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

Optimal. Leaf size=391 \[ -\frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \sqrt{a x+b x^3+c x^5}}+\frac{b \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}+\frac{x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}-\frac{b \sqrt{c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.248059, antiderivative size = 391, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1924, 1953, 1197, 1103, 1195} \[ \frac{b \sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{a^{3/4} \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt [4]{c} \sqrt{x} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt{a} \sqrt{c}\right ) \sqrt{a x+b x^3+c x^5}}+\frac{x^{3/2} \left (-2 a c+b^2+b c x^2\right )}{a \left (b^2-4 a c\right ) \sqrt{a x+b x^3+c x^5}}-\frac{b \sqrt{c} x^{3/2} \left (a+b x^2+c x^4\right )}{a \left (b^2-4 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}} \]

Antiderivative was successfully verified.

[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 1924

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 1953

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]

Rule 1197

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 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

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)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0
]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin{align*} \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{\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]  time = 1.08292, size = 463, normalized size = 1.18 \[ -\frac{\sqrt{x} \left (-i \left (b \sqrt{b^2-4 a c}+4 a c-b^2\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )-4 x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (-2 a c+b^2+b c x^2\right )+i b \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{4 a \left (b^2-4 a c\right ) \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.027, size = 533, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( c{x}^{4}+b{x}^{2}+a \right ) a \left ( 4\,ac-{b}^{2} \right ) }\sqrt{x \left ( c{x}^{4}+b{x}^{2}+a \right ) } \left ( -\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}{x}^{3}{b}^{2}c-\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}\sqrt{-4\,ac+{b}^{2}}{x}^{3}bc+c\sqrt{-2\,{\frac{{x}^{2}\sqrt{-4\,ac+{b}^{2}}-b{x}^{2}-2\,a}{a}}}\sqrt{{\frac{1}{a} \left ({x}^{2}\sqrt{-4\,ac+{b}^{2}}+b{x}^{2}+2\,a \right ) }}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{ac} \left ( b\sqrt{-4\,ac+{b}^{2}}-2\,ac+{b}^{2} \right ) }}} \right ) a\sqrt{-4\,ac+{b}^{2}}+bc\sqrt{-2\,{\frac{{x}^{2}\sqrt{-4\,ac+{b}^{2}}-b{x}^{2}-2\,a}{a}}}\sqrt{{\frac{1}{a} \left ({x}^{2}\sqrt{-4\,ac+{b}^{2}}+b{x}^{2}+2\,a \right ) }}a{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}},{\frac{\sqrt{2}}{2}\sqrt{{\frac{1}{ac} \left ( b\sqrt{-4\,ac+{b}^{2}}-2\,ac+{b}^{2} \right ) }}} \right ) +2\,\sqrt{{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}xabc+2\,\sqrt{{\frac{-b+\sqrt{-4\,ac+{b}^{2}}}{a}}}\sqrt{-4\,ac+{b}^{2}}xac-\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}x{b}^{3}-\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}\sqrt{-4\,ac+{b}^{2}}x{b}^{2} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{{\frac{1}{a} \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) }}}} \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{5} + b x^{3} + a x} \sqrt{x}}{c^{2} x^{9} + 2 \, b c x^{7} +{\left (b^{2} + 2 \, a c\right )} x^{5} + 2 \, a b x^{3} + a^{2} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{\left (x \left (a + b x^{2} + c x^{4}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{3}{2}}}{{\left (c x^{5} + b x^{3} + a x\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)