∫ (a+bx+cx2)5∕2 ______________________

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

10/693*(16*c^2*d^2+3*b^2*e^2-c*e*(-18*a*e+23*b*d)-7*c*e*(-b*e+2*c*d)*x)*(c*x^2+b*x+a)^(3/2)*(e*x+d)^(1/2)/c/e^

3+2/11*(c*x^2+b*x+a)^(5/2)*(e*x+d)^(1/2)/e+2/693*(128*c^4*d^4-4*b^4*e^4-4*c^3*d^2*e*(-69*a*e+76*b*d)-b^2*c*e^3

*(-27*a*e+7*b*d)+3*c^2*e^2*(60*a^2*e^2-124*a*b*d*e+65*b^2*d^2)-12*c*e*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-

2*a*e+b*d))*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/e^5-1/693*(-b*e+2*c*d)*(128*c^4*d^4+8*b^4*e^4+b^2*c*e^3*(

-93*a*e+29*b*d)-4*c^3*d^2*e*(-93*a*e+64*b*d)+3*c^2*e^2*(124*a^2*e^2-124*a*b*d*e+33*b^2*d^2))*EllipticE(1/2*((b

+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)

^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^3/e^6/(c*x^2

+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+4/693*(a*e^2-b*d*e+c*d^2)*(128*c^4*d^4+2*b^4*

e^4-4*c^3*d^2*e*(-69*a*e+64*b*d)+b^2*c*e^3*(-21*a*e+5*b*d)+3*c^2*e^2*(60*a^2*e^2-92*a*b*d*e+41*b^2*d^2))*Ellip

ticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+

(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*

d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^3/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 1.85, antiderivative size = 847, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {748, 828, 857, 732, 435, 430} \begin {gather*} \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 (\text {ArcSin}\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 (\text {ArcSin}\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}} \end {gather*}

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

(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])

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]

))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt

Q[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell

ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0

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]

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]

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])

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]

\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 ) \text {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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 33.08, size = 10879, normalized size = 12.84 \begin {gather*} \text {Result too large to show} \end {gather*}

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(12152\) vs. \(2(777)=1554\).
time = 0.88, size = 12153, normalized size = 14.35


method	result	size

elliptic	\(\text {Expression too large to display}\)	\(2544\)
risch	\(\text {Expression too large to display}\)	\(4982\)
default	\(\text {Expression too large to display}\)	\(12153\)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.75, size = 871, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left ({\left (256 \, c^{6} d^{6} - 768 \, b c^{5} d^{5} e + 6 \, {\left (121 \, b^{2} c^{4} + 156 \, a c^{5}\right )} d^{4} e^{2} - 4 \, {\left (43 \, b^{3} c^{3} + 468 \, a b c^{4}\right )} d^{3} e^{3} - 3 \, {\left (11 \, b^{4} c^{2} - 260 \, a b^{2} c^{3} - 416 \, a^{2} c^{4}\right )} d^{2} e^{4} - 3 \, {\left (3 \, b^{5} c - 52 \, a b^{3} c^{2} + 416 \, a^{2} b c^{3}\right )} d e^{5} - {\left (8 \, b^{6} - 105 \, a b^{4} c + 498 \, a^{2} b^{2} c^{2} - 1080 \, a^{3} c^{3}\right )} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right ) + 3 \, {\left (256 \, c^{6} d^{5} e - 640 \, b c^{5} d^{4} e^{2} + 2 \, {\left (227 \, b^{2} c^{4} + 372 \, a c^{5}\right )} d^{3} e^{3} - {\left (41 \, b^{3} c^{3} + 1116 \, a b c^{4}\right )} d^{2} e^{4} - {\left (13 \, b^{4} c^{2} - 186 \, a b^{2} c^{3} - 744 \, a^{2} c^{4}\right )} d e^{5} - {\left (8 \, b^{5} c - 93 \, a b^{3} c^{2} + 372 \, a^{2} b c^{3}\right )} e^{6}\right )} \sqrt {c} e^{\frac {1}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )} e^{\left (-2\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )} e^{\left (-3\right )}}{27 \, c^{3}}, \frac {{\left (c d + {\left (3 \, c x + b\right )} e\right )} e^{\left (-1\right )}}{3 \, c}\right )\right ) + 3 \, {\left (128 \, c^{6} d^{4} e^{2} + {\left (63 \, c^{6} x^{4} + 161 \, b c^{5} x^{3} - 4 \, b^{4} c^{2} + 42 \, a b^{2} c^{3} + 333 \, a^{2} c^{4} + {\left (113 \, b^{2} c^{4} + 216 \, a c^{5}\right )} x^{2} + {\left (3 \, b^{3} c^{3} + 347 \, a b c^{4}\right )} x\right )} e^{6} - {\left (70 \, c^{6} d x^{3} + 185 \, b c^{5} d x^{2} + {\left (139 \, b^{2} c^{4} + 262 \, a c^{5}\right )} d x + {\left (7 \, b^{3} c^{3} + 487 \, a b c^{4}\right )} d\right )} e^{5} + {\left (80 \, c^{6} d^{2} x^{2} + 224 \, b c^{5} d^{2} x + {\left (195 \, b^{2} c^{4} + 356 \, a c^{5}\right )} d^{2}\right )} e^{4} - 16 \, {\left (6 \, c^{6} d^{3} x + 19 \, b c^{5} d^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x + a} \sqrt {x e + d}\right )} e^{\left (-7\right )}}{2079 \, c^{4}} \end {gather*}

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

2/2079*((256*c^6*d^6 - 768*b*c^5*d^5*e + 6*(121*b^2*c^4 + 156*a*c^5)*d^4*e^2 - 4*(43*b^3*c^3 + 468*a*b*c^4)*d^

3*e^3 - 3*(11*b^4*c^2 - 260*a*b^2*c^3 - 416*a^2*c^4)*d^2*e^4 - 3*(3*b^5*c - 52*a*b^3*c^2 + 416*a^2*b*c^3)*d*e^

5 - (8*b^6 - 105*a*b^4*c + 498*a^2*b^2*c^2 - 1080*a^3*c^3)*e^6)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d

^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (

2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c) + 3*(256*c^6*d^5*e - 640*b*c^5*d^4*e^2 +

2*(227*b^2*c^4 + 372*a*c^5)*d^3*e^3 - (41*b^3*c^3 + 1116*a*b*c^4)*d^2*e^4 - (13*b^4*c^2 - 186*a*b^2*c^3 - 744

*a^2*c^4)*d*e^5 - (8*b^5*c - 93*a*b^3*c^2 + 372*a^2*b*c^3)*e^6)*sqrt(c)*e^(1/2)*weierstrassZeta(4/3*(c^2*d^2 -

b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^

3 - 9*a*b*c)*e^3)*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)*e^(-2)/c^2, -4/2

7*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*

x + b)*e)*e^(-1)/c)) + 3*(128*c^6*d^4*e^2 + (63*c^6*x^4 + 161*b*c^5*x^3 - 4*b^4*c^2 + 42*a*b^2*c^3 + 333*a^2*c

^4 + (113*b^2*c^4 + 216*a*c^5)*x^2 + (3*b^3*c^3 + 347*a*b*c^4)*x)*e^6 - (70*c^6*d*x^3 + 185*b*c^5*d*x^2 + (139

*b^2*c^4 + 262*a*c^5)*d*x + (7*b^3*c^3 + 487*a*b*c^4)*d)*e^5 + (80*c^6*d^2*x^2 + 224*b*c^5*d^2*x + (195*b^2*c^

4 + 356*a*c^5)*d^2)*e^4 - 16*(6*c^6*d^3*x + 19*b*c^5*d^3)*e^3)*sqrt(c*x^2 + b*x + a)*sqrt(x*e + d))*e^(-7)/c^4

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\sqrt {d + e x}}\, dx \end {gather*}

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

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{\sqrt {d+e\,x}} \,d x \end {gather*}

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

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

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3.25.55 \(\int \frac {(a+b x+c x^2)^{5/2}}{\sqrt {d+e x}} \, dx\) [2455]