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

Optimal. Leaf size=648 \[ \frac{24192 \sqrt{2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{1235 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{72576 a^4 x}{1235 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{36288 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1235 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2 \]

[Out]

(18144*a^3*x*(a - b*x^2)^(2/3))/1235 - (23544*a^2*x*(a - b*x^2)^(5/3))/6175 - (378*a*x*(a - b*x^2)^(5/3)*(3*a
+ b*x^2))/475 - (3*x*(a - b*x^2)^(5/3)*(3*a + b*x^2)^2)/25 - (72576*a^4*x)/(1235*((1 - Sqrt[3])*a^(1/3) - (a -
b*x^2)^(1/3))) - (36288*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(13/3)*(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]])/(1235*
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)]) + (24192*S
qrt[2]*3^(3/4)*a^(13/3)*(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]])/(1235*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)])

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Rubi [A]  time = 0.597054, antiderivative size = 648, normalized size of antiderivative = 1., 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 = {416, 528, 388, 195, 235, 304, 219, 1879} \[ -\frac{72576 a^4 x}{1235 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}+\frac{24192 \sqrt{2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1235 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{36288 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1235 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18144*a^3*x*(a - b*x^2)^(2/3))/1235 - (23544*a^2*x*(a - b*x^2)^(5/3))/6175 - (378*a*x*(a - b*x^2)^(5/3)*(3*a
+ b*x^2))/475 - (3*x*(a - b*x^2)^(5/3)*(3*a + b*x^2)^2)/25 - (72576*a^4*x)/(1235*((1 - Sqrt[3])*a^(1/3) - (a -
b*x^2)^(1/3))) - (36288*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(13/3)*(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]])/(1235*
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)]) + (24192*S
qrt[2]*3^(3/4)*a^(13/3)*(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]])/(1235*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 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 528

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

Rule 388

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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]

Rubi steps

\begin{align*} \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx &=-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac{3 \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \left (-78 a^2 b-42 a b^2 x^2\right ) \, dx}{25 b}\\ &=-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac{9 \int \left (a-b x^2\right )^{2/3} \left (1608 a^3 b^2+872 a^2 b^3 x^2\right ) \, dx}{475 b^2}\\ &=-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac{\left (42336 a^3\right ) \int \left (a-b x^2\right )^{2/3} \, dx}{1235}\\ &=\frac{18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac{\left (24192 a^4\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{1235}\\ &=\frac{18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac{\left (36288 a^4 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1235 b x}\\ &=\frac{18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac{\left (36288 a^4 \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 )}{1235 b x}-\frac{\left (36288 \sqrt{2 \left (2+\sqrt{3}\right )} a^{13/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1235 b x}\\ &=\frac{18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac{23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac{3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac{72576 a^4 x}{1235 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{36288 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \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 )}{1235 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{24192 \sqrt{2} 3^{3/4} a^{13/3} \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 )}{1235 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 = 5.04722, size = 99, normalized size = 0.15 \[ -\frac{3 \left (8992 a^2 b^2 x^5-40320 a^4 x \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+3390 a^3 b x^3-15255 a^4 x+2626 a b^3 x^7+247 b^4 x^9\right )}{6175 \sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(-15255*a^4*x + 3390*a^3*b*x^3 + 8992*a^2*b^2*x^5 + 2626*a*b^3*x^7 + 247*b^4*x^9 - 40320*a^4*x*(1 - (b*x^2
)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(6175*(a - b*x^2)^(1/3))

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Maple [F]  time = 0.032, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{2}+a \right ) ^{{\frac{2}{3}}} \left ( b{x}^{2}+3\,a \right ) ^{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)^3,x)

[Out]

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

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 27 \, a^{3}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}, x\right ) \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)^3,x, algorithm="fricas")

[Out]

integral((b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*b*x^2 + 27*a^3)*(-b*x^2 + a)^(2/3), x)

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Sympy [A]  time = 4.05953, size = 136, normalized size = 0.21 \begin{align*} 27 a^{\frac{11}{3}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + 9 a^{\frac{8}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + \frac{9 a^{\frac{5}{3}} b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac{a^{\frac{2}{3}} b^{3} x^{7}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7} \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)**3,x)

[Out]

27*a**(11/3)*x*hyper((-2/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + 9*a**(8/3)*b*x**3*hyper((-2/3, 3/2), (
5/2,), b*x**2*exp_polar(2*I*pi)/a) + 9*a**(5/3)*b**2*x**5*hyper((-2/3, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/
a)/5 + a**(2/3)*b**3*x**7*hyper((-2/3, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi)/a)/7

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

[Out]

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