\(\int \frac {(3 a+b x^2)^3}{(a-b x^2)^{4/3}} \, dx\) [340]

Optimal result

Integrand size = 24, antiderivative size = 609 \[ \int \frac {\left (3 a+b x^2\right )^3}{\left (a-b x^2\right )^{4/3}} \, dx=\frac {96 a^2 x}{\sqrt [3]{a-b x^2}}+\frac {417}{91} a x \left (a-b x^2\right )^{2/3}+\frac {3}{13} b x^3 \left (a-b x^2\right )^{2/3}+\frac {20088 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {10044 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {6696 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \] Output:

96*a^2*x/(-b*x^2+a)^(1/3)+417/91*a*x*(-b*x^2+a)^(2/3)+3/13*b*x^3*(-b*x^2+a 
)^(2/3)+20088*a^2*x/(91*(1-3^(1/2))*a^(1/3)-91*(-b*x^2+a)^(1/3))+10044/91* 
3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*a^(7/3)*(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/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)-6696/91* 
2^(1/2)*3^(3/4)*a^(7/3)*(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/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 = 15.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.12 \[ \int \frac {\left (3 a+b x^2\right )^3}{\left (a-b x^2\right )^{4/3}} \, dx=-\frac {3 x \left (-3051 a^2+132 a b x^2+7 b^2 x^4+2232 a^2 \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{91 \sqrt [3]{a-b x^2}} \] Input:

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

Output:

(-3*x*(-3051*a^2 + 132*a*b*x^2 + 7*b^2*x^4 + 2232*a^2*(1 - (b*x^2)/a)^(1/3 
)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(91*(a - b*x^2)^(1/3))
 

Rubi [A] (warning: unable to verify)

Time = 0.54 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {315, 27, 403, 27, 299, 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.

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-\frac {3 \int \frac {6 a b \left (b x^2+3 a\right ) \left (3 b x^2+a\right )}{\sqrt [3]{a-b x^2}}dx}{2 a b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \int \frac {\left (b x^2+3 a\right ) \left (3 b x^2+a\right )}{\sqrt [3]{a-b x^2}}dx\)

\(\Big \downarrow \) 403

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (-\frac {3 \int -\frac {2 a b \left (47 b x^2+33 a\right )}{3 \sqrt [3]{a-b x^2}}dx}{13 b}-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (\frac {2}{13} a \int \frac {47 b x^2+33 a}{\sqrt [3]{a-b x^2}}dx-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (\frac {2}{13} a \left (\frac {372}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx-\frac {141}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (\frac {2}{13} a \left (-\frac {558 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}-\frac {141}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (\frac {2}{13} a \left (-\frac {558 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}-\frac {141}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (\frac {2}{13} a \left (-\frac {558 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}-\frac {141}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {6 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}-9 \left (\frac {2}{13} a \left (-\frac {558 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}-\frac {141}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {9}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )\right )\)

Input:

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

Output:

(6*x*(3*a + b*x^2)^2)/(a - b*x^2)^(1/3) - 9*((-9*x*(a - b*x^2)^(2/3)*(3*a 
+ b*x^2))/13 + (2*a*((-141*x*(a - b*x^2)^(2/3))/7 - (558*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 - 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]])/(Sqr 
t[-(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]*Ell 
ipticF[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)]*Sqr 
t[-((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)
 

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 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 315

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

rule 403

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

integrate((b*x**2+3*a)**3/(-b*x**2+a)**(4/3),x)
 

Output:

Integral((3*a + b*x**2)**3/(a - b*x**2)**(4/3), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (3 a+b x^2\right )^3}{\left (a-b x^2\right )^{4/3}} \, dx=\left (\int \frac {x^{6}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) b^{3}+9 \left (\int \frac {x^{4}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) a \,b^{2}+27 \left (\int \frac {x^{2}}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) a^{2} b +27 \left (\int \frac {1}{\left (-b \,x^{2}+a \right )^{\frac {1}{3}} a -\left (-b \,x^{2}+a \right )^{\frac {1}{3}} b \,x^{2}}d x \right ) a^{3} \] Input:

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

Output:

int(x**6/((a - b*x**2)**(1/3)*a - (a - b*x**2)**(1/3)*b*x**2),x)*b**3 + 9* 
int(x**4/((a - b*x**2)**(1/3)*a - (a - b*x**2)**(1/3)*b*x**2),x)*a*b**2 + 
27*int(x**2/((a - b*x**2)**(1/3)*a - (a - b*x**2)**(1/3)*b*x**2),x)*a**2*b 
 + 27*int(1/((a - b*x**2)**(1/3)*a - (a - b*x**2)**(1/3)*b*x**2),x)*a**3