\(\int \frac {1}{\sqrt {x} (a x+b x^3+c x^5)^{3/2}} \, dx\) [119]

Optimal result

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

[Out]

2*(-3*a*c+b^2)*x^(3/2)*(c*x^4+b*x^2+a)*c^(1/2)/a^2/(-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*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*(-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)*(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^(7/4)/(-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))*(2*b^2-6*a*c+b*a^(1/2)*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^(7/4)/(-4*a*c+b^2)/(c*x^5+b*x^3+a*x)^(1/2)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1938, 1965, 1967, 1211, 1117, 1209} \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx=\frac {\sqrt [4]{c} \sqrt {x} \left (\sqrt {a} b \sqrt {c}-6 a c+2 b^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \sqrt [4]{c} \sqrt {x} \left (b^2-3 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}-\frac {2 \left (b^2-3 a c\right ) \sqrt {a x+b x^3+c x^5}}{a^2 x^{3/2} \left (b^2-4 a c\right )}+\frac {2 \sqrt {c} x^{3/2} \left (b^2-3 a c\right ) \left (a+b x^2+c x^4\right )}{a^2 \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {-2 a c+b^2+b c x^2}{a \sqrt {x} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \]

[In]

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

[Out]

(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*Sqrt[c]*(b^2 - 3*a*c)*x^(3/2)
*(a + b*x^2 + c*x^4))/(a^2*(b^2 - 4*a*c)*(Sqrt[a] + Sqrt[c]*x^2)*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^2*(b^2 - 4*a*c)*x^(3/2)) - (2*c^(1/4)*(b^2 - 3*a*c)*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^(7/4)*(b^2 - 4*a*c)*Sqrt[a*x + b*x^3 + c*x^5]) + (c^(1/4)*(2*b^2 + Sqrt[a]*b*Sqrt[c] - 6*
a*c)*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^(7/4)*(b^2 - 4*a*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 1965

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> 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] + Dist[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] && Eq
Q[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 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 {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 {\int \frac {-2 b^2+6 a c-b c x^2}{x^{3/2} \sqrt {a x+b x^3+c x^5}} \, dx}{a \left (b^2-4 a c\right )} \\ & = \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 \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 {\int \frac {\sqrt {x} \left (a b c+2 c \left (b^2-3 a c\right ) x^2\right )}{\sqrt {a x+b x^3+c x^5}} \, dx}{a^2 \left (b^2-4 a c\right )} \\ & = \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 \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 {\left (\sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {a b c+2 c \left (b^2-3 a c\right ) x^2}{\sqrt {a+b x^2+c x^4}} \, dx}{a^2 \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \\ & = \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 \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 {\left (2 \sqrt {c} \left (b^2-3 a c\right ) \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}{a^{3/2} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}}+\frac {\left (\left (\sqrt {a} b c^{3/2}+2 c \left (b^2-3 a c\right )\right ) \sqrt {x} \sqrt {a+b x^2+c x^4}\right ) \int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx}{a^{3/2} \sqrt {c} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \\ & = \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 \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^{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}} 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^{7/4} \left (b^2-4 a c\right ) \sqrt {a x+b x^3+c x^5}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 12.41 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.11 \[ \int \frac {1}{\sqrt {x} \left (a x+b x^3+c x^5\right )^{3/2}} \, dx=-\frac {2 \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \left (-4 a^2 c+2 b^2 x^2 \left (b+c x^2\right )+a \left (b^2-7 b c x^2-6 c^2 x^4\right )\right )-i \left (b^2-3 a c\right ) \left (-b+\sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (-b^3+4 a b c+b^2 \sqrt {b^2-4 a c}-3 a c \sqrt {b^2-4 a c}\right ) x \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{2 a^2 \left (b^2-4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x} \sqrt {x \left (a+b x^2+c x^4\right )}} \]

[In]

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

[Out]

-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])]*Sq
rt[(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 + 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])]*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)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*S
qrt[x]*Sqrt[x*(a + b*x^2 + c*x^4)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1135\) vs. \(2(454)=908\).

Time = 4.44 (sec) , antiderivative size = 1136, normalized size of antiderivative = 2.43

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

[In]

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

[Out]

-1/2*(x*(c*x^4+b*x^2+a))^(1/2)/x^(3/2)*(12*(-4*a*c+b^2)^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*a*c^2*x^4-4*(-
4*a*c+b^2)^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(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)*(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))*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)*(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^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)*(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))*x-12*(-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)*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)*(1/a*((-4*a*c+b^2)^(1/2)*x^2+b*x^2+2*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*(-4*a*c+b^2)^(1/2)*((-b+(-4*a
*c+b^2)^(1/2))/a)^(1/2)*a*b*c*x^2-4*(-4*a*c+b^2)^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*b^3*x^2+14*((-b+(-4*a
*c+b^2)^(1/2))/a)^(1/2)*a*b^2*c*x^2-4*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*b^4*x^2+8*(-4*a*c+b^2)^(1/2)*((-b+(-4*
a*c+b^2)^(1/2))/a)^(1/2)*a^2*c-2*(-4*a*c+b^2)^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*a*b^2+8*((-b+(-4*a*c+b^2
)^(1/2))/a)^(1/2)*a^2*b*c-2*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*a*b^3)/(c*x^4+b*x^2+a)/(4*a*c-b^2)/a^2/((-b+(-4*
a*c+b^2)^(1/2))/a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))

Fricas [F]

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

[In]

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

[Out]

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]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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