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

Optimal. Leaf size=425 \[ -\frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 c^{7/4} \sqrt{a x+b x^3+c x^5}}-\frac{2 b x^{3/2} \left (b^2-8 a c\right ) \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt [4]{a} b \sqrt{x} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (10 a c+b^2+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}} \]

[Out]

(-2*b*(b^2 - 8*a*c)*x^(3/2)*(a + b*x^2 + c*x^4))/(35*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[a*x + b*x^3 + c*x^5]
) + (Sqrt[x]*(b^2 + 10*a*c + 3*b*c*x^2)*Sqrt[a*x + b*x^3 + c*x^5])/(35*c) + (a*x + b*x^3 + c*x^5)^(3/2)/(7*Sqr
t[x]) + (2*a^(1/4)*b*(b^2 - 8*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])/(35*c^(7/4)*Sqrt[a*x + b*x^3 +
c*x^5]) - (a^(1/4)*(Sqrt[a]*Sqrt[c]*(b^2 - 20*a*c) + 2*b*(b^2 - 8*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])/(70*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5])

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Rubi [A]  time = 0.44855, antiderivative size = 425, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {1921, 1945, 1953, 1197, 1103, 1195} \[ -\frac{2 b x^{3/2} \left (b^2-8 a c\right ) \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt [4]{a} \sqrt{x} \left (\sqrt{a} \sqrt{c} \left (b^2-20 a c\right )+2 b \left (b^2-8 a c\right )\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{70 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{2 \sqrt [4]{a} b \sqrt{x} \left (b^2-8 a c\right ) \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{\frac{a+b x^2+c x^4}{\left (\sqrt{a}+\sqrt{c} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{35 c^{7/4} \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (10 a c+b^2+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*(b^2 - 8*a*c)*x^(3/2)*(a + b*x^2 + c*x^4))/(35*c^(3/2)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[a*x + b*x^3 + c*x^5]
) + (Sqrt[x]*(b^2 + 10*a*c + 3*b*c*x^2)*Sqrt[a*x + b*x^3 + c*x^5])/(35*c) + (a*x + b*x^3 + c*x^5)^(3/2)/(7*Sqr
t[x]) + (2*a^(1/4)*b*(b^2 - 8*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])/(35*c^(7/4)*Sqrt[a*x + b*x^3 +
c*x^5]) - (a^(1/4)*(Sqrt[a]*Sqrt[c]*(b^2 - 20*a*c) + 2*b*(b^2 - 8*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])/(70*c^(7/4)*Sqrt[a*x + b*x^3 + c*x^5])

Rule 1921

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a*
x^q + b*x^n + c*x^(2*n - q))^p)/(m + p*(2*n - q) + 1), x] + Dist[((n - q)*p)/(m + p*(2*n - q) + 1), Int[x^(m +
q)*(2*a + b*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] && GtQ[p, 0] && RationalQ[m, q] && Gt
Q[m + p*q + 1, -(n - q)] && NeQ[m + p*(2*n - q) + 1, 0]

Rule 1945

Int[(x_)^(m_.)*((c_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.))^(p_.)*((A_) + (B_.)*(x_)^(r_.)), x_Sym
bol] :> Simp[(x^(m + 1)*(b*B*(n - q)*p + A*c*(m + p*q + (n - q)*(2*p + 1) + 1) + B*c*(m + p*q + 2*(n - q)*p +
1)*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^p)/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)),
x] + Dist[((n - q)*p)/(c*(m + p*(2*n - q) + 1)*(m + p*q + (n - q)*(2*p + 1) + 1)), Int[x^(m + q)*Simp[2*a*A*c*
(m + p*q + (n - q)*(2*p + 1) + 1) - a*b*B*(m + p*q + 1) + (2*a*B*c*(m + p*q + 2*(n - q)*p + 1) + A*b*c*(m + p*
q + (n - q)*(2*p + 1) + 1) - b^2*B*(m + p*q + (n - q)*p + 1))*x^(n - q), x]*(a*x^q + b*x^n + c*x^(2*n - q))^(p
- 1), 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] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q, -(n - q) - 1] && NeQ[m + p*(2*n - q) +
1, 0] && NeQ[m + p*q + (n - q)*(2*p + 1) + 1, 0]

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{\left (a x+b x^3+c x^5\right )^{3/2}}{x^{3/2}} \, dx &=\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}}+\frac{3}{7} \int \frac{\left (2 a+b x^2\right ) \sqrt{a x+b x^3+c x^5}}{\sqrt{x}} \, dx\\ &=\frac{\sqrt{x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}}+\frac{\int \frac{\sqrt{x} \left (-a \left (b^2-20 a c\right )-2 b \left (b^2-8 a c\right ) x^2\right )}{\sqrt{a x+b x^3+c x^5}} \, dx}{35 c}\\ &=\frac{\sqrt{x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}}+\frac{\left (\sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{-a \left (b^2-20 a c\right )-2 b \left (b^2-8 a c\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c \sqrt{a x+b x^3+c x^5}}\\ &=\frac{\sqrt{x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}}+\frac{\left (2 \sqrt{a} b \left (b^2-8 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}{35 c^{3/2} \sqrt{a x+b x^3+c x^5}}-\frac{\left (\sqrt{a} \left (\sqrt{a} \left (b^2-20 a c\right )+\frac{2 b \left (b^2-8 a c\right )}{\sqrt{c}}\right ) \sqrt{x} \sqrt{a+b x^2+c x^4}\right ) \int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx}{35 c \sqrt{a x+b x^3+c x^5}}\\ &=-\frac{2 b \left (b^2-8 a c\right ) x^{3/2} \left (a+b x^2+c x^4\right )}{35 c^{3/2} \left (\sqrt{a}+\sqrt{c} x^2\right ) \sqrt{a x+b x^3+c x^5}}+\frac{\sqrt{x} \left (b^2+10 a c+3 b c x^2\right ) \sqrt{a x+b x^3+c x^5}}{35 c}+\frac{\left (a x+b x^3+c x^5\right )^{3/2}}{7 \sqrt{x}}+\frac{2 \sqrt [4]{a} b \left (b^2-8 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 )}{35 c^{7/4} \sqrt{a x+b x^3+c x^5}}-\frac{\sqrt [4]{a} \left (\sqrt{a} \left (b^2-20 a c\right )+\frac{2 b \left (b^2-8 a c\right )}{\sqrt{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 )}{70 c^{5/4} \sqrt{a x+b x^3+c x^5}}\\ \end{align*}

Mathematica [C]  time = 1.7485, size = 540, normalized size = 1.27 \[ \frac{\sqrt{x} \left (i \left (-20 a^2 c^2+b^3 \sqrt{b^2-4 a c}+9 a b^2 c-8 a b c \sqrt{b^2-4 a c}-b^4\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{2} x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+2 c x \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \left (15 a^2 c+a \left (b^2+23 b c x^2+20 c^2 x^4\right )+x^2 \left (9 b^2 c x^2+b^3+13 b c^2 x^4+5 c^3 x^6\right )\right )-i b \left (b^2-8 a c\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x^2}{\sqrt{b^2-4 a c}+b}} \sqrt{\frac{-2 \sqrt{b^2-4 a c}+2 b+4 c x^2}{b-\sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\sqrt{2} \sqrt{\frac{c}{b+\sqrt{b^2-4 a c}}} x\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )\right )}{70 c^2 \sqrt{\frac{c}{\sqrt{b^2-4 a c}+b}} \sqrt{x \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[x]*(2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(15*a^2*c + a*(b^2 + 23*b*c*x^2 + 20*c^2*x^4) + x^2*(b^3 + 9*b
^2*c*x^2 + 13*b*c^2*x^4 + 5*c^3*x^6)) - I*b*(b^2 - 8*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])]*Ellip
ticE[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^4 + 9*a*b^2*c - 20*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 8*a*b*c*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])]*El
lipticF[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])]
))/(70*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[x*(a + b*x^2 + c*x^4)])

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Maple [B]  time = 0.023, size = 1394, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/70*(x*(c*x^4+b*x^2+a))^(1/2)*(-10*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^9*b*c^3-10*(-4*a*c+b^2)^(1/2)*(1/a*
(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^9*c^3-26*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^7*b^2*c^2-26*(-4*a*c+b^2)^(1/2
)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^7*b*c^2-40*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^5*a*b*c^2-40*(-4*a*c+
b^2)^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^5*a*c^2-18*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^5*b^3*c-18*(
-4*a*c+b^2)^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^5*b^2*c-46*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^3*a*b
^2*c-46*(-4*a*c+b^2)^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^3*a*b*c-2*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)
*x^3*b^4-2*(-4*a*c+b^2)^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x^3*b^3+12*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))*(-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^2*b*c-20*(-4*a*c+b^2)^(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))*(-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^2*c-3*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))*(-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*b^3+(-4*a*c+b^2)^(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))*(-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*b^2-
32*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*(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^2*b*
c+4*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*(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*b^3
-30*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x*a^2*b*c-30*(-4*a*c+b^2)^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x*
a^2*c-2*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x*a*b^3-2*(-4*a*c+b^2)^(1/2)*(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)*x
*a*b^2)/x^(1/2)/(c*x^4+b*x^2+a)/c/(1/a*(-b+(-4*a*c+b^2)^(1/2)))^(1/2)/(b+(-4*a*c+b^2)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c x^{5} + b x^{3} + a x}{\left (c x^{4} + b x^{2} + a\right )}}{\sqrt{x}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*x^5 + b*x^3 + a*x)*(c*x^4 + b*x^2 + a)/sqrt(x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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