\(\int \frac {A+B x+C x^2+D x^3}{(a+b x^2)^{10/3}} \, dx\) [71]

Optimal result

Integrand size = 27, antiderivative size = 682 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=-\frac {3 (a (b B-a D)-b (A b-a C) x)}{14 a b^2 \left (a+b x^2\right )^{7/3}}-\frac {3 \left (14 a^2 D-b (11 A b+3 a C) x\right )}{112 a^2 b^2 \left (a+b x^2\right )^{4/3}}+\frac {15 (11 A b+3 a C) x}{224 a^3 b \sqrt [3]{a+b x^2}}+\frac {15 (11 A b+3 a C) x}{224 a^3 b \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {15 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (11 A b+3 a C) \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{448 a^{8/3} b^2 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {5\ 3^{3/4} (11 A b+3 a C) \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{112 \sqrt {2} a^{8/3} b^2 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

1/14*(-3*a*(B*b-D*a)+3*b*(A*b-C*a)*x)/a/b^2/(b*x^2+a)^(7/3)-3/112*(14*a^2* 
D-b*(11*A*b+3*C*a)*x)/a^2/b^2/(b*x^2+a)^(4/3)+15/224*(11*A*b+3*C*a)*x/a^3/ 
b/(b*x^2+a)^(1/3)+15/224*(11*A*b+3*C*a)*x/a^3/b/((1-3^(1/2))*a^(1/3)-(b*x^ 
2+a)^(1/3))-15/448*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(11*A*b+3*C*a)*(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))/a^( 
8/3)/b^2/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)+5/224*3^(3/4)*(11*A*b+3*C*a)*(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))*2^(1/2)/a^(8/3)/b^2/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 = 10.10 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.22 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\frac {-36 a^4 D+165 A b^4 x^5+9 a b^3 x^3 \left (44 A+5 C x^2\right )+9 a^2 b^2 x \left (31 A+12 C x^2\right )-3 a^3 b (16 B+x (-5 C+28 D x))-5 b (11 A b+3 a C) x \left (a+b x^2\right )^2 \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},-\frac {b x^2}{a}\right )}{224 a^3 b^2 \left (a+b x^2\right )^{7/3}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^(10/3),x]
 

Output:

(-36*a^4*D + 165*A*b^4*x^5 + 9*a*b^3*x^3*(44*A + 5*C*x^2) + 9*a^2*b^2*x*(3 
1*A + 12*C*x^2) - 3*a^3*b*(16*B + x*(-5*C + 28*D*x)) - 5*b*(11*A*b + 3*a*C 
)*x*(a + b*x^2)^2*(1 + (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, - 
((b*x^2)/a)])/(224*a^3*b^2*(a + b*x^2)^(7/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.63 (sec) , antiderivative size = 710, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2345, 27, 454, 215, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx\)

\(\Big \downarrow \) 2345

\(\displaystyle -\frac {3 \int -\frac {b \left (11 A+\frac {3 a C}{b}\right )+14 a D x}{3 b \left (b x^2+a\right )^{7/3}}dx}{14 a}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {11 A b+3 a C+14 a D x}{\left (b x^2+a\right )^{7/3}}dx}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 454

\(\displaystyle \frac {\frac {5 (3 a C+11 A b) \int \frac {1}{\left (b x^2+a\right )^{4/3}}dx}{8 a}-\frac {3 \left (14 a^2 D-b x (3 a C+11 A b)\right )}{8 a b \left (a+b x^2\right )^{4/3}}}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {\frac {5 (3 a C+11 A b) \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {\int \frac {1}{\sqrt [3]{b x^2+a}}dx}{2 a}\right )}{8 a}-\frac {3 \left (14 a^2 D-b x (3 a C+11 A b)\right )}{8 a b \left (a+b x^2\right )^{4/3}}}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {\frac {5 (3 a C+11 A b) \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{4 a b x}\right )}{8 a}-\frac {3 \left (14 a^2 D-b x (3 a C+11 A b)\right )}{8 a b \left (a+b x^2\right )^{4/3}}}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {\frac {5 (3 a C+11 A b) \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{4 a b x}\right )}{8 a}-\frac {3 \left (14 a^2 D-b x (3 a C+11 A b)\right )}{8 a b \left (a+b x^2\right )^{4/3}}}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {\frac {5 (3 a C+11 A b) \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{4 a b x}\right )}{8 a}-\frac {3 \left (14 a^2 D-b x (3 a C+11 A b)\right )}{8 a b \left (a+b x^2\right )^{4/3}}}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {\frac {5 (3 a C+11 A b) \left (\frac {3 x}{2 a \sqrt [3]{a+b x^2}}-\frac {3 \sqrt {b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{4 a b x}\right )}{8 a}-\frac {3 \left (14 a^2 D-b x (3 a C+11 A b)\right )}{8 a b \left (a+b x^2\right )^{4/3}}}{14 a b}-\frac {3 \left (a \left (B-\frac {a D}{b}\right )-x (A b-a C)\right )}{14 a b \left (a+b x^2\right )^{7/3}}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/(a + b*x^2)^(10/3),x]
 

Output:

(-3*(a*(B - (a*D)/b) - (A*b - a*C)*x))/(14*a*b*(a + b*x^2)^(7/3)) + ((-3*( 
14*a^2*D - b*(11*A*b + 3*a*C)*x))/(8*a*b*(a + b*x^2)^(4/3)) + (5*(11*A*b + 
 3*a*C)*((3*x)/(2*a*(a + b*x^2)^(1/3)) - (3*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 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2]*E 
llipticE[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)*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)])))/(4*a*b 
*x)))/(8*a))/(14*a*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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (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 454

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

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 2345

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot 
ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 
 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b 
*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   In 
t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] 
/; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -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]

\[\int \frac {D x^{3}+C \,x^{2}+B x +A}{\left (b \,x^{2}+a \right )^{\frac {10}{3}}}d x\]

Input:

int((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(10/3),x)
 

Output:

int((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(10/3),x)
 

Fricas [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (b x^{2} + a\right )}^{\frac {10}{3}}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(10/3),x, algorithm="fricas")
 

Output:

integral((D*x^3 + C*x^2 + B*x + A)*(b*x^2 + a)^(2/3)/(b^4*x^8 + 4*a*b^3*x^ 
6 + 6*a^2*b^2*x^4 + 4*a^3*b*x^2 + a^4), x)
 

Sympy [A] (verification not implemented)

Time = 13.95 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.39 \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\frac {A x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {10}{3} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac {10}{3}}} + B \left (\begin {cases} - \frac {3}{14 a^{2} b \sqrt [3]{a + b x^{2}} + 28 a b^{2} x^{2} \sqrt [3]{a + b x^{2}} + 14 b^{3} x^{4} \sqrt [3]{a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{2}}{2 a^{\frac {10}{3}}} & \text {otherwise} \end {cases}\right ) + \frac {C x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {10}{3} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac {10}{3}}} + D \left (\begin {cases} - \frac {9 a}{56 a^{2} b^{2} \sqrt [3]{a + b x^{2}} + 112 a b^{3} x^{2} \sqrt [3]{a + b x^{2}} + 56 b^{4} x^{4} \sqrt [3]{a + b x^{2}}} - \frac {21 b x^{2}}{56 a^{2} b^{2} \sqrt [3]{a + b x^{2}} + 112 a b^{3} x^{2} \sqrt [3]{a + b x^{2}} + 56 b^{4} x^{4} \sqrt [3]{a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{\frac {10}{3}}} & \text {otherwise} \end {cases}\right ) \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(b*x**2+a)**(10/3),x)
 

Output:

A*x*hyper((1/2, 10/3), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(10/3) + B*Pie 
cewise((-3/(14*a**2*b*(a + b*x**2)**(1/3) + 28*a*b**2*x**2*(a + b*x**2)**( 
1/3) + 14*b**3*x**4*(a + b*x**2)**(1/3)), Ne(b, 0)), (x**2/(2*a**(10/3)), 
True)) + C*x**3*hyper((3/2, 10/3), (5/2,), b*x**2*exp_polar(I*pi)/a)/(3*a* 
*(10/3)) + D*Piecewise((-9*a/(56*a**2*b**2*(a + b*x**2)**(1/3) + 112*a*b** 
3*x**2*(a + b*x**2)**(1/3) + 56*b**4*x**4*(a + b*x**2)**(1/3)) - 21*b*x**2 
/(56*a**2*b**2*(a + b*x**2)**(1/3) + 112*a*b**3*x**2*(a + b*x**2)**(1/3) + 
 56*b**4*x**4*(a + b*x**2)**(1/3)), Ne(b, 0)), (x**4/(4*a**(10/3)), True))
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (b x^{2} + a\right )}^{\frac {10}{3}}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(10/3),x, algorithm="maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)/(b*x^2 + a)^(10/3), x)
 

Giac [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{{\left (b x^{2} + a\right )}^{\frac {10}{3}}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(10/3),x, algorithm="giac")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)/(b*x^2 + a)^(10/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{{\left (b\,x^2+a\right )}^{10/3}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/(a + b*x^2)^(10/3),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/(a + b*x^2)^(10/3), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2+D x^3}{\left (a+b x^2\right )^{10/3}} \, dx=\left (\int \frac {x^{3}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{3} x^{6}}d x \right ) d +\left (\int \frac {x^{2}}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{3} x^{6}}d x \right ) c +\left (\int \frac {x}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{3} x^{6}}d x \right ) b +\left (\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{3}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a^{2} b \,x^{2}+3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} a \,b^{2} x^{4}+\left (b \,x^{2}+a \right )^{\frac {1}{3}} b^{3} x^{6}}d x \right ) a \] Input:

int((D*x^3+C*x^2+B*x+A)/(b*x^2+a)^(10/3),x)
 

Output:

int(x**3/((a + b*x**2)**(1/3)*a**3 + 3*(a + b*x**2)**(1/3)*a**2*b*x**2 + 3 
*(a + b*x**2)**(1/3)*a*b**2*x**4 + (a + b*x**2)**(1/3)*b**3*x**6),x)*d + i 
nt(x**2/((a + b*x**2)**(1/3)*a**3 + 3*(a + b*x**2)**(1/3)*a**2*b*x**2 + 3* 
(a + b*x**2)**(1/3)*a*b**2*x**4 + (a + b*x**2)**(1/3)*b**3*x**6),x)*c + in 
t(x/((a + b*x**2)**(1/3)*a**3 + 3*(a + b*x**2)**(1/3)*a**2*b*x**2 + 3*(a + 
 b*x**2)**(1/3)*a*b**2*x**4 + (a + b*x**2)**(1/3)*b**3*x**6),x)*b + int(1/ 
((a + b*x**2)**(1/3)*a**3 + 3*(a + b*x**2)**(1/3)*a**2*b*x**2 + 3*(a + b*x 
**2)**(1/3)*a*b**2*x**4 + (a + b*x**2)**(1/3)*b**3*x**6),x)*a