerQ[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 1967

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(j_.)))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Sym
bol] :> Dist[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] /; FreeQ[{a, b, c, A, B, m, n, q}, x
] && EqQ[j, n - q] && EqQ[r, 2*n - q] && PosQ[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q
, 1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\int \frac {\sqrt {x} \left (2 a c+b c x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{a \left (b^2-4 a c\right )} \\ & = \frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {2 a c+b c x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \\ & = \frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\left (b \sqrt {c} \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\left (\left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt {c} \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{\sqrt {a} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \\ & = \frac {x^{3/2} \left (b^2-2 a c+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {b \sqrt {c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \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 {b \sqrt [4]{c} \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 \tan ^{-1}\left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {\sqrt [4]{c} \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}} F\left (2 \tan ^{-1}\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^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) \sqrt {a x+b x^3+c x^5}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 11.66 (sec) , antiderivative size = 463, normalized size of antiderivative = 1.18 \[ \int \frac {x^{3/2}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=-\frac {\sqrt {x} \left (-4 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (b^2-2 a c+b c x^2\right )+i b \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{4 a \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x \left (a+b x^2+c x^4\right )}} \]

[In]

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

[Out]

-1/4*(Sqrt[x]*(-4*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(b^2 - 2*a*c + b*c*x^2) + I*b*(-b + Sqrt[b^2 - 4*a*c])*Sqr
t[(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 - S
qrt[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^2 + 4*a*c + b*Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqr
t[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*(b^2 - 4*a*c)*Sqr
t[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])

Maple [A] (verified)

Time = 0.52 (sec) , antiderivative size = 533, normalized size of antiderivative = 1.36

method	result	size
default	\(\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (-\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b c \,x^{3}-\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} c \,x^{3}+c \sqrt {-\frac {2 \left (\sqrt {-4 a c +b^{2}}\, x^{2}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, x^{2}+b \,x^{2}+2 a}{a}}\, F\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) a \sqrt {-4 a c +b^{2}}+b c \sqrt {-\frac {2 \left (\sqrt {-4 a c +b^{2}}\, x^{2}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, x^{2}+b \,x^{2}+2 a}{a}}\, a E\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right )+2 \sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a c x -\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} x +2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b c x -\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{3} x \right )}{\sqrt {x}\, \left (c \,x^{4}+b \,x^{2}+a \right ) a \left (4 a c -b^{2}\right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(533\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.23 \[ \int \frac {x^{3/2}}{\left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (b^{2} c x^{6} + b^{3} x^{4} + a b^{2} x^{2} - {\left (b c^{2} x^{6} + b^{2} c x^{4} + a b c x^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (b^{2} c + 2 \, b c^{2}\right )} x^{6} + {\left (b^{3} + 2 \, b^{2} c\right )} x^{4} + {\left (a b^{2} + 2 \, a b c\right )} x^{2} - {\left ({\left (b c^{2} - 2 \, c^{3}\right )} x^{6} + {\left (b^{2} c - 2 \, b c^{2}\right )} x^{4} + {\left (a b c - 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - 2 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (2 \, a c^{2} x^{2} + a b c\right )} \sqrt {x}}{2 \, {\left ({\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} x^{6} + {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} x^{4} + {\left (a^{2} b^{2} c - 4 \, a^{3} c^{2}\right )} x^{2}\right )}} \]

[In]

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

[Out]

1/2*(sqrt(1/2)*(b^2*c*x^6 + b^3*x^4 + a*b^2*x^2 - (b*c^2*x^6 + b^2*c*x^4 + a*b*c*x^2)*sqrt((b^2 - 4*a*c)/c^2))
*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) -
b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - sqrt(1/2)*((b^2*c + 2*b*c^2)*x^6 + (b^3 +
2*b^2*c)*x^4 + (a*b^2 + 2*a*b*c)*x^2 - ((b*c^2 - 2*c^3)*x^6 + (b^2*c - 2*b*c^2)*x^4 + (a*b*c - 2*a*c^2)*x^2)*s
qrt((b^2 - 4*a*c)/c^2))*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(arcsin(sqrt(1/2)*sqrt((c*sq
rt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) - 2*sqrt(c*x^5 + b*x^
3 + a*x)*(2*a*c^2*x^2 + a*b*c)*sqrt(x))/((a*b^2*c^2 - 4*a^2*c^3)*x^6 + (a*b^3*c - 4*a^2*b*c^2)*x^4 + (a^2*b^2*
c - 4*a^3*c^2)*x^2)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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