\(\int \frac {(d+e x)^2}{(a+b x+c x^2)^{7/3}} \, dx\) [696]

Optimal result

Integrand size = 22, antiderivative size = 150 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 e (10 c d-b e+8 c e x)}{40 c^2 \left (a+b x+c x^2\right )^{4/3}}+\frac {4\ 2^{2/3} \left (\left (b^2+6 a c\right ) e^2+10 c d (c d-b e)\right ) (b+2 c x) \left (-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}\right )^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {7}{3},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{5 c^3 \left (a+b x+c x^2\right )^{7/3}} \] Output:

-3/40*e*(8*c*e*x-b*e+10*c*d)/c^2/(c*x^2+b*x+a)^(4/3)+4/5*2^(2/3)*((6*a*c+b 
^2)*e^2+10*c*d*(-b*e+c*d))*(2*c*x+b)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(7/3) 
*hypergeom([1/2, 7/3],[3/2],(2*c*x+b)^2/(-4*a*c+b^2))/c^3/(c*x^2+b*x+a)^(7 
/3)
 

Mathematica [A] (verified)

Time = 10.86 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.99 \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\frac {3 c \left (-b^3 \left (d^2+8 d e x-3 e^2 x^2\right )+b^2 \left (2 a e (-3 d+7 e x)+2 c x \left (4 d^2-15 d e x+e^2 x^2\right )\right )+4 c \left (5 c^2 d^2 x^3+a^2 e (-4 d+e x)+a c x \left (7 d^2+3 e^2 x^2\right )\right )+2 b \left (5 a^2 e^2+5 c^2 d x^2 (3 d-2 e x)+a c \left (7 d^2-14 d e x+9 e^2 x^2\right )\right )\right )-2^{2/3} \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right ) (b+2 c x) (a+x (b+c x)) \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \] Input:

Integrate[(d + e*x)^2/(a + b*x + c*x^2)^(7/3),x]
 

Output:

(3*c*(-(b^3*(d^2 + 8*d*e*x - 3*e^2*x^2)) + b^2*(2*a*e*(-3*d + 7*e*x) + 2*c 
*x*(4*d^2 - 15*d*e*x + e^2*x^2)) + 4*c*(5*c^2*d^2*x^3 + a^2*e*(-4*d + e*x) 
 + a*c*x*(7*d^2 + 3*e^2*x^2)) + 2*b*(5*a^2*e^2 + 5*c^2*d*x^2*(3*d - 2*e*x) 
 + a*c*(7*d^2 - 14*d*e*x + 9*e^2*x^2))) - 2^(2/3)*(10*c^2*d^2 + b^2*e^2 + 
2*c*e*(-5*b*d + 3*a*e))*(b + 2*c*x)*(a + x*(b + 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)])/(4*c*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3))
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1211\) vs. \(2(150)=300\).

Time = 1.01 (sec) , antiderivative size = 1211, normalized size of antiderivative = 8.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {1164, 27, 1159, 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.

\(\displaystyle \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx\)

\(\Big \downarrow \) 1164

\(\displaystyle -\frac {3 \int \frac {2 \left (5 c d^2-e (4 b d-3 a e)+e (2 c d-b e) x\right )}{3 \left (c x^2+b x+a\right )^{4/3}}dx}{4 \left (b^2-4 a c\right )}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {5 c d^2-e (4 b d-3 a e)+e (2 c d-b e) x}{\left (c x^2+b x+a\right )^{4/3}}dx}{2 \left (b^2-4 a c\right )}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 1159

\(\displaystyle -\frac {\frac {\left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \int \frac {1}{\sqrt [3]{c x^2+b x+a}}dx}{b^2-4 a c}+\frac {3 \left (-x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+4 a c d e+4 b^2 d e\right )}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{2 \left (b^2-4 a c\right )}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 1095

\(\displaystyle -\frac {\frac {3 \sqrt {(b+2 c x)^2} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \int \frac {\sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{\left (b^2-4 a c\right ) (b+2 c x)}+\frac {3 \left (-x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+4 a c d e+4 b^2 d e\right )}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{2 \left (b^2-4 a c\right )}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 832

\(\displaystyle -\frac {\frac {3 \sqrt {(b+2 c x)^2} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c} \int \frac {1}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}\right )}{\left (b^2-4 a c\right ) (b+2 c x)}+\frac {3 \left (-x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+4 a c d e+4 b^2 d e\right )}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{2 \left (b^2-4 a c\right )}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 759

\(\displaystyle -\frac {\frac {3 \sqrt {(b+2 c x)^2} \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right ) \left (\frac {\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\sqrt {b^2-4 a c+4 c \left (c x^2+b x+a\right )}}d\sqrt [3]{c x^2+b x+a}}{2^{2/3} \sqrt [3]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt {\frac {-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [3]{2} \sqrt [4]{3} c^{2/3} \sqrt {\frac {\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \sqrt {4 c \left (a+b x+c x^2\right )-4 a c+b^2}}\right )}{\left (b^2-4 a c\right ) (b+2 c x)}+\frac {3 \left (-x \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-5 b \left (a e^2+c d^2\right )+4 a c d e+4 b^2 d e\right )}{\left (b^2-4 a c\right ) \sqrt [3]{a+b x+c x^2}}}{2 \left (b^2-4 a c\right )}-\frac {3 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{4 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{4/3}}\)

\(\Big \downarrow \) 2416

\(\displaystyle -\frac {3 (d+e x) (b d-2 a e+(2 c d-b e) x)}{4 \left (b^2-4 a c\right ) \left (c x^2+b x+a\right )^{4/3}}-\frac {\frac {3 \left (4 d e b^2-5 \left (c d^2+a e^2\right ) b+4 a c d e-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{\left (b^2-4 a c\right ) \sqrt [3]{c x^2+b x+a}}+\frac {3 \left (10 c^2 d^2+b^2 e^2-2 c e (5 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 )}{\left (b^2-4 a c\right ) (b+2 c x)}}{2 \left (b^2-4 a c\right )}\)

Input:

Int[(d + e*x)^2/(a + b*x + c*x^2)^(7/3),x]
 

Output:

(-3*(d + e*x)*(b*d - 2*a*e + (2*c*d - b*e)*x))/(4*(b^2 - 4*a*c)*(a + b*x + 
 c*x^2)^(4/3)) - ((3*(4*b^2*d*e + 4*a*c*d*e - 5*b*(c*d^2 + a*e^2) - (10*c^ 
2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*x))/((b^2 - 4*a*c)*(a + b*x + c*x 
^2)^(1/3)) + (3*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*Sqrt[(b + 2 
*c*x)^2]*(((2^(1/3)*Sqrt[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)*Sqrt[((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 + Sq...
 

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 1159

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

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]

\[\int \frac {\left (e x +d \right )^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}d x\]

Input:

int((e*x+d)^2/(c*x^2+b*x+a)^(7/3),x)
 

Output:

int((e*x+d)^2/(c*x^2+b*x+a)^(7/3),x)
 

Fricas [F]

\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate((e*x+d)^2/(c*x^2+b*x+a)^(7/3),x, algorithm="fricas")
 

Output:

integral((e^2*x^2 + 2*d*e*x + d^2)*(c*x^2 + b*x + a)^(2/3)/(c^3*x^6 + 3*b* 
c^2*x^5 + 3*(b^2*c + a*c^2)*x^4 + 3*a^2*b*x + (b^3 + 6*a*b*c)*x^3 + a^3 + 
3*(a*b^2 + a^2*c)*x^2), x)
 

Sympy [F]

\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {\left (d + e x\right )^{2}}{\left (a + b x + c x^{2}\right )^{\frac {7}{3}}}\, dx \] Input:

integrate((e*x+d)**2/(c*x**2+b*x+a)**(7/3),x)
 

Output:

Integral((d + e*x)**2/(a + b*x + c*x**2)**(7/3), x)
 

Maxima [F]

\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate((e*x+d)^2/(c*x^2+b*x+a)^(7/3),x, algorithm="maxima")
 

Output:

integrate((e*x + d)^2/(c*x^2 + b*x + a)^(7/3), x)
 

Giac [F]

\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int { \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate((e*x+d)^2/(c*x^2+b*x+a)^(7/3),x, algorithm="giac")
 

Output:

integrate((e*x + d)^2/(c*x^2 + b*x + a)^(7/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \] Input:

int((d + e*x)^2/(a + b*x + c*x^2)^(7/3),x)
 

Output:

int((d + e*x)^2/(a + b*x + c*x^2)^(7/3), x)
 

Reduce [F]

\[ \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\left (\int \frac {x^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a b x +2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a c \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} b^{2} x^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} b c \,x^{3}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} c^{2} x^{4}}d x \right ) e^{2}+2 \left (\int \frac {x}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a b x +2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a c \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} b^{2} x^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} b c \,x^{3}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} c^{2} x^{4}}d x \right ) d e +\left (\int \frac {1}{\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a b x +2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} a c \,x^{2}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} b^{2} x^{2}+2 \left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} b c \,x^{3}+\left (c \,x^{2}+b x +a \right )^{\frac {1}{3}} c^{2} x^{4}}d x \right ) d^{2} \] Input:

int((e*x+d)^2/(c*x^2+b*x+a)^(7/3),x)
 

Output:

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