3.2.28 \(\int \frac {1}{\sqrt [3]{a-b x^2} (3 a+b x^2)^3} \, dx\) [128]

Optimal. Leaf size=818 \[ \frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}+\frac {5 x \left (a-b x^2\right )^{2/3}}{288 a^3 \left (3 a+b x^2\right )}-\frac {5 x}{288 a^3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} a^{17/6} \sqrt {b}}-\frac {5 \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 )}{192\ 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 {5 \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 )}{144 \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}}} \]

[Out]

1/48*x*(-b*x^2+a)^(2/3)/a^2/(b*x^2+3*a)^2+5/288*x*(-b*x^2+a)^(2/3)/a^3/(b*x^2+3*a)-5/288*x/a^3/(-(-b*x^2+a)^(1
/3)+a^(1/3)*(1-3^(1/2)))+5/288*arctanh(x*b^(1/2)/a^(1/6)/(a^(1/3)+2^(1/3)*(-b*x^2+a)^(1/3)))*2^(1/3)/a^(17/6)/
b^(1/2)-5/864*arctanh(x*b^(1/2)/a^(1/2))*2^(1/3)/a^(17/6)/b^(1/2)+5/864*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(-b*x^
2+a)^(1/3))*3^(1/2)/x/b^(1/2))*2^(1/3)/a^(17/6)*3^(1/2)/b^(1/2)+5/864*arctan(3^(1/2)*a^(1/2)/x/b^(1/2))*2^(1/3
)/a^(17/6)*3^(1/2)/b^(1/2)+5/864*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticF((-(-b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/
(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/(-
(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*3^(3/4)/a^(8/3)/b/x*2^(1/2)/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3)
)/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)-5/576*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticE((-(-b*x^2+a)^(1/
3)+a^(1/3)*(1+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I*3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1
/3)+(-b*x^2+a)^(2/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))*3^(1/4)/a^(8/
3)/b/x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)

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Rubi [A]
time = 0.47, antiderivative size = 818, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {425, 541, 544, 241, 310, 225, 1893, 402} \begin {gather*} \frac {5 \left (a-b x^2\right )^{2/3} x}{288 a^3 \left (b x^2+3 a\right )}-\frac {5 x}{288 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}{48 a^2 \left (b x^2+3 a\right )^2}+\frac {5 \text {ArcTan}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {5 \text {ArcTan}\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 )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} a^{17/6} \sqrt {b}}-\frac {5 \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 (\text {ArcSin}\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 )}{192\ 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 {5 \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 (\text {ArcSin}\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 )}{144 \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a - b*x^2)^(2/3))/(48*a^2*(3*a + b*x^2)^2) + (5*x*(a - b*x^2)^(2/3))/(288*a^3*(3*a + b*x^2)) - (5*x)/(288*
a^3*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (5*ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(144*2^(2/3)*Sqrt
[3]*a^(17/6)*Sqrt[b]) + (5*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(144*2
^(2/3)*Sqrt[3]*a^(17/6)*Sqrt[b]) - (5*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])/(432*2^(2/3)*a^(17/6)*Sqrt[b]) + (5*ArcTan
h[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/(144*2^(2/3)*a^(17/6)*Sqrt[b]) - (5*Sqrt[2 + S
qrt[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 - Sqr
t[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]])/(192*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)]) + (5*(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[Ar
cSin[((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]]
)/(144*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 225

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

Rule 241

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 310

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 + Sqrt[3]))*(s/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 402

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan
[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[Sqr
t[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]

Rule 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, n, p, q, x]

Rule 541

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 544

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 1893

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]/(r^2
*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/(
(1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqr
t[3])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^3} \, dx &=\frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}-\frac {\int \frac {-15 a b+\frac {5 b^2 x^2}{3}}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx}{48 a^2 b}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}+\frac {5 x \left (a-b x^2\right )^{2/3}}{288 a^3 \left (3 a+b x^2\right )}+\frac {\int \frac {100 a^2 b^2+\frac {20}{3} a b^3 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{1152 a^4 b^2}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}+\frac {5 x \left (a-b x^2\right )^{2/3}}{288 a^3 \left (3 a+b x^2\right )}+\frac {5 \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{864 a^3}+\frac {5 \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{72 a^2}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}+\frac {5 x \left (a-b x^2\right )^{2/3}}{288 a^3 \left (3 a+b x^2\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} a^{17/6} \sqrt {b}}-\frac {\left (5 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{576 a^3 b x}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}+\frac {5 x \left (a-b x^2\right )^{2/3}}{288 a^3 \left (3 a+b x^2\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {\left (5 \sqrt {-b x^2}\right ) \text {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 )}{576 a^3 b x}-\frac {\left (5 \sqrt {\frac {1}{2} \left (2+\sqrt {3}\right )} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{288 a^{8/3} b x}\\ &=\frac {x \left (a-b x^2\right )^{2/3}}{48 a^2 \left (3 a+b x^2\right )^2}+\frac {5 x \left (a-b x^2\right )^{2/3}}{288 a^3 \left (3 a+b x^2\right )}-\frac {5 x}{288 a^3 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} \sqrt {3} a^{17/6} \sqrt {b}}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{432\ 2^{2/3} a^{17/6} \sqrt {b}}+\frac {5 \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 )}{144\ 2^{2/3} a^{17/6} \sqrt {b}}-\frac {5 \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 )}{192\ 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 {5 \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 )}{144 \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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 10.13, size = 255, normalized size = 0.31 \begin {gather*} \frac {x \left (\frac {27 a \left (a-b x^2\right ) \left (21 a+5 b x^2\right )}{\left (3 a+b x^2\right )^2}+5 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 )+\frac {6075 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 (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 )+2 b x^2 \left (-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 )+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 )\right )\right )}\right )}{7776 a^4 \sqrt [3]{a-b x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*((27*a*(a - b*x^2)*(21*a + 5*b*x^2))/(3*a + b*x^2)^2 + 5*b*x^2*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1,
5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + (6075*a^3*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/((3*a + b*x
^2)*(9*a*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2, 5/2, (b*x^2)/
a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a])))))/(7776*a^4*(a - b*x^2)^(1/3))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}} \left (b \,x^{2}+3 a \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^2 + 3*a)^3*(-b*x^2 + a)^(1/3)), x)

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{a - b x^{2}} \left (3 a + b x^{2}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a - b*x**2)**(1/3)*(3*a + b*x**2)**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/((b*x^2 + 3*a)^3*(-b*x^2 + a)^(1/3)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a-b\,x^2\right )}^{1/3}\,{\left (b\,x^2+3\,a\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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