### 3.2447 $$\int (d+e x)^{3/2} (a+b x+c x^2)^{3/2} \, dx$$

Optimal. Leaf size=816 $\frac{2 e \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}}$

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 + 8*b^4*e^4 - c^3*d^2*e*(19*b*d - 42*a*e) - b^2*c*e^3*(19*b*d + 21*a*e) + 3*c^2*e^
2*(2*b^2*d^2 + 17*a*b*d*e - 10*a^2*e^2) - 3*c*e*(2*c*d - b*e)*(c^2*d^2 + 8*b^2*e^2 - c*e*(b*d + 31*a*e))*x)*Sq
rt[a + b*x + c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 - 6*b^2*e^2 + c*e*(13*b*d - 3*a*e) + 14*c*e*(2
*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*c) - (8*
Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c^2*d^2 - 2*b^2*e^2 - c*e*(b*d - 9*a*e))*(c^2*d^2 + b^2*e^2 - c*e*(b*
d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4
*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(11
55*c^4*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^
2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 21*a*e) + b^2*c*e^3*(13*b*d
+ 51*a*e) + 3*c^2*e^2*(b^2*d^2 - 28*a*b*d*e - 20*a^2*e^2))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])
*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b
^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(1155*c^4*e^4*Sqrt[d + e
*x]*Sqrt[a + b*x + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.83596, antiderivative size = 816, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {742, 814, 843, 718, 424, 419} $\frac{2 e \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 c}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{231 c^2 e}+\frac{2 \sqrt{d+e x} \left (8 c^4 d^4-c^3 e (19 b d-42 a e) d^2+8 b^4 e^4-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b e d-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{1155 c^3 e^3}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^4 d^4-4 c^3 e (8 b d-21 a e) d^2-8 b^4 e^4+b^2 c e^3 (13 b d+51 a e)+3 c^2 e^2 \left (b^2 d^2-28 a b e d-20 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{c x^2+b x+a}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*c^4*d^4 + 8*b^4*e^4 - c^3*d^2*e*(19*b*d - 42*a*e) - b^2*c*e^3*(19*b*d + 21*a*e) + 3*c^2*e^
2*(2*b^2*d^2 + 17*a*b*d*e - 10*a^2*e^2) - 3*c*e*(2*c*d - b*e)*(c^2*d^2 + 8*b^2*e^2 - c*e*(b*d + 31*a*e))*x)*Sq
rt[a + b*x + c*x^2])/(1155*c^3*e^3) + (2*Sqrt[d + e*x]*(c^2*d^2 - 6*b^2*e^2 + c*e*(13*b*d - 3*a*e) + 14*c*e*(2
*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(231*c^2*e) + (2*e*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*c) - (8*
Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c^2*d^2 - 2*b^2*e^2 - c*e*(b*d - 9*a*e))*(c^2*d^2 + b^2*e^2 - c*e*(b*
d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4
*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(11
55*c^4*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (2*Sqrt[2]*Sqrt[b^
2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 21*a*e) + b^2*c*e^3*(13*b*d
+ 51*a*e) + 3*c^2*e^2*(b^2*d^2 - 28*a*b*d*e - 20*a^2*e^2))*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])
*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b
^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(1155*c^4*e^4*Sqrt[d + e
*x]*Sqrt[a + b*x + c*x^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 814

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*(a + b*x + c*x^
2)^p)/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), x] - Dist[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)), Int[(d + e*x)^m*(a
+ b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2*a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p -
c*d - 2*c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c^2*d^2*(1 + 2*p) - c*e*(b*
d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0
] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && GtQ[p, 0] && (IntegerQ[p] ||  !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])
) &&  !ILtQ[m + 2*p, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 843

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

Rule 718

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(2*Rt[b^2 - 4*a*c, 2]
*(d + e*x)^m*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))])/(c*Sqrt[a + b*x + c*x^2]*((2*c*(d + e*x))/(2*c*d -
b*e - e*Rt[b^2 - 4*a*c, 2]))^m), Subst[Int[(1 + (2*e*Rt[b^2 - 4*a*c, 2]*x^2)/(2*c*d - b*e - e*Rt[b^2 - 4*a*c,
2]))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], 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] && EqQ[m^2, 1/4]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac{2 \int \frac{\left (\frac{1}{2} \left (11 c d^2-e (5 b d+a e)\right )+3 e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 c}\\ &=\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{4 \int \frac{\left (\frac{3}{4} e \left (b c^2 d^3+13 b^2 c d^2 e-58 a c^2 d^2 e-6 b^3 d e^2+27 a b c d e^2-2 a b^2 e^3+6 a^2 c e^3\right )+\frac{3}{4} e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{231 c^2 e^2}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}+\frac{8 \int \frac{-\frac{3}{8} e \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )-3 e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3465 c^3 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{1155 c^3 e^4}+\frac{\left (8 \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac{3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{3465 c^3 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{\left (8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}}{\sqrt{1-x^2}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{a+b x+c x^2}}+\frac{\left (16 \sqrt{2} \sqrt{b^2-4 a c} \left (3 d e (2 c d-b e) \left (c^2 d^2-b c d e+b^2 e^2-3 a c e^2\right ) \left (c^2 d^2-b c d e-2 b^2 e^2+9 a c e^2\right )-\frac{3}{8} e^2 \left (2 (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) \left (\frac{1}{2} b d (4 c d-b e)-a e \left (c d+\frac{b e}{2}\right )\right )+5 c e (b d-2 a e) \left (6 b^3 d e^2+2 a c e \left (29 c d^2-3 a e^2\right )-b c d \left (c d^2+27 a e^2\right )-b^2 \left (13 c d^2 e-2 a e^3\right )\right )\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-b e-\sqrt{b^2-4 a c} e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2} \sqrt{1+\frac{2 \sqrt{b^2-4 a c} e x^2}{2 c d-b e-\sqrt{b^2-4 a c} e}}} \, dx,x,\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )}{3465 c^4 e^5 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^4 d^4+8 b^4 e^4-c^3 d^2 e (19 b d-42 a e)-b^2 c e^3 (19 b d+21 a e)+3 c^2 e^2 \left (2 b^2 d^2+17 a b d e-10 a^2 e^2\right )-3 c e (2 c d-b e) \left (c^2 d^2+8 b^2 e^2-c e (b d+31 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{1155 c^3 e^3}+\frac{2 \sqrt{d+e x} \left (c^2 d^2-6 b^2 e^2+c e (13 b d-3 a e)+14 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{231 c^2 e}+\frac{2 e \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 c}-\frac{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^4 d^4-32 b c^3 d^3 e+3 b^2 c^2 d^2 e^2+84 a c^3 d^2 e^2+13 b^3 c d e^3-84 a b c^2 d e^3-8 b^4 e^4+51 a b^2 c e^4-60 a^2 c^2 e^4\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+\sqrt{b^2-4 a c}+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{1155 c^4 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 13.9613, size = 10848, normalized size = 13.29 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.407, size = 11933, normalized size = 14.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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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}{2}}{\left (e x + d\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out