\(\int (a+b x^2)^{2/3} (A+B x^2+C x^4) \, dx\) [124]

Optimal result

Integrand size = 24, antiderivative size = 676 \[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\frac {3 \left (247 A-\frac {3 a (19 b B-9 a C)}{b^2}\right ) x \left (a+b x^2\right )^{2/3}}{1729}+\frac {3 (19 b B-9 a C) x \left (a+b x^2\right )^{5/3}}{247 b^2}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}-\frac {12 a \left (247 A b^2-57 a b B+27 a^2 C\right ) x}{1729 b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}+\frac {6 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (247 A b^2-57 a b B+27 a^2 C\right ) \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{1729 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {4 \sqrt {2} 3^{3/4} a^{4/3} \left (247 A b^2-57 a b B+27 a^2 C\right ) \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{1729 b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/1729*(247*A-3*a*(19*B*b-9*C*a)/b^2)*x*(b*x^2+a)^(2/3)+3/247*(19*B*b-9*C* 
a)*x*(b*x^2+a)^(5/3)/b^2+3/19*C*x^3*(b*x^2+a)^(5/3)/b-12/1729*a*(247*A*b^2 
-57*B*a*b+27*C*a^2)*x/b^2/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))+6/1729*3^( 
1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^(4/3)*(247*A*b^2-57*B*a*b+27*C*a^2)*(a^(1 
/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3))/(( 
1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*a^(1/3 
)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2))/b^ 
3/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/ 
3))^2)^(1/2)-4/1729*2^(1/2)*3^(3/4)*a^(4/3)*(247*A*b^2-57*B*a*b+27*C*a^2)* 
(a^(1/3)-(b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(b*x^2+a)^(1/3)+(b*x^2+a)^(2/3 
))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3))^2)^(1/2)*EllipticF(((1+3^(1/2))*a 
^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a)^(1/3)),2*I-I*3^(1/2 
))/b^3/x/(-a^(1/3)*(a^(1/3)-(b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(b*x^2+a 
)^(1/3))^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.69 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.14 \[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\frac {x \left (a+b x^2\right )^{2/3} \left (-3 \left (a+b x^2\right ) \left (-19 b B+9 a C-13 b C x^2\right )+\frac {\left (247 A b^2+3 a (-19 b B+9 a C)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{\left (1+\frac {b x^2}{a}\right )^{2/3}}\right )}{247 b^2} \] Input:

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

Output:

(x*(a + b*x^2)^(2/3)*(-3*(a + b*x^2)*(-19*b*B + 9*a*C - 13*b*C*x^2) + ((24 
7*A*b^2 + 3*a*(-19*b*B + 9*a*C))*Hypergeometric2F1[-2/3, 1/2, 3/2, -((b*x^ 
2)/a)])/(1 + (b*x^2)/a)^(2/3)))/(247*b^2)
 

Rubi [A] (warning: unable to verify)

Time = 0.56 (sec) , antiderivative size = 677, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1473, 27, 299, 211, 233, 833, 760, 2418}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx\)

\(\Big \downarrow \) 1473

\(\displaystyle \frac {3 \int \frac {1}{3} \left (b x^2+a\right )^{2/3} \left ((19 b B-9 a C) x^2+19 A b\right )dx}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \left (b x^2+a\right )^{2/3} \left ((19 b B-9 a C) x^2+19 A b\right )dx}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {\frac {\left (247 A b^2-3 a (19 b B-9 a C)\right ) \int \left (b x^2+a\right )^{2/3}dx}{13 b}+\frac {3 x \left (a+b x^2\right )^{5/3} (19 b B-9 a C)}{13 b}}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {\left (247 A b^2-3 a (19 b B-9 a C)\right ) \left (\frac {4}{7} a \int \frac {1}{\sqrt [3]{b x^2+a}}dx+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )}{13 b}+\frac {3 x \left (a+b x^2\right )^{5/3} (19 b B-9 a C)}{13 b}}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {\frac {\left (247 A b^2-3 a (19 b B-9 a C)\right ) \left (\frac {6 a \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )}{13 b}+\frac {3 x \left (a+b x^2\right )^{5/3} (19 b B-9 a C)}{13 b}}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {\frac {\left (247 A b^2-3 a (19 b B-9 a C)\right ) \left (\frac {6 a \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )}{13 b}+\frac {3 x \left (a+b x^2\right )^{5/3} (19 b B-9 a C)}{13 b}}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {\frac {\left (247 A b^2-3 a (19 b B-9 a C)\right ) \left (\frac {6 a \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )}{13 b}+\frac {3 x \left (a+b x^2\right )^{5/3} (19 b B-9 a C)}{13 b}}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {\frac {\left (247 A b^2-3 a (19 b B-9 a C)\right ) \left (\frac {6 a \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{7 b x}+\frac {3}{7} x \left (a+b x^2\right )^{2/3}\right )}{13 b}+\frac {3 x \left (a+b x^2\right )^{5/3} (19 b B-9 a C)}{13 b}}{19 b}+\frac {3 C x^3 \left (a+b x^2\right )^{5/3}}{19 b}\)

Input:

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

Output:

(3*C*x^3*(a + b*x^2)^(5/3))/(19*b) + ((3*(19*b*B - 9*a*C)*x*(a + b*x^2)^(5 
/3))/(13*b) + ((247*A*b^2 - 3*a*(19*b*B - 9*a*C))*((3*x*(a + b*x^2)^(2/3)) 
/7 + (6*a*Sqrt[b*x^2]*((-2*Sqrt[b*x^2])/((1 - Sqrt[3])*a^(1/3) - (a + b*x^ 
2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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 - Sq 
rt[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]])/(Sqrt[b*x^2]*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)]) - (2*Sqrt[2 - Sqrt[3]]*( 
1 + Sqrt[3])*a^(1/3)*(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]])/(3^(1/4)*Sq 
rt[b*x^2]*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)])))/(7*b*x)))/(13*b))/(19*b)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 233

Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[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 299

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x 
*((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 
*p + 3))   Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && NeQ[2*p + 3, 0]
 

rule 760

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]*Sqrt[(- 
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 833

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

rule 1473

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), 
x_Symbol] :> Simp[c^p*x^(4*p - 1)*((d + e*x^2)^(q + 1)/(e*(4*p + 2*q + 1))) 
, x] + Simp[1/(e*(4*p + 2*q + 1))   Int[(d + e*x^2)^q*ExpandToSum[e*(4*p + 
2*q + 1)*(a + b*x^2 + c*x^4)^p - d*c^p*(4*p - 1)*x^(4*p - 2) - e*c^p*(4*p + 
 2*q + 1)*x^(4*p), x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b^2 - 
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] &&  !LtQ[q, -1]
 

rule 2418

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[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] + S 
imp[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 - S 
qrt[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*Sqrt[3])*a*d^3, 0]
 

Maple [F]

\[\int \left (b \,x^{2}+a \right )^{\frac {2}{3}} \left (C \,x^{4}+x^{2} B +A \right )d x\]

Input:

int((b*x^2+a)^(2/3)*(C*x^4+B*x^2+A),x)
 

Output:

int((b*x^2+a)^(2/3)*(C*x^4+B*x^2+A),x)
 

Fricas [F]

\[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

integral((C*x^4 + B*x^2 + A)*(b*x^2 + a)^(2/3), x)
 

Sympy [A] (verification not implemented)

Time = 1.31 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.14 \[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=A a^{\frac {2}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {B a^{\frac {2}{3}} x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + \frac {C a^{\frac {2}{3}} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{5} \] Input:

integrate((b*x**2+a)**(2/3)*(C*x**4+B*x**2+A),x)
 

Output:

A*a**(2/3)*x*hyper((-2/3, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a) + B*a**( 
2/3)*x**3*hyper((-2/3, 3/2), (5/2,), b*x**2*exp_polar(I*pi)/a)/3 + C*a**(2 
/3)*x**5*hyper((-2/3, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/5
 

Maxima [F]

\[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

integrate((C*x^4 + B*x^2 + A)*(b*x^2 + a)^(2/3), x)
 

Giac [F]

\[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\int { {\left (C x^{4} + B x^{2} + A\right )} {\left (b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \] Input:

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

Output:

integrate((C*x^4 + B*x^2 + A)*(b*x^2 + a)^(2/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\int {\left (b\,x^2+a\right )}^{2/3}\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] Input:

int((a + b*x^2)^(2/3)*(A + B*x^2 + C*x^4),x)
 

Output:

int((a + b*x^2)^(2/3)*(A + B*x^2 + C*x^4), x)
 

Reduce [F]

\[ \int \left (a+b x^2\right )^{2/3} \left (A+B x^2+C x^4\right ) \, dx=\frac {-108 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a^{2} c x +969 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a \,b^{2} x +84 \left (b \,x^{2}+a \right )^{\frac {2}{3}} a b c \,x^{3}+399 \left (b \,x^{2}+a \right )^{\frac {2}{3}} b^{3} x^{3}+273 \left (b \,x^{2}+a \right )^{\frac {2}{3}} b^{2} c \,x^{5}+108 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{3} c +760 \left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}}}d x \right ) a^{2} b^{2}}{1729 b^{2}} \] Input:

int((b*x^2+a)^(2/3)*(C*x^4+B*x^2+A),x)
 

Output:

( - 108*(a + b*x**2)**(2/3)*a**2*c*x + 969*(a + b*x**2)**(2/3)*a*b**2*x + 
84*(a + b*x**2)**(2/3)*a*b*c*x**3 + 399*(a + b*x**2)**(2/3)*b**3*x**3 + 27 
3*(a + b*x**2)**(2/3)*b**2*c*x**5 + 108*int((a + b*x**2)**(2/3)/(a + b*x** 
2),x)*a**3*c + 760*int((a + b*x**2)**(2/3)/(a + b*x**2),x)*a**2*b**2)/(172 
9*b**2)