3.2448 \(\int \sqrt{d+e x} (a+b x+c x^2)^{3/2} \, dx\)

Optimal. Leaf size=712 \[ \frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\sqrt{b^2-4 a c}+b+2 c x}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 e \sqrt{b^2-4 a c}}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}\right )}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (-28 a^2 e^2-20 a b d e+3 b^2 d^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{315 c^2 e^3}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c e} \]

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 4*b^3*e^3 - 3*c^2*d*e*(5*b*d - 8*a*e) + 3*b*c*e^2*(b*d + 3*a*e) - 6*c*e*(c^2*d^2
+ 2*b^2*e^2 - c*e*(b*d + 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(a
+ b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c
]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3*c^2*e^2*(3*b^2*d^2 -
20*a*b*d*e - 28*a^2*e^2))*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)])/(315*c^3*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]) + (8*S
qrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*Sqr
t[(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[Ar
cSin[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)])/(315*c^3*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.27277, antiderivative size = 712, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {734, 832, 814, 843, 718, 424, 419} \[ -\frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (-28 a^2 e^2-20 a b d e+3 b^2 d^2\right )+b^2 c e^3 (57 a e+7 b d)-4 c^3 d^2 e (8 b d-15 a e)-8 b^4 e^4+16 c^4 d^4\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}+\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (-6 c e x \left (-c e (7 a e+b d)+2 b^2 e^2+c^2 d^2\right )-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (3 a e+b d)-4 b^3 e^3+8 c^3 d^3\right )}{315 c^2 e^3}+\frac{8 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{21 c e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[d + e*x]*(8*c^3*d^3 - 4*b^3*e^3 - 3*c^2*d*e*(5*b*d - 8*a*e) + 3*b*c*e^2*(b*d + 3*a*e) - 6*c*e*(c^2*d^2
+ 2*b^2*e^2 - c*e*(b*d + 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(315*c^2*e^3) - (2*(2*c*d - b*e)*Sqrt[d + e*x]*(a
+ b*x + c*x^2)^(3/2))/(21*c*e) + (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c
]*(16*c^4*d^4 - 8*b^4*e^4 - 4*c^3*d^2*e*(8*b*d - 15*a*e) + b^2*c*e^3*(7*b*d + 57*a*e) + 3*c^2*e^2*(3*b^2*d^2 -
20*a*b*d*e - 28*a^2*e^2))*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)])/(315*c^3*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]) + (8*S
qrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d - 3*a*e))*Sqr
t[(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[Ar
cSin[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)])/(315*c^3*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 734

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

Rule 832

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

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 \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2} \, dx &=\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{\int \sqrt{d+e x} (b d-2 a e+(2 c d-b e) x) \sqrt{a+b x+c x^2} \, dx}{3 e}\\ &=-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{2 \int \frac{\left (\frac{1}{2} \left (b c d^2+3 b^2 d e-16 a c d e+a b e^2\right )+\left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{21 c e}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-4 b^3 e^3-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (b d+3 a e)-6 c e \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}+\frac{4 \int \frac{\frac{1}{4} \left (5 c e (b d-2 a e) \left (b c d^2+3 b^2 d e-16 a c d e+a b e^2\right )-4 \left (c^2 d^2+2 b^2 e^2-c e (b d+7 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 )\right )-\frac{1}{4} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{315 c^2 e^3}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-4 b^3 e^3-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (b d+3 a e)-6 c e \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{\left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{315 c^2 e^4}+\frac{\left (4 \left (-\frac{1}{4} d \left (-16 c^4 d^4+8 b^4 e^4+4 c^3 d^2 e (8 b d-15 a e)-b^2 c e^3 (7 b d+57 a e)-3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right )+\frac{1}{4} e \left (5 c e (b d-2 a e) \left (b c d^2+3 b^2 d e-16 a c d e+a b e^2\right )-4 \left (c^2 d^2+2 b^2 e^2-c e (b d+7 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 )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{315 c^2 e^4}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-4 b^3 e^3-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (b d+3 a e)-6 c e \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\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 )}{315 c^3 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 (8 \sqrt{2} \sqrt{b^2-4 a c} \left (-\frac{1}{4} d \left (-16 c^4 d^4+8 b^4 e^4+4 c^3 d^2 e (8 b d-15 a e)-b^2 c e^3 (7 b d+57 a e)-3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\right )+\frac{1}{4} e \left (5 c e (b d-2 a e) \left (b c d^2+3 b^2 d e-16 a c d e+a b e^2\right )-4 \left (c^2 d^2+2 b^2 e^2-c e (b d+7 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 )\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 )}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (8 c^3 d^3-4 b^3 e^3-3 c^2 d e (5 b d-8 a e)+3 b c e^2 (b d+3 a e)-6 c e \left (c^2 d^2+2 b^2 e^2-c e (b d+7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{315 c^2 e^3}-\frac{2 (2 c d-b e) \sqrt{d+e x} \left (a+b x+c x^2\right )^{3/2}}{21 c e}+\frac{2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} \left (16 c^4 d^4-8 b^4 e^4-4 c^3 d^2 e (8 b d-15 a e)+b^2 c e^3 (7 b d+57 a e)+3 c^2 e^2 \left (3 b^2 d^2-20 a b d e-28 a^2 e^2\right )\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 )}{315 c^3 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{8 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (2 c^2 d^2-2 b c d e-b^2 e^2+6 a c e^2\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 )}{315 c^3 e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 13.6839, size = 7541, normalized size = 10.59 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.332, size = 9177, normalized size = 12.9 \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)^(1/2)*(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}} \sqrt{e x + d}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d + e x} \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)**(1/2)*(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(sqrt(d + e*x)*(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)^(1/2)*(c*x^2+b*x+a)^(3/2),x, algorithm="giac")

[Out]

Timed out