3.2.34 \(\int \frac {1}{(a-b x^2)^{4/3} (3 a+b x^2)^3} \, dx\) [134]

Optimal. Leaf size=849 \[ \frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {19 x}{1152 a^4 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {19 \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 )}{768\ 3^{3/4} a^{11/3} 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 {19 \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 )}{576 \sqrt {2} \sqrt [4]{3} a^{11/3} 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/a^2/(-b*x^2+a)^(1/3)/(b*x^2+3*a)^2+17/192*x/a^3/(-b*x^2+a)^(1/3)/(b*x^2+3*a)-19/1152*x*(-b*x^2+a)^(2/3)
/a^4/(b*x^2+3*a)+19/1152*x/a^4/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))+7/576*arctanh(x*b^(1/2)/a^(1/6)/(a^(1/3
)+2^(1/3)*(-b*x^2+a)^(1/3)))*2^(1/3)/a^(23/6)/b^(1/2)-7/1728*arctanh(x*b^(1/2)/a^(1/2))*2^(1/3)/a^(23/6)/b^(1/
2)+7/1728*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^(23/6)*3^(1/2)/b^(1/2
)+7/1728*arctan(3^(1/2)*a^(1/2)/x/b^(1/2))*2^(1/3)/a^(23/6)*3^(1/2)/b^(1/2)-19/3456*(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^(1
1/3)/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)+19/2304
*(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^(11/3)/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)

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 849, normalized size of antiderivative = 1.00, 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 = {425, 541, 544, 241, 310, 225, 1893, 402} \begin {gather*} -\frac {19 \left (a-b x^2\right )^{2/3} x}{1152 a^4 \left (b x^2+3 a\right )}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}+\frac {19 x}{1152 a^4 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )^2}+\frac {7 \text {ArcTan}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}+\frac {7 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {19 \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 )}{768\ 3^{3/4} a^{11/3} 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 {19 \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 )}{576 \sqrt {2} \sqrt [4]{3} a^{11/3} 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)^(4/3)*(3*a + b*x^2)^3),x]

[Out]

x/(48*a^2*(a - b*x^2)^(1/3)*(3*a + b*x^2)^2) + (17*x)/(192*a^3*(a - b*x^2)^(1/3)*(3*a + b*x^2)) - (19*x*(a - b
*x^2)^(2/3))/(1152*a^4*(3*a + b*x^2)) + (19*x)/(1152*a^4*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (7*Arc
Tan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(288*2^(2/3)*Sqrt[3]*a^(23/6)*Sqrt[b]) + (7*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/
3) - 2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(288*2^(2/3)*Sqrt[3]*a^(23/6)*Sqrt[b]) - (7*ArcTanh[(Sqrt[b]*x)
/Sqrt[a]])/(864*2^(2/3)*a^(23/6)*Sqrt[b]) + (7*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/
3)))])/(288*2^(2/3)*a^(23/6)*Sqrt[b]) + (19*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]])/(768
*3^(3/4)*a^(11/3)*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)]) - (19*(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]])/(576*Sqrt[2]*3^(1/4)*a^(11/3)*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}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^3} \, dx &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}-\frac {\int \frac {-15 a b+\frac {11 b^2 x^2}{3}}{\left (a-b x^2\right )^{4/3} \left (3 a+b x^2\right )^2} \, dx}{48 a^2 b}\\ &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {\int \frac {-6 a^2 b^2-\frac {170}{9} a b^3 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2} \, dx}{128 a^4 b^2}\\ &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {\int \frac {\frac {296 a^3 b^3}{3}-\frac {152}{9} a^2 b^4 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{3072 a^6 b^3}\\ &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}-\frac {19 \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{3456 a^4}+\frac {7 \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{144 a^3}\\ &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {\left (19 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{2304 a^4 b x}\\ &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {7 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}-\frac {\left (19 \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 )}{2304 a^4 b x}+\frac {\left (19 \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 )}{1152 a^{11/3} b x}\\ &=\frac {x}{48 a^2 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )^2}+\frac {17 x}{192 a^3 \sqrt [3]{a-b x^2} \left (3 a+b x^2\right )}-\frac {19 x \left (a-b x^2\right )^{2/3}}{1152 a^4 \left (3 a+b x^2\right )}+\frac {19 x}{1152 a^4 \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \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 )}{288\ 2^{2/3} \sqrt {3} a^{23/6} \sqrt {b}}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{864\ 2^{2/3} a^{23/6} \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 )}{288\ 2^{2/3} a^{23/6} \sqrt {b}}+\frac {19 \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 )}{768\ 3^{3/4} a^{11/3} 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 {19 \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 )}{576 \sqrt {2} \sqrt [4]{3} a^{11/3} 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 = 10.17, size = 256, normalized size = 0.30 \begin {gather*} \frac {x \left (-19 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 {27 a \left (273 a^2+140 a b x^2+19 b^2 x^4+\frac {333 a^2 \left (3 a+b x^2\right ) 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 )}{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 )}{\left (3 a+b x^2\right )^2}\right )}{31104 a^5 \sqrt [3]{a-b x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^3),x]

[Out]

(x*(-19*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] + (27*a*(273*a^2 + 1
40*a*b*x^2 + 19*b^2*x^4 + (333*a^2*(3*a + b*x^2)*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*A
ppellF1[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]))))/(3*a + b*x^2)^2))/(31104*a^5*(a - b*x^2)^
(1/3))

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}} \left (b \,x^{2}+3 a \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-b*x^2+a)^(4/3)/(b*x^2+3*a)^3,x)

[Out]

int(1/(-b*x^2+a)^(4/3)/(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)^(4/3)/(b*x^2+3*a)^3,x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + 3*a)^3*(-b*x^2 + a)^(4/3)), 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)^(4/3)/(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}{\left (a - b x^{2}\right )^{\frac {4}{3}} \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)**(4/3)/(b*x**2+3*a)**3,x)

[Out]

Integral(1/((a - b*x**2)**(4/3)*(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)^(4/3)/(b*x^2+3*a)^3,x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + 3*a)^3*(-b*x^2 + a)^(4/3)), 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 )}^{4/3}\,{\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)^(4/3)*(3*a + b*x^2)^3),x)

[Out]

int(1/((a - b*x^2)^(4/3)*(3*a + b*x^2)^3), x)

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