\(\int \frac {(d+e x)^2}{(a+b x+c x^2)^{4/3}} \, dx\) [691]

Optimal result

Integrand size = 22, antiderivative size = 147 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{4/3}} \, dx=-\frac {3 e (c d-b e-c e x)}{c^2 \sqrt [3]{a+b x+c x^2}}+\frac {2\ 2^{2/3} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (b+2 c x) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{4/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^3 \left (a+b x+c x^2\right )^{4/3}} \] Output:

-3*e*(-c*e*x-b*e+c*d)/c^2/(c*x^2+b*x+a)^(1/3)+2*2^(2/3)*(c^2*d^2+b^2*e^2-c 
*e*(3*a*e+b*d))*(2*c*x+b)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(4/3)*hypergeom( 
[1/2, 4/3],[3/2],(2*c*x+b)^2/(-4*a*c+b^2))/c^3/(c*x^2+b*x+a)^(4/3)
 

Mathematica [A] (verified)

Time = 10.31 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.17 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{4/3}} \, dx=\frac {-3 c \left (a b e^2+2 c^2 d^2 x+b^2 e^2 x+b c d (d-2 e x)-2 a c e (2 d+e x)\right )+2^{2/3} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (b+2 c x) \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{c^2 \left (b^2-4 a c\right ) \sqrt [3]{a+x (b+c x)}} \] Input:

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

Output:

(-3*c*(a*b*e^2 + 2*c^2*d^2*x + b^2*e^2*x + b*c*d*(d - 2*e*x) - 2*a*c*e*(2* 
d + e*x)) + 2^(2/3)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(b + 2*c*x)*(( 
c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2 
, (b + 2*c*x)^2/(b^2 - 4*a*c)])/(c^2*(b^2 - 4*a*c)*(a + x*(b + c*x))^(1/3) 
)
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1143\) vs. \(2(147)=294\).

Time = 0.97 (sec) , antiderivative size = 1143, normalized size of antiderivative = 7.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1164, 27, 1160, 1095, 832, 759, 2416}

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.

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

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {3 \int -\frac {2 \left (c d^2+e (b d-3 a e)+2 e (2 c d-b e) x\right )}{3 \sqrt [3]{c x^2+b x+a}}dx}{b^2-4 a c}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 \int \frac {c d^2+e (b d-3 a e)+2 e (2 c d-b e) x}{\sqrt [3]{c x^2+b x+a}}dx}{b^2-4 a c}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\)

\(\Big \downarrow \) 1160

\(\displaystyle \frac {2 \left (\frac {\left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \int \frac {1}{\sqrt [3]{c x^2+b x+a}}dx}{c}+\frac {3 e \left (a+b x+c x^2\right )^{2/3} (2 c d-b e)}{2 c}\right )}{b^2-4 a c}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\)

\(\Big \downarrow \) 1095

\(\displaystyle \frac {2 \left (\frac {3 \sqrt {(b+2 c x)^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \int \frac {\sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{c (b+2 c x)}+\frac {3 e \left (a+b x+c x^2\right )^{2/3} (2 c d-b e)}{2 c}\right )}{b^2-4 a c}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\)

\(\Big \downarrow \) 832

\(\displaystyle \frac {2 \left (\frac {3 \sqrt {(b+2 c x)^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}\right )}{c (b+2 c x)}+\frac {3 e \left (a+b x+c x^2\right )^{2/3} (2 c d-b e)}{2 c}\right )}{b^2-4 a c}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}\)

\(\Big \downarrow \) 759

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

\(\Big \downarrow \) 2416

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

Input:

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

Output:

(-3*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c 
*x^2)^(1/3)) + (2*((3*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(2/3))/(2*c) + (3* 
(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[(b + 2*c*x)^2]*(((2^(1/3)*Sqr 
t[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)])/(c^(1/3)*((1 + Sqrt[3])*(b^2 - 4*a 
*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))) - (3^(1/4)*Sqrt[2 - 
Sqrt[3]]*(b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b 
*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c)^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a* 
c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/ 
3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2) 
^(1/3))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c 
^(1/3)*(a + b*x + c*x^2)^(1/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/ 
3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2^(2/3)*c^(1/3)*Sq 
rt[((b^2 - 4*a*c)^(1/3)*((b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + 
c*x^2)^(1/3)))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b 
*x + c*x^2)^(1/3))^2]*Sqrt[b^2 - 4*a*c + 4*c*(a + b*x + c*x^2)]))/(2^(2/3) 
*c^(1/3)) - ((1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^(1/3)*((b^2 - 4 
*a*c)^(1/3) + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))*Sqrt[((b^2 - 4*a*c) 
^(2/3) - 2^(2/3)*c^(1/3)*(b^2 - 4*a*c)^(1/3)*(a + b*x + c*x^2)^(1/3) + 2*2 
^(1/3)*c^(2/3)*(a + b*x + c*x^2)^(2/3))/((1 + Sqrt[3])*(b^2 - 4*a*c)^(1/3) 
 + 2^(2/3)*c^(1/3)*(a + b*x + c*x^2)^(1/3))^2]*EllipticF[ArcSin[((1 - S...
 

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 759

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] & 
& PosQ[a]
 

rule 832

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] && PosQ[a]
 

rule 1095

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[3*(Sqrt[(b 
+ 2*c*x)^2]/(b + 2*c*x))   Subst[Int[x^(3*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4 
*c*x^3], x], x, (a + b*x + c*x^2)^(1/3)], x] /; FreeQ[{a, b, c}, x] && Inte 
gerQ[3*p]
 

rule 1160

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

rule 1164

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m - 1)*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b*x 
+ c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a* 
c))   Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2* 
c*d^2*(2*p + 3) + e*(b*e - 2*d*c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p 
+ 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 1] && Int 
QuadraticQ[a, b, c, d, e, m, p, x]
 

rule 2416

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 + Sqrt 
[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] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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