3.2460 \(\int \frac{(a+b x+c x^2)^{5/2}}{(d+e x)^{11/2}} \, dx\)

Optimal. Leaf size=923 \[ -\frac{2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 e^3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{63 e^5 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right ),-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

[Out]

(-2*(128*c^4*d^5 - 2*a*b^3*e^5 - 4*c^3*d^3*e*(60*b*d - 49*a*e) - b*c*e^3*(b^2*d^2 + 9*a*b*d*e - 24*a^2*e^2) +
3*c^2*d*e^2*(37*b^2*d^2 - 52*a*b*d*e + 12*a^2*e^2) + e*(160*c^4*d^4 - 2*b^4*e^4 - 4*c^3*d^2*e*(80*b*d - 69*a*e
) - b^2*c*e^3*(11*b*d - 27*a*e) + 3*c^2*e^2*(57*b^2*d^2 - 92*a*b*d*e + 28*a^2*e^2))*x)*Sqrt[a + b*x + c*x^2])/
(63*e^5*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(16*c^2*d^3 - b*e^2*(2*b*d - 5*a*e) - c*d*e*(11*b*d -
4*a*e) + e*(26*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(13*b*d - 7*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3*(c*d^2 - b*d*
e + a*e^2)*(d + e*x)^(7/2)) - (2*(a + b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]
*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 -
76*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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)^2*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]*(2*c*d - b*e)*(128*c^2*d^2 - b^2*e^2 - 4*c*e*(32*b*d - 33*a*e))*
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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.18827, antiderivative size = 923, 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 = {732, 810, 843, 718, 424, 419} \[ -\frac{2 \left (c x^2+b x+a\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \left (16 c^2 d^3-c e (11 b d-4 a e) d-b e^2 (2 b d-5 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (c x^2+b x+a\right )^{3/2}}{63 e^3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (128 c^4 d^5-4 c^3 e (60 b d-49 a e) d^3+3 c^2 e^2 \left (37 b^2 d^2-52 a b e d+12 a^2 e^2\right ) d-2 a b^3 e^5-b c e^3 \left (b^2 d^2+9 a b e d-24 a^2 e^2\right )+e \left (160 c^4 d^4-4 c^3 e (80 b d-69 a e) d^2-2 b^4 e^4-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b e d+28 a^2 e^2\right )\right ) x\right ) \sqrt{c x^2+b x+a}}{63 e^5 \left (c d^2-b e d+a e^2\right )^2 (d+e x)^{3/2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^4 d^4-4 c^3 e (64 b d-57 a e) d^2-b^4 e^4-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b e d+28 a^2 e^2\right )\right ) \sqrt{d+e x} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{c x^2+b x+a}}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt{\frac{c (d+e x)}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{c \left (c x^2+b x+a\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{63 e^6 \left (c d^2-b e d+a e^2\right ) \sqrt{d+e x} \sqrt{c x^2+b x+a}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

(-2*(128*c^4*d^5 - 2*a*b^3*e^5 - 4*c^3*d^3*e*(60*b*d - 49*a*e) - b*c*e^3*(b^2*d^2 + 9*a*b*d*e - 24*a^2*e^2) +
3*c^2*d*e^2*(37*b^2*d^2 - 52*a*b*d*e + 12*a^2*e^2) + e*(160*c^4*d^4 - 2*b^4*e^4 - 4*c^3*d^2*e*(80*b*d - 69*a*e
) - b^2*c*e^3*(11*b*d - 27*a*e) + 3*c^2*e^2*(57*b^2*d^2 - 92*a*b*d*e + 28*a^2*e^2))*x)*Sqrt[a + b*x + c*x^2])/
(63*e^5*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^(3/2)) - (2*(16*c^2*d^3 - b*e^2*(2*b*d - 5*a*e) - c*d*e*(11*b*d -
4*a*e) + e*(26*c^2*d^2 + 3*b^2*e^2 - 2*c*e*(13*b*d - 7*a*e))*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3*(c*d^2 - b*d*
e + a*e^2)*(d + e*x)^(7/2)) - (2*(a + b*x + c*x^2)^(5/2))/(9*e*(d + e*x)^(9/2)) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]
*(128*c^4*d^4 - b^4*e^4 - 4*c^3*d^2*e*(64*b*d - 57*a*e) - b^2*c*e^3*(7*b*d - 15*a*e) + 3*c^2*e^2*(45*b^2*d^2 -
76*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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)^2*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]*(2*c*d - b*e)*(128*c^2*d^2 - b^2*e^2 - 4*c*e*(32*b*d - 33*a*e))*
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)])/(63*e^6*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 732

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 + 1)), x] - Dist[p/(e*(m + 1)), Int[(d + e*x)^(m + 1)*(b + 2*c*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] && (IntegerQ[p] || LtQ[m, -1]) && NeQ[m, -1] &&  !ILtQ[m + 2*p + 1, 0] && IntQua
draticQ[a, b, c, d, e, m, p, x]

Rule 810

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

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}}{(d+e x)^{11/2}} \, dx &=-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{5 \int \frac{(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx}{9 e}\\ &=-\frac{2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{2 \int \frac{\left (\frac{1}{2} \left (11 b^2 c d e+20 a c^2 d e+2 b^3 e^2-8 b c \left (2 c d^2+3 a e^2\right )\right )-\frac{1}{2} c \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt{a+b x+c x^2}}{(d+e x)^{5/2}} \, dx}{21 e^3 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{4 \int \frac{-\frac{1}{4} c \left (b^4 d e^3+32 a c^2 d e \left (2 c d^2+3 a e^2\right )+12 b^2 c d e \left (20 c d^2+19 a e^2\right )-b^3 \left (111 c d^2 e^2-a e^4\right )-4 b c \left (32 c^2 d^4+81 a c d^2 e^2+33 a^2 e^4\right )\right )+\frac{1}{2} c \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) x}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{63 e^5 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}-\frac{\left (c (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right )\right ) \int \frac{1}{\sqrt{d+e x} \sqrt{a+b x+c x^2}} \, dx}{63 e^6 \left (c d^2-b d e+a e^2\right )}+\frac{\left (2 c \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right )\right ) \int \frac{\sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx}{63 e^6 \left (c d^2-b d e+a e^2\right )^2}\\ &=-\frac{2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{\left (2 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 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 )}{63 e^6 \left (c d^2-b d e+a e^2\right )^2 \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 (2 \sqrt{2} \sqrt{b^2-4 a c} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\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 )}{63 e^6 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ &=-\frac{2 \left (128 c^4 d^5-2 a b^3 e^5-4 c^3 d^3 e (60 b d-49 a e)-b c e^3 \left (b^2 d^2+9 a b d e-24 a^2 e^2\right )+3 c^2 d e^2 \left (37 b^2 d^2-52 a b d e+12 a^2 e^2\right )+e \left (160 c^4 d^4-2 b^4 e^4-4 c^3 d^2 e (80 b d-69 a e)-b^2 c e^3 (11 b d-27 a e)+3 c^2 e^2 \left (57 b^2 d^2-92 a b d e+28 a^2 e^2\right )\right ) x\right ) \sqrt{a+b x+c x^2}}{63 e^5 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac{2 \left (16 c^2 d^3-b e^2 (2 b d-5 a e)-c d e (11 b d-4 a e)+e \left (26 c^2 d^2+3 b^2 e^2-2 c e (13 b d-7 a e)\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{63 e^3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{7/2}}-\frac{2 \left (a+b x+c x^2\right )^{5/2}}{9 e (d+e x)^{9/2}}+\frac{2 \sqrt{2} \sqrt{b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 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 )}{63 e^6 \left (c d^2-b d e+a e^2\right )^2 \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} (2 c d-b e) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\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 )}{63 e^6 \left (c d^2-b d e+a e^2\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [C]  time = 14.0085, size = 8108, normalized size = 8.78 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(11/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.576, size = 44994, normalized size = 48.8 \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)^(11/2),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}}}{{\left (e x + d\right )}^{\frac{11}{2}}}\,{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)^(11/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(11/2), 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}}{e^{6} x^{6} + 6 \, d e^{5} x^{5} + 15 \, d^{2} e^{4} x^{4} + 20 \, d^{3} e^{3} x^{3} + 15 \, d^{4} e^{2} x^{2} + 6 \, d^{5} e x + d^{6}}, 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)^(11/2),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)/(e^6*x^
6 + 6*d*e^5*x^5 + 15*d^2*e^4*x^4 + 20*d^3*e^3*x^3 + 15*d^4*e^2*x^2 + 6*d^5*e*x + d^6), 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)**(11/2),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)^(11/2),x, algorithm="giac")

[Out]

Timed out