### 3.2516 $$\int (d+e x)^3 (a+b x+c x^2)^{3/4} \, dx$$

Optimal. Leaf size=703 $-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right ),\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}-\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}$

[Out]

((2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(120*c^4)
+ (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/4))/(11*c) + (e*(312*c^2*d^2 + 55*b^2*e^2 - 2*c*e*(121*b*d + 24*a*e)
+ 70*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/4))/(462*c^3) - (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(12*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(80*c^(9/2)*(1 + (2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d +
2*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 +
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/
4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(80*Sqrt[2]*c^(19/4)*(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2
*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x
^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt
[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(160*Sqrt[2]*c^(19/4)*(b + 2*c*x))

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Rubi [A]  time = 0.866685, antiderivative size = 703, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.318, Rules used = {742, 779, 612, 623, 305, 220, 1196} $\frac{e \left (a+b x+c x^2\right )^{7/4} \left (-2 c e (24 a e+121 b d)+55 b^2 e^2+70 c e x (2 c d-b e)+312 c^2 d^2\right )}{462 c^3}+\frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/4} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{120 c^4}-\frac{\sqrt{b^2-4 a c} (b+2 c x) \sqrt [4]{a+b x+c x^2} (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right )}{80 c^{9/2} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )}-\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{\left (b^2-4 a c\right )^{7/4} \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right )^2}} \left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}+1\right ) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{c x^2+b x+a}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(3/4))/(120*c^4)
+ (2*e*(d + e*x)^2*(a + b*x + c*x^2)^(7/4))/(11*c) + (e*(312*c^2*d^2 + 55*b^2*e^2 - 2*c*e*(121*b*d + 24*a*e)
+ 70*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(7/4))/(462*c^3) - (Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(12*c^2*d^2 +
5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*(b + 2*c*x)*(a + b*x + c*x^2)^(1/4))/(80*c^(9/2)*(1 + (2*Sqrt[c]*Sqrt[a + b
*x + c*x^2])/Sqrt[b^2 - 4*a*c])) + ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d +
2*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 +
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticE[2*ArcTan[(Sqrt[2]*c^(1/4)*(a + b*x + c*x^2)^(1/
4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(80*Sqrt[2]*c^(19/4)*(b + 2*c*x)) - ((b^2 - 4*a*c)^(7/4)*(2*c*d - b*e)*(12*c^2
*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e))*Sqrt[(b + 2*c*x)^2/((b^2 - 4*a*c)*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x
^2])/Sqrt[b^2 - 4*a*c])^2)]*(1 + (2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c])*EllipticF[2*ArcTan[(Sqrt
[2]*c^(1/4)*(a + b*x + c*x^2)^(1/4))/(b^2 - 4*a*c)^(1/4)], 1/2])/(160*Sqrt[2]*c^(19/4)*(b + 2*c*x))

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 779

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

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 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

\begin{align*} \int (d+e x)^3 \left (a+b x+c x^2\right )^{3/4} \, dx &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{2 \int (d+e x) \left (\frac{1}{4} \left (22 c d^2-7 b d e-8 a e^2\right )+\frac{15}{4} e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/4} \, dx}{11 c}\\ &=\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}+\frac{\left ((2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right )\right ) \int \left (a+b x+c x^2\right )^{3/4} \, dx}{24 c^3}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right )\right ) \int \frac{1}{\sqrt [4]{a+b x+c x^2}} \, dx}{160 c^4}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{40 c^4 (b+2 c x)}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\left (\left (b^2-4 a c\right )^{3/2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 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^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{80 c^{9/2} (b+2 c x)}+\frac{\left (\left (b^2-4 a c\right )^{3/2} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{(b+2 c x)^2}\right ) \operatorname{Subst}\left (\int \frac{1-\frac{2 \sqrt{c} x^2}{\sqrt{b^2-4 a c}}}{\sqrt{b^2-4 a c+4 c x^4}} \, dx,x,\sqrt [4]{a+b x+c x^2}\right )}{80 c^{9/2} (b+2 c x)}\\ &=\frac{(2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \left (a+b x+c x^2\right )^{3/4}}{120 c^4}+\frac{2 e (d+e x)^2 \left (a+b x+c x^2\right )^{7/4}}{11 c}+\frac{e \left (312 c^2 d^2+55 b^2 e^2-2 c e (121 b d+24 a e)+70 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{7/4}}{462 c^3}-\frac{\sqrt{b^2-4 a c} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) (b+2 c x) \sqrt [4]{a+b x+c x^2}}{80 c^{9/2} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}+\frac{\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{80 \sqrt{2} c^{19/4} (b+2 c x)}-\frac{\left (b^2-4 a c\right )^{7/4} (2 c d-b e) \left (12 c^2 d^2+5 b^2 e^2-4 c e (3 b d+2 a e)\right ) \sqrt{\frac{(b+2 c x)^2}{\left (b^2-4 a c\right ) \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )^2}} \left (1+\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} \sqrt [4]{a+b x+c x^2}}{\sqrt [4]{b^2-4 a c}}\right )|\frac{1}{2}\right )}{160 \sqrt{2} c^{19/4} (b+2 c x)}\\ \end{align*}

Mathematica [C]  time = 0.636513, size = 234, normalized size = 0.33 $\frac{77 (b+2 c x) (2 c d-b e) \left (-4 c e (2 a e+3 b d)+5 b^2 e^2+12 c^2 d^2\right ) \left (8 c (a+x (b+c x))-3 \sqrt{2} \left (b^2-4 a c\right ) \sqrt [4]{\frac{c (a+x (b+c x))}{4 a c-b^2}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{(b+2 c x)^2}{b^2-4 a c}\right )\right )+160 c^2 e (a+x (b+c x))^2 \left (-2 c e (24 a e+121 b d+35 b e x)+55 b^2 e^2+4 c^2 d (78 d+35 e x)\right )+13440 c^4 e (d+e x)^2 (a+x (b+c x))^2}{73920 c^5 \sqrt [4]{a+x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(13440*c^4*e*(d + e*x)^2*(a + x*(b + c*x))^2 + 160*c^2*e*(a + x*(b + c*x))^2*(55*b^2*e^2 + 4*c^2*d*(78*d + 35*
e*x) - 2*c*e*(121*b*d + 24*a*e + 35*b*e*x)) + 77*(2*c*d - b*e)*(12*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(3*b*d + 2*a*e)
)*(b + 2*c*x)*(8*c*(a + x*(b + c*x)) - 3*Sqrt[2]*(b^2 - 4*a*c)*((c*(a + x*(b + c*x)))/(-b^2 + 4*a*c))^(1/4)*Hy
pergeometric2F1[1/4, 1/2, 3/2, (b + 2*c*x)^2/(b^2 - 4*a*c)]))/(73920*c^5*(a + x*(b + c*x))^(1/4))

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Maple [F]  time = 1.04, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{4}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((e*x+d)^3*(c*x^2+b*x+a)^(3/4),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{3}{4}}{\left (e x + d\right )}^{3}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError