\(\int \frac {(c+d x^2)^2}{(a+b x^2)^{4/3}} \, dx\) [381]

Optimal result

Integrand size = 21, antiderivative size = 662 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {3 (b c-a d)^2 x}{2 a b^2 \sqrt [3]{a+b x^2}}+\frac {3 d^2 x \left (a+b x^2\right )^{2/3}}{7 b^2}+\frac {3 \left (7 b^2 c^2-42 a b c d+27 a^2 d^2\right ) x}{14 a b^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (7 b^2 c^2-42 a b c d+27 a^2 d^2\right ) \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt {3}\right )}{28 a^{2/3} b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {3^{3/4} \left (7 b^2 c^2-42 a b c d+27 a^2 d^2\right ) \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),-7+4 \sqrt {3}\right )}{7 \sqrt {2} a^{2/3} b^3 x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}} \] Output:

3/2*(-a*d+b*c)^2*x/a/b^2/(b*x^2+a)^(1/3)+3/7*d^2*x*(b*x^2+a)^(2/3)/b^2+3/1 
4*(27*a^2*d^2-42*a*b*c*d+7*b^2*c^2)*x/a/b^2/((1-3^(1/2))*a^(1/3)-(b*x^2+a) 
^(1/3))-3/28*3^(1/4)*(1/2*6^(1/2)+1/2*2^(1/2))*(27*a^2*d^2-42*a*b*c*d+7*b^ 
2*c^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))/a^(2/3)/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)+1/14*3^(3/4)*(27*a^2*d^2-42*a*b*c*d+7*b^2 
*c^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))*2^(1/2)/a^(2/3)/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] (warning: unable to verify)

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

Time = 10.13 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.26 \[ \int \frac {\left (c+d x^2\right )^2}{\left (a+b x^2\right )^{4/3}} \, dx=\frac {x \sqrt [3]{1+\frac {b x^2}{a}} \operatorname {Gamma}\left (\frac {1}{3}\right ) \left (21 a \left (15 c^2+10 c d x^2+3 d^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{2},-\frac {b x^2}{a}\right )-32 b x^2 \left (2 c^2+3 c d x^2+d^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{3},\frac {9}{2},-\frac {b x^2}{a}\right )-16 b x^2 \left (c+d x^2\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {7}{3};1,\frac {9}{2};-\frac {b x^2}{a}\right )\right )}{945 a^2 \sqrt [3]{a+b x^2} \operatorname {Gamma}\left (\frac {4}{3}\right )} \] Input:

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

Output:

(x*(1 + (b*x^2)/a)^(1/3)*Gamma[1/3]*(21*a*(15*c^2 + 10*c*d*x^2 + 3*d^2*x^4 
)*Hypergeometric2F1[1/2, 4/3, 7/2, -((b*x^2)/a)] - 32*b*x^2*(2*c^2 + 3*c*d 
*x^2 + d^2*x^4)*Hypergeometric2F1[3/2, 7/3, 9/2, -((b*x^2)/a)] - 16*b*x^2* 
(c + d*x^2)^2*HypergeometricPFQ[{3/2, 2, 7/3}, {1, 9/2}, -((b*x^2)/a)]))/( 
945*a^2*(a + b*x^2)^(1/3)*Gamma[4/3])
 

Rubi [A] (warning: unable to verify)

Time = 0.58 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {315, 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.

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

\(\Big \downarrow \) 315

\(\displaystyle \frac {3 \int -\frac {d (7 b c-9 a d) x^2+c (b c-3 a d)}{3 \sqrt [3]{b x^2+a}}dx}{2 a b}+\frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}-\frac {\int \frac {d (7 b c-9 a d) x^2+c (b c-3 a d)}{\sqrt [3]{b x^2+a}}dx}{2 a b}\)

\(\Big \downarrow \) 299

\(\displaystyle \frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}-\frac {\frac {\left (27 a^2 d^2-42 a b c d+7 b^2 c^2\right ) \int \frac {1}{\sqrt [3]{b x^2+a}}dx}{7 b}+\frac {3 d x \left (a+b x^2\right )^{2/3} (7 b c-9 a d)}{7 b}}{2 a b}\)

\(\Big \downarrow \) 233

\(\displaystyle \frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}-\frac {\frac {3 \sqrt {b x^2} \left (27 a^2 d^2-42 a b c d+7 b^2 c^2\right ) \int \frac {\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}}{14 b^2 x}+\frac {3 d x \left (a+b x^2\right )^{2/3} (7 b c-9 a d)}{7 b}}{2 a b}\)

\(\Big \downarrow \) 833

\(\displaystyle \frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}-\frac {\frac {3 \sqrt {b x^2} \left (27 a^2 d^2-42 a b c d+7 b^2 c^2\right ) \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}\right )}{14 b^2 x}+\frac {3 d x \left (a+b x^2\right )^{2/3} (7 b c-9 a d)}{7 b}}{2 a b}\)

\(\Big \downarrow \) 760

\(\displaystyle \frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}-\frac {\frac {3 \sqrt {b x^2} \left (27 a^2 d^2-42 a b c d+7 b^2 c^2\right ) \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\sqrt {b x^2}}d\sqrt [3]{b x^2+a}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\right )}{14 b^2 x}+\frac {3 d x \left (a+b x^2\right )^{2/3} (7 b c-9 a d)}{7 b}}{2 a b}\)

\(\Big \downarrow \) 2418

\(\displaystyle \frac {3 x \left (c+d x^2\right ) (b c-a d)}{2 a b \sqrt [3]{a+b x^2}}-\frac {\frac {3 \sqrt {b x^2} \left (27 a^2 d^2-42 a b c d+7 b^2 c^2\right ) \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac {2 \sqrt {b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )}{14 b^2 x}+\frac {3 d x \left (a+b x^2\right )^{2/3} (7 b c-9 a d)}{7 b}}{2 a b}\)

Input:

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

Output:

(3*(b*c - a*d)*x*(c + d*x^2))/(2*a*b*(a + b*x^2)^(1/3)) - ((3*d*(7*b*c - 9 
*a*d)*x*(a + b*x^2)^(2/3))/(7*b) + (3*(7*b^2*c^2 - 42*a*b*c*d + 27*a^2*d^2 
)*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*Sqr 
t[3]])/(Sqrt[b*x^2]*Sqrt[-((a^(1/3)*(a^(1/3) - (a + b*x^2)^(1/3)))/((1 - S 
qrt[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 - Sq 
rt[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)])))/(14*b^2*x))/(2*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 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 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d^2*x^4 + 2*c*d*x^2 + c^2)*(b*x^2 + a)^(2/3)/(b^2*x^4 + 2*a*b*x^ 
2 + a^2), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(x**4/((a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*d**2 + 2* 
int(x**2/((a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*c*d + int 
(1/((a + b*x**2)**(1/3)*a + (a + b*x**2)**(1/3)*b*x**2),x)*c**2