3.2483 \(\int (d+e x)^2 (a+b x+c x^2)^{4/3} \, dx\)

Optimal. Leaf size=638 \[ \frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}}\right ),-7-4 \sqrt{3}\right )}{935 c^{13/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}+\frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{374 c^3}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{935 c^4}+\frac{15 e \left (a+b x+c x^2\right )^{7/3} (2 c d-b e)}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c} \]

[Out]

(-3*(b^2 - 4*a*c)*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(935*c^
4) + (3*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(4/3))/(374*c^3) + (15*e
*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/3))/(119*c^2) + (3*e*(d + e*x)*(a + b*x + c*x^2)^(7/3))/(17*c) + (2^(1/3)*
3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^2*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*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]])/(935*c^(13/3)*(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])

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Rubi [A]  time = 1.07187, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {742, 640, 623, 321, 218} \[ \frac{3 (b+2 c x) \left (a+b x+c x^2\right )^{4/3} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{374 c^3}-\frac{3 \left (b^2-4 a c\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right )}{935 c^4}+\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right ) \sqrt{\frac{-2^{2/3} \sqrt [3]{c} \sqrt [3]{b^2-4 a c} \sqrt [3]{a+b x+c x^2}+\left (b^2-4 a c\right )^{2/3}+2 \sqrt [3]{2} c^{2/3} \left (a+b x+c x^2\right )^{2/3}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}} \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}{\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{c x^2+b x+a}}\right )|-7-4 \sqrt{3}\right )}{935 c^{13/3} (b+2 c x) \sqrt{\frac{\sqrt [3]{b^2-4 a c} \left (\sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{b^2-4 a c}+2^{2/3} \sqrt [3]{c} \sqrt [3]{a+b x+c x^2}\right )^2}}}+\frac{15 e \left (a+b x+c x^2\right )^{7/3} (2 c d-b e)}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*(b^2 - 4*a*c)*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/3))/(935*c^
4) + (3*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(4/3))/(374*c^3) + (15*e
*(2*c*d - b*e)*(a + b*x + c*x^2)^(7/3))/(119*c^2) + (3*e*(d + e*x)*(a + b*x + c*x^2)^(7/3))/(17*c) + (2^(1/3)*
3^(3/4)*Sqrt[2 + Sqrt[3]]*(b^2 - 4*a*c)^2*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*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]])/(935*c^(13/3)*(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 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, 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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 640

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] + Dist[(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[2*c*d - b*e, 0] && NeQ[p, -1]

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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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]

Rubi steps

\begin{align*} \int (d+e x)^2 \left (a+b x+c x^2\right )^{4/3} \, dx &=\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac{3 \int \left (\frac{1}{3} \left (17 c d^2-3 e \left (\frac{7 b d}{3}+a e\right )\right )+\frac{10}{3} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{4/3} \, dx}{17 c}\\ &=\frac{15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac{\left (3 \left (-\frac{10}{3} b e (2 c d-b e)+\frac{2}{3} c \left (17 c d^2-3 e \left (\frac{7 b d}{3}+a e\right )\right )\right )\right ) \int \left (a+b x+c x^2\right )^{4/3} \, dx}{34 c^2}\\ &=\frac{15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac{\left (9 \left (-\frac{10}{3} b e (2 c d-b e)+\frac{2}{3} c \left (17 c d^2-3 e \left (\frac{7 b d}{3}+a e\right )\right )\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^6}{\sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{34 c^2 (b+2 c x)}\\ &=\frac{3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac{15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}-\frac{\left (9 \left (b^2-4 a c\right ) \left (-\frac{10}{3} b e (2 c d-b e)+\frac{2}{3} c \left (17 c d^2-3 e \left (\frac{7 b d}{3}+a e\right )\right )\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^3}{\sqrt{b^2-4 a c+4 c x^3}} \, dx,x,\sqrt [3]{a+b x+c x^2}\right )}{187 c^3 (b+2 c x)}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{935 c^4}+\frac{3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac{15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac{\left (9 \left (b^2-4 a c\right )^2 \left (-\frac{10}{3} b e (2 c d-b e)+\frac{2}{3} c \left (17 c d^2-3 e \left (\frac{7 b d}{3}+a e\right )\right )\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 )}{1870 c^4 (b+2 c x)}\\ &=-\frac{3 \left (b^2-4 a c\right ) \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \sqrt [3]{a+b x+c x^2}}{935 c^4}+\frac{3 \left (17 c^2 d^2+5 b^2 e^2-c e (17 b d+3 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{4/3}}{374 c^3}+\frac{15 e (2 c d-b e) \left (a+b x+c x^2\right )^{7/3}}{119 c^2}+\frac{3 e (d+e x) \left (a+b x+c x^2\right )^{7/3}}{17 c}+\frac{\sqrt [3]{2} 3^{3/4} \sqrt{2+\sqrt{3}} \left (b^2-4 a c\right )^2 \left (17 c^2 d^2+5 b^2 e^2-c e (17 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 )}{935 c^{13/3} (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]  time = 0.383738, size = 164, normalized size = 0.26 \[ \frac{3 (a+x (b+c x))^{4/3} \left (\frac{14 \sqrt [3]{2} (b+2 c x) \left (-c e (3 a e+17 b d)+5 b^2 e^2+17 c^2 d^2\right ) \, _2F_1\left (-\frac{4}{3},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )}{3 c^2 \left (-\frac{c (a+x (b+c x))}{b^2-4 a c}\right )^{4/3}}-\frac{160 e (a+x (b+c x)) (b e-2 c d)}{c}+224 e (d+e x) (a+x (b+c x))\right )}{3808 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(a + x*(b + c*x))^(4/3)*((-160*e*(-2*c*d + b*e)*(a + x*(b + c*x)))/c + 224*e*(d + e*x)*(a + x*(b + c*x)) +
(14*2^(1/3)*(17*c^2*d^2 + 5*b^2*e^2 - c*e*(17*b*d + 3*a*e))*(b + 2*c*x)*Hypergeometric2F1[-4/3, 1/2, 3/2, (b +
2*c*x)^2/(b^2 - 4*a*c)])/(3*c^2*(-((c*(a + x*(b + c*x)))/(b^2 - 4*a*c)))^(4/3))))/(3808*c)

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Maple [F]  time = 1.145, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{4}{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}{\left (e x + d\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c e^{2} x^{4} +{\left (2 \, c d e + b e^{2}\right )} x^{3} + a d^{2} +{\left (c d^{2} + 2 \, b d e + a e^{2}\right )} x^{2} +{\left (b d^{2} + 2 \, a d e\right )} x\right )}{\left (c x^{2} + b x + a\right )}^{\frac{1}{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{4}{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{4}{3}}{\left (e x + d\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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