### 3.2455 $$\int \frac{(a+b x+c x^2)^{5/2}}{\sqrt{d+e x}} \, dx$$

Optimal. Leaf size=847 $\frac{2 \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{693 c^2 e^5}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-4 c^3 e (64 b d-69 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (41 b^2 d^2-92 a b e d+60 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{c x^2+b x+a}}$

[Out]

(2*Sqrt[d + e*x]*(128*c^4*d^4 - 4*b^4*e^4 - 4*c^3*d^2*e*(76*b*d - 69*a*e) - b^2*c*e^3*(7*b*d - 27*a*e) + 3*c^2
*e^2*(65*b^2*d^2 - 124*a*b*d*e + 60*a^2*e^2) - 12*c*e*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)
)*x)*Sqrt[a + b*x + c*x^2])/(693*c^2*e^5) + (10*Sqrt[d + e*x]*(16*c^2*d^2 + 3*b^2*e^2 - c*e*(23*b*d - 18*a*e)
- 7*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(693*c*e^3) + (2*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*
e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^4*d^4 + 8*b^4*e^4 + b^2*c*e^3*(29*b*d - 93*a*e) - 4*c^3*d
^2*e*(64*b*d - 93*a*e) + 3*c^2*e^2*(33*b^2*d^2 - 124*a*b*d*e + 124*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)])/(693*c^3*e^6*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^4*d^
4 + 2*b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 69*a*e) + b^2*c*e^3*(5*b*d - 21*a*e) + 3*c^2*e^2*(41*b^2*d^2 - 92*a*b*d*
e + 60*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)])/(693*c^3*e^6*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 2.70071, antiderivative size = 847, 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 = {734, 814, 843, 718, 424, 419} $\frac{2 \sqrt{d+e x} \left (c x^2+b x+a\right )^{5/2}}{11 e}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (c x^2+b x+a\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 c^3 e (76 b d-69 a e) d^2-4 b^4 e^4-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b e d+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{c x^2+b x+a}}{693 c^2 e^5}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4-4 c^3 e (64 b d-93 a e) d^2+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b e d+124 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (128 c^4 d^4-4 c^3 e (64 b d-69 a e) d^2+2 b^4 e^4+b^2 c e^3 (5 b d-21 a e)+3 c^2 e^2 \left (41 b^2 d^2-92 a b e d+60 a^2 e^2\right )\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{c x^2+b x+a}}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + b*x + c*x^2)^(5/2)/Sqrt[d + e*x],x]

[Out]

(2*Sqrt[d + e*x]*(128*c^4*d^4 - 4*b^4*e^4 - 4*c^3*d^2*e*(76*b*d - 69*a*e) - b^2*c*e^3*(7*b*d - 27*a*e) + 3*c^2
*e^2*(65*b^2*d^2 - 124*a*b*d*e + 60*a^2*e^2) - 12*c*e*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e)
)*x)*Sqrt[a + b*x + c*x^2])/(693*c^2*e^5) + (10*Sqrt[d + e*x]*(16*c^2*d^2 + 3*b^2*e^2 - c*e*(23*b*d - 18*a*e)
- 7*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(693*c*e^3) + (2*Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2))/(11*
e) - (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(128*c^4*d^4 + 8*b^4*e^4 + b^2*c*e^3*(29*b*d - 93*a*e) - 4*c^3*d
^2*e*(64*b*d - 93*a*e) + 3*c^2*e^2*(33*b^2*d^2 - 124*a*b*d*e + 124*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)])/(693*c^3*e^6*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sq
rt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(128*c^4*d^
4 + 2*b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 69*a*e) + b^2*c*e^3*(5*b*d - 21*a*e) + 3*c^2*e^2*(41*b^2*d^2 - 92*a*b*d*
e + 60*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)])/(693*c^3*e^6*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 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 \frac{\left (a+b x+c x^2\right )^{5/2}}{\sqrt{d+e x}} \, dx &=\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{5 \int \frac{(b d-2 a e+(2 c d-b e) x) \left (a+b x+c x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx}{11 e}\\ &=\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}+\frac{10 \int \frac{\left (\frac{1}{2} \left (9 c e (b d-2 a e)^2-2 (2 c d-b e) \left (b d \left (4 c d-\frac{3 b e}{2}\right )-a e \left (c d+\frac{b e}{2}\right )\right )\right )-2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{d+e x}} \, dx}{231 c e^3}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{4 \int \frac{\frac{1}{4} \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (8 b c d^2-3 b^2 d e-2 a c d e-a b e^2\right )\right )\right )+\frac{1}{4} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{693 c^2 e^5}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{\left ((2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{693 c^2 e^6}-\frac{\left (4 \left (-\frac{1}{4} d (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right )+\frac{1}{4} e \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (8 b c d^2-3 b^2 d e-2 a c d e-a b e^2\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{693 c^2 e^6}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{\left (\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 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 )}{693 c^3 e^6 \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 (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 a^2 e^2\right )\right )+\frac{1}{4} e \left (4 (2 c d-b e) \left (4 b c d^2-b^2 d e-2 a c d e-a b e^2\right ) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+5 c e (b d-2 a e) \left (9 c e (b d-2 a e)^2-(2 c d-b e) \left (8 b c d^2-3 b^2 d e-2 a c d e-a 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 )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=\frac{2 \sqrt{d+e x} \left (128 c^4 d^4-4 b^4 e^4-4 c^3 d^2 e (76 b d-69 a e)-b^2 c e^3 (7 b d-27 a e)+3 c^2 e^2 \left (65 b^2 d^2-124 a b d e+60 a^2 e^2\right )-12 c e (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{693 c^2 e^5}+\frac{10 \sqrt{d+e x} \left (16 c^2 d^2+3 b^2 e^2-c e (23 b d-18 a e)-7 c e (2 c d-b e) x\right ) \left (a+b x+c x^2\right )^{3/2}}{693 c e^3}+\frac{2 \sqrt{d+e x} \left (a+b x+c x^2\right )^{5/2}}{11 e}-\frac{\sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^4 d^4+8 b^4 e^4+b^2 c e^3 (29 b d-93 a e)-4 c^3 d^2 e (64 b d-93 a e)+3 c^2 e^2 \left (33 b^2 d^2-124 a b d e+124 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 )}{693 c^3 e^6 \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{4 \sqrt{2} \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (128 c^4 d^4-256 b c^3 d^3 e+123 b^2 c^2 d^2 e^2+276 a c^3 d^2 e^2+5 b^3 c d e^3-276 a b c^2 d e^3+2 b^4 e^4-21 a b^2 c e^4+180 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 )}{693 c^3 e^6 \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 13.9757, size = 10879, normalized size = 12.84 $\text{Result too large to show}$

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/Sqrt[d + e*x],x]

[Out]

Result too large to show

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Maple [B]  time = 0.401, size = 12152, normalized size = 14.4 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),x)

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}}}{\sqrt{e x + d}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/sqrt(e*x + d), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\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((c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [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((c*x**2+b*x+a)**(5/2)/(e*x+d)**(1/2),x)

[Out]

Timed out

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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((c*x^2+b*x+a)^(5/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

Timed out