3.115 \(\int \frac{(a-b x^2)^{2/3}}{(3 a+b x^2)^4} \, dx\)

Optimal. Leaf size=849 \[ \frac{\left (a-b x^2\right )^{2/3} x}{144 a^3 \left (b x^2+3 a\right )}-\frac{x}{144 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\left (a-b x^2\right )^{2/3} x}{54 a^2 \left (b x^2+3 a\right )^2}+\frac{\left (a-b x^2\right )^{2/3} x}{18 a \left (b x^2+3 a\right )^3}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3888\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{1296\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}+\frac{\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

[Out]

(x*(a - b*x^2)^(2/3))/(18*a*(3*a + b*x^2)^3) + (x*(a - b*x^2)^(2/3))/(54*a^2*(3*a + b*x^2)^2) + (x*(a - b*x^2)
^(2/3))/(144*a^3*(3*a + b*x^2)) - x/(144*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (7*ArcTan[(Sqrt[3]
*Sqrt[a])/(Sqrt[b]*x)])/(1296*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + (7*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3
)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(1296*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (7*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])
/(3888*2^(2/3)*a^(17/6)*Sqrt[b]) + (7*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(1
296*2^(2/3)*a^(17/6)*Sqrt[b]) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a -
b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3
])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(96*3^(3/4)*a^(
8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + ((a
^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1
/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3
) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(72*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^
2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

________________________________________________________________________________________

Rubi [A]  time = 0.650621, antiderivative size = 849, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {412, 527, 530, 235, 304, 219, 1879, 393} \[ \frac{\left (a-b x^2\right )^{2/3} x}{144 a^3 \left (b x^2+3 a\right )}-\frac{x}{144 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{\left (a-b x^2\right )^{2/3} x}{54 a^2 \left (b x^2+3 a\right )^2}+\frac{\left (a-b x^2\right )^{2/3} x}{18 a \left (b x^2+3 a\right )^3}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3888\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{1296\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 3^{3/4} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x}+\frac{\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a-b x^2} \sqrt [3]{a}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} a^{8/3} b \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} x} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(2/3)/(3*a + b*x^2)^4,x]

[Out]

(x*(a - b*x^2)^(2/3))/(18*a*(3*a + b*x^2)^3) + (x*(a - b*x^2)^(2/3))/(54*a^2*(3*a + b*x^2)^2) + (x*(a - b*x^2)
^(2/3))/(144*a^3*(3*a + b*x^2)) - x/(144*a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (7*ArcTan[(Sqrt[3]
*Sqrt[a])/(Sqrt[b]*x)])/(1296*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) + (7*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3
)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(1296*2^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (7*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])
/(3888*2^(2/3)*a^(17/6)*Sqrt[b]) + (7*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(1
296*2^(2/3)*a^(17/6)*Sqrt[b]) - (Sqrt[2 + Sqrt[3]]*(a^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a -
b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqrt[3
])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(96*3^(3/4)*a^(
8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)]) + ((a
^(1/3) - (a - b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3)*(a - b*x^2)^(1/3) + (a - b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1
/3) - (a - b*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))/((1 - Sqrt[3])*a^(1/3
) - (a - b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(72*Sqrt[2]*3^(1/4)*a^(8/3)*b*x*Sqrt[-((a^(1/3)*(a^(1/3) - (a - b*x^
2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2)])

Rule 412

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^q)/(a*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(n*(p
+ 1) + 1) + d*(n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p,
-1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, n, p, q, x]

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 530

Int[(((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[f/d,
Int[(a + b*x^n)^p, x], x] + Dist[(d*e - c*f)/d, Int[(a + b*x^n)^p/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, p, n}, x]

Rule 235

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Dist[(3*Sqrt[b*x^2])/(2*b*x), Subst[Int[x/Sqrt[-a + x^3], x], x
, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b}, x]

Rule 304

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, -Dist[(S
qrt[2]*s)/(Sqrt[2 - Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a
+ b*x^3], x], x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 219

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 - Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 + Sqrt[3
])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[-((s*(s + r*x))/((1 - S
qrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b}, x] && NegQ[a]

Rule 1879

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Simplify[((1 + Sqrt[3])*d)/c]]
, s = Denom[Simplify[((1 + Sqrt[3])*d)/c]]}, Simp[(2*d*s^3*Sqrt[a + b*x^3])/(a*r^2*((1 - Sqrt[3])*s + r*x)), x
] + Simp[(3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]*Elli
pticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[-((s
*(s + r*x))/((1 - Sqrt[3])*s + r*x)^2)]), x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rule 393

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-(b/a), 2]}, Simp[(q*ArcT
an[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (Simp[(q*ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a +
b*x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] - Simp[(q*ArcTanh[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[(
Sqrt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^4} \, dx &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}-\frac{\int \frac{-5 a+\frac{11 b x^2}{3}}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^3} \, dx}{18 a}\\ &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}+\frac{x \left (a-b x^2\right )^{2/3}}{54 a^2 \left (3 a+b x^2\right )^2}+\frac{\int \frac{64 a^2 b-\frac{80}{3} a b^2 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx}{864 a^3 b}\\ &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}+\frac{x \left (a-b x^2\right )^{2/3}}{54 a^2 \left (3 a+b x^2\right )^2}+\frac{x \left (a-b x^2\right )^{2/3}}{144 a^3 \left (3 a+b x^2\right )}-\frac{\int \frac{-368 a^3 b^2-48 a^2 b^3 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{20736 a^5 b^2}\\ &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}+\frac{x \left (a-b x^2\right )^{2/3}}{54 a^2 \left (3 a+b x^2\right )^2}+\frac{x \left (a-b x^2\right )^{2/3}}{144 a^3 \left (3 a+b x^2\right )}+\frac{\int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{432 a^3}+\frac{7 \int \frac{1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{648 a^2}\\ &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}+\frac{x \left (a-b x^2\right )^{2/3}}{54 a^2 \left (3 a+b x^2\right )^2}+\frac{x \left (a-b x^2\right )^{2/3}}{144 a^3 \left (3 a+b x^2\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3888\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{1296\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{\sqrt{-b x^2} \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{288 a^3 b x}\\ &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}+\frac{x \left (a-b x^2\right )^{2/3}}{54 a^2 \left (3 a+b x^2\right )^2}+\frac{x \left (a-b x^2\right )^{2/3}}{144 a^3 \left (3 a+b x^2\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3888\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{1296\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{\sqrt{-b x^2} \operatorname{Subst}\left (\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{288 a^3 b x}-\frac{\left (\sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{144 a^{8/3} b x}\\ &=\frac{x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )^3}+\frac{x \left (a-b x^2\right )^{2/3}}{54 a^2 \left (3 a+b x^2\right )^2}+\frac{x \left (a-b x^2\right )^{2/3}}{144 a^3 \left (3 a+b x^2\right )}-\frac{x}{144 a^3 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}+\frac{7 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{1296\ 2^{2/3} \sqrt{3} a^{17/6} \sqrt{b}}-\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3888\ 2^{2/3} a^{17/6} \sqrt{b}}+\frac{7 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{1296\ 2^{2/3} a^{17/6} \sqrt{b}}-\frac{\sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{96\ 3^{3/4} a^{8/3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{72 \sqrt{2} \sqrt [4]{3} a^{8/3} b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\\ \end{align*}

Mathematica [C]  time = 0.180592, size = 265, normalized size = 0.31 \[ \frac{x \left (\frac{621 a^3 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}+\frac{9 a \left (a-b x^2\right ) \left (75 a^2+26 a b x^2+3 b^2 x^4\right )}{\left (3 a+b x^2\right )^3}+b x^2 \sqrt [3]{1-\frac{b x^2}{a}} F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}{3888 a^4 \sqrt [3]{a-b x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a - b*x^2)^(2/3)/(3*a + b*x^2)^4,x]

[Out]

(x*((9*a*(a - b*x^2)*(75*a^2 + 26*a*b*x^2 + 3*b^2*x^4))/(3*a + b*x^2)^3 + b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1
[3/2, 1/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + (621*a^3*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)]
)/((3*a + b*x^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -(b*x^2)/(3*a)] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2,
5/2, (b*x^2)/a, -(b*x^2)/(3*a)] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -(b*x^2)/(3*a)])))))/(3888*a^4*(a - b*
x^2)^(1/3))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( b{x}^{2}+3\,a \right ) ^{4}} \left ( -b{x}^{2}+a \right ) ^{{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(2/3)/(b*x^2+3*a)^4,x)

[Out]

int((-b*x^2+a)^(2/3)/(b*x^2+3*a)^4,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(2/3)/(b*x^2+3*a)^4,x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(2/3)/(b*x^2 + 3*a)^4, x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(2/3)/(b*x^2+3*a)^4,x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+a)**(2/3)/(b*x**2+3*a)**4,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(2/3)/(b*x^2+3*a)^4,x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(2/3)/(b*x^2 + 3*a)^4, x)