\(\int \frac {(d+e x)^3}{(a+b x+c x^2)^{7/3}} \, dx\) [2489]

Optimal result

Integrand size = 22, antiderivative size = 1224 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 (d+e x)^2 (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 (10 b c d \left (c d^2+3 a e^2\right )-8 a c e \left (2 c d^2+3 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{5/3} \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}} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 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 (\arcsin \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^{5/3} \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} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 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}} \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]{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^{5/3} \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}}} \]

[Out]

-3/4*(e*x+d)^2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(4/3)+3/4*(10*b*c*d*(3*a*e^2+c*d^2)-8*a*c
*e*(3*a*e^2+2*c*d^2)-b^2*(-a*e^3+11*c*d^2*e)+(-b*e+2*c*d)*(10*c^2*d^2-b^2*e^2-2*c*e*(-7*a*e+5*b*d))*x)/c/(-4*a
*c+b^2)^2/(c*x^2+b*x+a)^(1/3)-3/4*(-b*e+2*c*d)*(5*c^2*d^2-b^2*e^2-c*e*(-9*a*e+5*b*d))*(2*c*x+b)*2^(2/3)/c^(5/3
)/(-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)*(-b*e+2*c*d)
*(5*c^2*d^2-b^2*e^2-c*e*(-9*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^(5/3)/(-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)*(-b*e+2*c*d)*(5*c^2*d^2-b^2*e^2-c*e*(-9*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^(5/3)/(-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] (warning: unable to verify)

Time = 1.09 (sec) , antiderivative size = 1224, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {752, 791, 637, 309, 224, 1891} \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=-\frac {3 (b d-2 a e+(2 c d-b e) x) (d+e x)^2}{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}} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 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 (\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 )}{4 \sqrt [3]{2} c^{5/3} \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]{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}}}-\frac {3^{3/4} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 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}} \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 )}{2^{5/6} c^{5/3} \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]{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}}}+\frac {3 \left (-\left (\left (11 c d^2 e-a e^3\right ) b^2\right )+10 c d \left (c d^2+3 a e^2\right ) b-8 a c e \left (2 c d^2+3 a e^2\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{c x^2+b x+a}}-\frac {3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{5/3} \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 )} \]

[In]

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

[Out]

(-3*(d + e*x)^2*(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*(10*b*c*d*(c*d
^2 + 3*a*e^2) - 8*a*c*e*(2*c*d^2 + 3*a*e^2) - b^2*(11*c*d^2*e - a*e^3) + (2*c*d - b*e)*(10*c^2*d^2 - b^2*e^2 -
2*c*e*(5*b*d - 7*a*e))*x))/(4*c*(b^2 - 4*a*c)^2*(a + b*x + c*x^2)^(1/3)) - (3*(2*c*d - b*e)*(5*c^2*d^2 - b^2*
e^2 - c*e*(5*b*d - 9*a*e))*(b + 2*c*x))/(2*2^(1/3)*c^(5/3)*(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 - Sqrt[3]]*(2*c*d - b*e)*(5*c^2*d^2 - b^2*e^2
- c*e*(5*b*d - 9*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^(5/3)*(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)*(2*c*d - b*e)*(5*c^2*d^2 - b^2*e^2 - c*e*(
5*b*d - 9*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 + S
qrt[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^(5/3)*(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 224

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]*Sq
rt[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 309

Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Dist[(-(
1 - Sqrt[3]))*(s/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 637

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 752

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 791

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(2
*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2
)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*
p + 3))/(c*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &&
NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]

Rule 1891

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]/(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] && EqQ[b*c^3 - 2*(5 - 3*Sqrt[3
])*a*d^3, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {3 (d+e x)^2 (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 {(d+e x) \left (\frac {1}{3} \left (10 c d^2-11 b d e+12 a e^2\right )-\frac {1}{3} e (2 c d-b e) x\right )}{\left (a+b x+c x^2\right )^{4/3}} \, dx}{4 \left (b^2-4 a c\right )} \\ & = -\frac {3 (d+e x)^2 (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 (10 b c d \left (c d^2+3 a e^2\right )-8 a c e \left (2 c d^2+3 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left ((2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right )\right ) \int \frac {1}{\sqrt [3]{a+b x+c x^2}} \, dx}{2 c \left (b^2-4 a c\right )^2} \\ & = -\frac {3 (d+e x)^2 (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 (10 b c d \left (c d^2+3 a e^2\right )-8 a c e \left (2 c d^2+3 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {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 c \left (b^2-4 a c\right )^2 (b+2 c x)} \\ & = -\frac {3 (d+e x)^2 (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 (10 b c d \left (c d^2+3 a e^2\right )-8 a c e \left (2 c d^2+3 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {\left (3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {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} c^{4/3} \left (b^2-4 a c\right )^2 (b+2 c x)}+\frac {\left (3 \left (1-\sqrt {3}\right ) (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) \sqrt {(b+2 c x)^2}\right ) \text {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\ 2^{2/3} c^{4/3} \left (b^2-4 a c\right )^{5/3} (b+2 c x)} \\ & = -\frac {3 (d+e x)^2 (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 (10 b c d \left (c d^2+3 a e^2\right )-8 a c e \left (2 c d^2+3 a e^2\right )-b^2 \left (11 c d^2 e-a e^3\right )+(2 c d-b e) \left (10 c^2 d^2-b^2 e^2-2 c e (5 b d-7 a e)\right ) x\right )}{4 c \left (b^2-4 a c\right )^2 \sqrt [3]{a+b x+c x^2}}-\frac {3 (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 a e)\right ) (b+2 c x)}{2 \sqrt [3]{2} c^{5/3} \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}} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 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^{5/3} \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} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2-c e (5 b d-9 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^{5/3} \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*}

Mathematica [C] (verified)

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

Time = 11.00 (sec) , antiderivative size = 403, normalized size of antiderivative = 0.33 \[ \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^{7/3}} \, dx=\frac {3 c \left (b^4 e^3 x^2+b^2 \left (a^2 e^3+a c e \left (-9 d^2+42 d e x-11 e^2 x^2\right )+c^2 d x \left (8 d^2-45 d e x+6 e^2 x^2\right )\right )+4 c \left (-6 a^3 e^3+5 c^3 d^3 x^3+a^2 c e \left (-6 d^2+3 d e x-8 e^2 x^2\right )+a c^2 d x \left (7 d^2+9 e^2 x^2\right )\right )+2 b c \left (a^2 e^2 (15 d-19 e x)+15 c^2 d^2 x^2 (d-e x)+a c \left (7 d^3-21 d^2 e x+27 d e^2 x^2-9 e^3 x^3\right )\right )+b^3 \left (2 a e^3 x-c \left (d^3+12 d^2 e x-9 d e^2 x^2-2 e^3 x^3\right )\right )\right )-2^{2/3} (2 c d-b e) \left (5 c^2 d^2-b^2 e^2+c e (-5 b d+9 a e)\right ) (b+2 c x) (a+x (b+c x)) \sqrt [3]{\frac {c (a+x (b+c x))}{-b^2+4 a c}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {(b+2 c x)^2}{b^2-4 a c}\right )}{4 c^2 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{4/3}} \]

[In]

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

[Out]

(3*c*(b^4*e^3*x^2 + b^2*(a^2*e^3 + a*c*e*(-9*d^2 + 42*d*e*x - 11*e^2*x^2) + c^2*d*x*(8*d^2 - 45*d*e*x + 6*e^2*
x^2)) + 4*c*(-6*a^3*e^3 + 5*c^3*d^3*x^3 + a^2*c*e*(-6*d^2 + 3*d*e*x - 8*e^2*x^2) + a*c^2*d*x*(7*d^2 + 9*e^2*x^
2)) + 2*b*c*(a^2*e^2*(15*d - 19*e*x) + 15*c^2*d^2*x^2*(d - e*x) + a*c*(7*d^3 - 21*d^2*e*x + 27*d*e^2*x^2 - 9*e
^3*x^3)) + b^3*(2*a*e^3*x - c*(d^3 + 12*d^2*e*x - 9*d*e^2*x^2 - 2*e^3*x^3))) - 2^(2/3)*(2*c*d - b*e)*(5*c^2*d^
2 - b^2*e^2 + c*e*(-5*b*d + 9*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^2*(b^2 - 4*a*c)^2*(a + x*(b + c*x))^(4/3)
)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(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(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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