3.2490 \(\int \frac {(d+e x)^2}{(a+b x+c x^2)^{7/3}} \, dx\)

Optimal. Leaf size=1153 \[ -\frac {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)}{2 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^2 \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 {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 )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\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 {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \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 (\sin ^{-1}\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 )}{4 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^{5/3} \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}} (b+2 c x)}-\frac {3^{3/4} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \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}} F\left (\sin ^{-1}\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^{5/6} c^{2/3} \left (b^2-4 a c\right )^{5/3} \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}} (b+2 c x)} \]

[Out]

-3/4*(e*x+d)*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(4/3)-3/2*(4*b^2*d*e+4*a*c*d*e-5*b*(a*e^2+c
*d^2)-(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*x)/(-4*a*c+b^2)^2/(c*x^2+b*x+a)^(1/3)-3/4*(10*c^2*d^2+b^2*e^2-
2*c*e*(-3*a*e+5*b*d))*(2*c*x+b)*2^(2/3)/c^(2/3)/(-4*a*c+b^2)^2/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^
2)^(1/3)*(1+3^(1/2)))-1/2*3^(3/4)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3
)*(c*x^2+b*x+a)^(1/3))*EllipticF((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1-3^(1/2)))/(2^(2/3)
*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(((-4*a*c+b^2)^(2/3)-2^(2/3)*c^(1/
3)*(-4*a*c+b^2)^(1/3)*(c*x^2+b*x+a)^(1/3)+2*2^(1/3)*c^(2/3)*(c*x^2+b*x+a)^(2/3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a
)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)*2^(1/6)/c^(2/3)/(-4*a*c+b^2)^(5/3)/(2*c*x+b)/((-4*a*c+b^2)^(1
/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)
^(1/3)*(1+3^(1/2)))^2)^(1/2)+3/8*3^(1/4)*(10*c^2*d^2+b^2*e^2-2*c*e*(-3*a*e+5*b*d))*((-4*a*c+b^2)^(1/3)+2^(2/3)
*c^(1/3)*(c*x^2+b*x+a)^(1/3))*EllipticE((2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1-3^(1/2)))/(
2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2))),I*3^(1/2)+2*I)*(1/2*6^(1/2)-1/2*2^(1/2))*(
((-4*a*c+b^2)^(2/3)-2^(2/3)*c^(1/3)*(-4*a*c+b^2)^(1/3)*(c*x^2+b*x+a)^(1/3)+2*2^(1/3)*c^(2/3)*(c*x^2+b*x+a)^(2/
3))/(2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)*2^(2/3)/c^(2/3)/(-4*a*c+b^2)
^(5/3)/(2*c*x+b)/((-4*a*c+b^2)^(1/3)*((-4*a*c+b^2)^(1/3)+2^(2/3)*c^(1/3)*(c*x^2+b*x+a)^(1/3))/(2^(2/3)*c^(1/3)
*(c*x^2+b*x+a)^(1/3)+(-4*a*c+b^2)^(1/3)*(1+3^(1/2)))^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 1.37, antiderivative size = 1153, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {738, 638, 623, 303, 218, 1877} \[ -\frac {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)}{2 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^2 \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 {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 )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\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 {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \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 (\sin ^{-1}\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 )}{4 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^{5/3} \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}} (b+2 c x)}-\frac {3^{3/4} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \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}} F\left (\sin ^{-1}\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^{5/6} c^{2/3} \left (b^2-4 a c\right )^{5/3} \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}} (b+2 c x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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))/(2*(b^2 - 4*a*c)^2*(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))*(b + 2*c*x))/(2*2^(1/3)*c^(2/3)*(b^2 - 4*a
*c)^2*((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*3^(1/4)*Sqrt[2 - Sqr
t[3]]*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*((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]])/(4*2^(1/3)*c^(2/3)*(b^2 - 4
*a*c)^(5/3)*(b + 2*c*x)*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]) - (3^(3/4)*(10*c^2*d^2
+ b^2*e^2 - 2*c*e*(5*b*d - 3*a*e))*((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 - 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^(5/6)*c^(2/3)*(b^2 - 4*a*c)^(5/3)*(b + 2*c
*x)*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])

Rule 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((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]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 303

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(Sq
rt[2]*s)/(Sqrt[2 + Sqrt[3]]*r), Int[1/Sqrt[a + b*x^3], x], x] + Dist[1/r, Int[((1 - Sqrt[3])*s + r*x)/Sqrt[a +
b*x^3], x], x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 623

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{d = Denominator[p]}, Dist[(d*Sqrt[(b + 2*c*x)
^2])/(b + 2*c*x), Subst[Int[x^(d*(p + 1) - 1)/Sqrt[b^2 - 4*a*c + 4*c*x^d], x], x, (a + b*x + c*x^2)^(1/d)], x]
/; 3 <= d <= 4] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && RationalQ[p]

Rule 638

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)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((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] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> 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] + Dist[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] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 1877

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[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
] - Simp[(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]*Elli
pticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(r^2*Sqrt[a + b*x^3]*Sqrt[(s*(
s + r*x))/((1 + Sqrt[3])*s + r*x)^2]), x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps

\begin {align*} \int \frac {(d+e x)^2}{\left (a+b x+c x^2\right )^{7/3}} \, dx &=-\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 (a+b x+c x^2\right )^{4/3}}-\frac {3 \int \frac {\frac {2}{3} \left (5 c d^2-e (4 b d-3 a e)\right )+\frac {2}{3} e (2 c d-b e) x}{\left (a+b x+c x^2\right )^{4/3}} \, dx}{4 \left (b^2-4 a c\right )}\\ &=-\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 (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \int \frac {1}{\sqrt [3]{a+b x+c x^2}} \, dx}{2 \left (b^2-4 a c\right )^2}\\ &=-\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 (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (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}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2 \left (b^2-4 a c\right )^2 (b+2 c x)}\\ &=-\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 (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (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}\right ) \operatorname {Subst}\left (\int \frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} x}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2\ 2^{2/3} \sqrt [3]{c} \left (b^2-4 a c\right )^2 (b+2 c x)}-\frac {\left (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}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{2 \sqrt [6]{2} \sqrt {2+\sqrt {3}} \sqrt [3]{c} \left (b^2-4 a c\right )^{5/3} (b+2 c x)}\\ &=-\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 (a+b x+c x^2\right )^{4/3}}-\frac {3 \left (4 b^2 d e+4 a c d e-5 b \left (c d^2+a e^2\right )-\left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) x\right )}{2 \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {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)}{2 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^2 \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 )}+\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \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 {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+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}} E\left (\sin ^{-1}\left (\frac {\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}}{\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 )|-7-4 \sqrt {3}\right )}{4 \sqrt [3]{2} c^{2/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \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}}}-\frac {3^{3/4} \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right ) \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 {\left (b^2-4 a c\right )^{2/3}-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+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}} F\left (\sin ^{-1}\left (\frac {\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}}{\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 )|-7-4 \sqrt {3}\right )}{2^{5/6} c^{2/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x) \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}}}\\ \end {align*}

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Mathematica [C]  time = 0.79, size = 342, normalized size = 0.30 \[ \frac {3 c \left (2 b \left (5 a^2 e^2+a c \left (7 d^2-14 d e x+9 e^2 x^2\right )+5 c^2 d x^2 (3 d-2 e x)\right )+4 c \left (a^2 e (e x-4 d)+a c x \left (7 d^2+3 e^2 x^2\right )+5 c^2 d^2 x^3\right )+b^2 \left (2 a e (7 e x-3 d)+2 c x \left (4 d^2-15 d e x+e^2 x^2\right )\right )-\left (b^3 \left (d^2+8 d e x-3 e^2 x^2\right )\right )\right )-2^{2/3} (b+2 c x) \sqrt [3]{\frac {c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac {1}{3},\frac {1}{2};\frac {3}{2};\frac {(b+2 c x)^2}{b^2-4 a c}\right ) \left (6 a^2 c e^2+a \left (b^2 e^2+2 b c e (3 e x-5 d)+2 c^2 \left (5 d^2+3 e^2 x^2\right )\right )+x (b+c x) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right )\right )}{4 c \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)*(b + 2*c*x)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/3)*(
6*a^2*c*e^2 + (10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*x*(b + c*x) + a*(b^2*e^2 + 2*b*c*e*(-5*d + 3*e*x) + 2*c^2*(5
*d^2 + 3*e^2*x^2)))*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))

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fricas [F]  time = 1.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} {\left (c x^{2} + b x + a\right )}^{\frac {2}{3}}}{c^{3} x^{6} + 3 \, b c^{2} x^{5} + 3 \, {\left (b^{2} c + a c^{2}\right )} x^{4} + 3 \, a^{2} b x + {\left (b^{3} + 6 \, a b c\right )} x^{3} + a^{3} + 3 \, {\left (a b^{2} + a^{2} c\right )} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 2.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (e x +d \right )^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {7}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x + d\right )}^{2}}{{\left (c x^{2} + b x + a\right )}^{\frac {7}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d+e\,x\right )}^2}{{\left (c\,x^2+b\,x+a\right )}^{7/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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