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

Optimal. Leaf size=765 \[ -\frac {32 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt {3}-7\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt {b}}+\frac {48 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {96 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {3}{7} x \left (a-b x^2\right )^{2/3} \]

[Out]

-3/7*x*(-b*x^2+a)^(2/3)+96/7*a*x/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))+4*2^(1/3)*a^(7/6)*arctanh(x*b^(1/2)/a
^(1/6)/(a^(1/3)+2^(1/3)*(-b*x^2+a)^(1/3)))/b^(1/2)-4/3*2^(1/3)*a^(7/6)*arctanh(x*b^(1/2)/a^(1/2))/b^(1/2)+4/3*
2^(1/3)*a^(7/6)*arctan(a^(1/6)*(a^(1/3)-2^(1/3)*(-b*x^2+a)^(1/3))*3^(1/2)/x/b^(1/2))*3^(1/2)/b^(1/2)+4/3*2^(1/
3)*a^(7/6)*arctan(3^(1/2)*a^(1/2)/x/b^(1/2))*3^(1/2)/b^(1/2)-32/7*3^(3/4)*a^(4/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*E
llipticF((-(-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))*2^(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)/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)+48/7*3^(1/4)*a^(4/3)*(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))/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 = 0.44, antiderivative size = 765, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {416, 530, 235, 304, 219, 1879, 393} \[ \frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt {b}}-\frac {32 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {48 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}-\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {96 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {3}{7} x \left (a-b x^2\right )^{2/3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*x*(a - b*x^2)^(2/3))/7 + (96*a*x)/(7*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) + (4*2^(1/3)*a^(7/6)*Arc
Tan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) + (4*2^(1/3)*a^(7/6)*ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) -
2^(1/3)*(a - b*x^2)^(1/3)))/(Sqrt[b]*x)])/(Sqrt[3]*Sqrt[b]) - (4*2^(1/3)*a^(7/6)*ArcTanh[(Sqrt[b]*x)/Sqrt[a]])
/(3*Sqrt[b]) + (4*2^(1/3)*a^(7/6)*ArcTanh[(Sqrt[b]*x)/(a^(1/6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))])/Sqrt[b
] + (48*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(4/3)*(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]])/(7*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)]) - (32*Sqrt[2]*3^(3/4)*a^(4/3)*(
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]*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]])/(7*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 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 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 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]

Rule 416

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1)*(c
+ d*x^n)^(q - 1))/(b*(n*(p + q) + 1)), x] + Dist[1/(b*(n*(p + q) + 1)), Int[(a + b*x^n)^p*(c + d*x^n)^(q - 2)
*Simp[c*(b*c*(n*(p + q) + 1) - a*d) + d*(b*c*(n*(p + 2*q - 1) + 1) - a*d*(n*(q - 1) + 1))*x^n, x], x], x] /; F
reeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 1] && NeQ[n*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntB
inomialQ[a, b, c, d, n, p, q, x]

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 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]

Rubi steps

\begin {align*} \int \frac {\left (a-b x^2\right )^{5/3}}{3 a+b x^2} \, dx &=-\frac {3}{7} x \left (a-b x^2\right )^{2/3}+\frac {3 \int \frac {\frac {16 a^2 b}{3}-\frac {32}{3} a b^2 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{7 b}\\ &=-\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {1}{7} (32 a) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx+\left (16 a^2\right ) \int \frac {1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx\\ &=-\frac {3}{7} x \left (a-b x^2\right )^{2/3}+\frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \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 )}{\sqrt {3} \sqrt {b}}-\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \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 )}{\sqrt {b}}+\frac {\left (48 a \sqrt {-b x^2}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=-\frac {3}{7} x \left (a-b x^2\right )^{2/3}+\frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \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 )}{\sqrt {3} \sqrt {b}}-\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \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 )}{\sqrt {b}}-\frac {\left (48 a \sqrt {-b x^2}\right ) \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 )}{7 b x}+\frac {\left (48 \sqrt {2 \left (2+\sqrt {3}\right )} a^{4/3} \sqrt {-b x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=-\frac {3}{7} x \left (a-b x^2\right )^{2/3}+\frac {96 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{\sqrt {3} \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \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 )}{\sqrt {3} \sqrt {b}}-\frac {4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 \sqrt {b}}+\frac {4 \sqrt [3]{2} a^{7/6} \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 )}{\sqrt {b}}+\frac {48 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \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 )}{7 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 {32 \sqrt {2} 3^{3/4} a^{4/3} \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}} 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 )}{7 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 = 0.15, size = 231, normalized size = 0.30 \[ \frac {x \left (27 \left (\frac {48 a^3 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )-F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )+9 a F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}-a+b x^2\right )-32 b x^2 \sqrt [3]{1-\frac {b x^2}{a}} F_1\left (\frac {3}{2};\frac {1}{3},1;\frac {5}{2};\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}{63 \sqrt [3]{a-b x^2}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x*(-32*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 + b*x^2 + (
48*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]))))))/(63*(a - b*x^2)^(1/3))

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {5}{3}}}{b x^{2} + 3 \, a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{2}+a \right )^{\frac {5}{3}}}{b \,x^{2}+3 a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-b x^{2} + a\right )}^{\frac {5}{3}}}{b x^{2} + 3 \, a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a-b\,x^2\right )}^{5/3}}{b\,x^2+3\,a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a - b x^{2}\right )^{\frac {5}{3}}}{3 a + b x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a - b*x**2)**(5/3)/(3*a + b*x**2), x)

________________________________________________________________________________________