\(\int \frac {(a+b x+c x^2)^{5/2}}{(d+e x)^{3/2}} \, dx\) [2456]

Optimal result

Integrand size = 24, antiderivative size = 716 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \left (32 c^2 d^2+b^2 e^2-4 c e (8 b d-7 a e)\right ) x\right ) \sqrt {a+b x+c x^2}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (128 c^4 d^4-b^4 e^4-4 c^3 d^2 e (64 b d-57 a e)-b^2 c e^3 (7 b d-15 a e)+3 c^2 e^2 \left (45 b^2 d^2-76 a b d e+28 a^2 e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \left (128 c^2 d^2-b^2 e^2-4 c e (32 b d-33 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

-2*(c*x^2+b*x+a)^(5/2)/e/(e*x+d)^(1/2)-10/63*(-14*c*e*x-15*b*e+16*c*d)*(c*x^2+b*x+a)^(3/2)*(e*x+d)^(1/2)/e^3-2
/63*(128*c^3*d^3-b^3*e^3+3*b*c*e^2*(-36*a*e+37*b*d)-12*c^2*d*e*(-11*a*e+20*b*d)-3*c*e*(32*c^2*d^2+b^2*e^2-4*c*
e*(-7*a*e+8*b*d))*x)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e^5+2/63*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(-57*a*e+64
*b*d)-b^2*c*e^3*(-15*a*e+7*b*d)+3*c^2*e^2*(28*a^2*e^2-76*a*b*d*e+45*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^2/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)-2/63*(-b*e+2*c*d)*(a*e^2-b*d*e+c*d^2)*(128*c^2*d^2-b^2*e^2-4
*c*e*(-33*a*e+32*b*d))*EllipticF(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^2/e^6/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 716, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {746, 828, 857, 732, 435, 430} \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (3 c^2 e^2 \left (28 a^2 e^2-76 a b d e+45 b^2 d^2\right )-b^2 c e^3 (7 b d-15 a e)-4 c^3 d^2 e (64 b d-57 a e)-b^4 e^4+128 c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \left (-4 c e (32 b d-33 a e)-b^2 e^2+128 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-3 c e x \left (-4 c e (8 b d-7 a e)+b^2 e^2+32 c^2 d^2\right )-12 c^2 d e (20 b d-11 a e)+3 b c e^2 (37 b d-36 a e)-b^3 e^3+128 c^3 d^3\right )}{63 c e^5}-\frac {10 \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} (-15 b e+16 c d-14 c e x)}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}} \]

[In]

Int[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

(-2*Sqrt[d + e*x]*(128*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(37*b*d - 36*a*e) - 12*c^2*d*e*(20*b*d - 11*a*e) - 3*c*e*
(32*c^2*d^2 + b^2*e^2 - 4*c*e*(8*b*d - 7*a*e))*x)*Sqrt[a + b*x + c*x^2])/(63*c*e^5) - (10*Sqrt[d + e*x]*(16*c*
d - 15*b*e - 14*c*e*x)*(a + b*x + c*x^2)^(3/2))/(63*e^3) - (2*(a + b*x + c*x^2)^(5/2))/(e*Sqrt[d + e*x]) + (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*c^2*e^6*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)*(c*d^2 - b*d*e + a*e^2)*(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*c^2*e^6*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 430

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])

Rule 435

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
]

Rule 732

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 746

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 828

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 857

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {5 \int \frac {(b+2 c x) \left (a+b x+c x^2\right )^{3/2}}{\sqrt {d+e x}} \, dx}{e} \\ & = -\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}-\frac {10 \int \frac {\left (\frac {1}{2} c \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+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}}{\sqrt {d+e x}} \, dx}{21 c e^3} \\ & = -\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \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}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {4 \int \frac {\frac {1}{4} c \left (2 \left (32 c^2 d^2+b^2 e^2-4 c e (8 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 )+5 c e (b d-2 a e) \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a e^2\right )\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 c^2 e^5} \\ & = -\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \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}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\frac {\left (2 \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 c e^6}+\frac {\left (4 \left (-\frac {1}{2} c d \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 )+\frac {1}{4} c e \left (2 \left (32 c^2 d^2+b^2 e^2-4 c e (8 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 )+5 c e (b d-2 a e) \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a e^2\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{63 c^2 e^6} \\ & = -\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \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}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\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 ) \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 )}{63 c^2 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}{2} c d \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 )+\frac {1}{4} c e \left (2 \left (32 c^2 d^2+b^2 e^2-4 c e (8 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 )+5 c e (b d-2 a e) \left (15 b^2 d e+4 a c d e-16 b \left (c d^2+a 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 )}{63 c^3 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (128 c^3 d^3-b^3 e^3+3 b c e^2 (37 b d-36 a e)-12 c^2 d e (20 b d-11 a e)-3 c e \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}}{63 c e^5}-\frac {10 \sqrt {d+e x} (16 c d-15 b e-14 c e x) \left (a+b x+c x^2\right )^{3/2}}{63 e^3}-\frac {2 \left (a+b x+c x^2\right )^{5/2}}{e \sqrt {d+e x}}+\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 c^2 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 {2 \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 (128 c^2 d^2-128 b c d e-b^2 e^2+132 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 )}{63 c^2 e^6 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.22 (sec) , antiderivative size = 1140, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\frac {\frac {4 e^2 \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 ) (a+x (b+c x))}{\sqrt {d+e x}}-\frac {2 c e^2 (a+x (b+c x)) \left (63 c \left (c d^2+e (-b d+a e)\right )^2+\left (65 c^3 d^3-b^3 e^3+3 b c e^2 (16 b d-19 a e)+2 c^2 d e (-57 b d+43 a e)\right ) (d+e x)-c e \left (33 c^2 d^2-50 b c d e+15 b^2 e^2+28 a c e^2\right ) x (d+e x)+c^2 e^2 (17 c d-19 b e) x^2 (d+e x)-7 c^3 e^3 x^3 (d+e x)\right )}{\sqrt {d+e x}}-\frac {i (d+e x) \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (-\left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \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 ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )+\left (-b^5 e^5+b^4 e^4 \left (-6 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b^3 c e^3 \left (24 c d^2+16 a e^2+7 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )-b^2 c e^2 \left (16 c^2 d^3+135 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+15 a e^2 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+4 b \left (-12 a^2 c^2 e^5+64 c^3 d^3 e \sqrt {\left (b^2-4 a c\right ) e^2}+3 a c^2 d e^3 \left (-8 c d+19 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )-4 c^2 \left (32 c^2 d^4 \sqrt {\left (b^2-4 a c\right ) e^2}+3 a^2 e^4 \left (-8 c d+7 \sqrt {\left (b^2-4 a c\right ) e^2}\right )+a c d^2 e^2 \left (-16 c d+57 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{\sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}}{63 c^2 e^7 \sqrt {a+x (b+c x)}} \]

[In]

Integrate[(a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x]

[Out]

((4*e^2*(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))*(a + x*(b + c*x)))/Sqrt[d + e*x] - (2*c*e^2*(a + x*(b + c*x))*(63*c*(c*d^2
+ e*(-(b*d) + a*e))^2 + (65*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(16*b*d - 19*a*e) + 2*c^2*d*e*(-57*b*d + 43*a*e))*(d
+ e*x) - c*e*(33*c^2*d^2 - 50*b*c*d*e + 15*b^2*e^2 + 28*a*c*e^2)*x*(d + e*x) + c^2*e^2*(17*c*d - 19*b*e)*x^2*
(d + e*x) - 7*c^3*e^3*x^3*(d + e*x)))/Sqrt[d + e*x] - (I*(d + e*x)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2
*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqr
t[(b^2 - 4*a*c)*e^2])*(d + e*x))]*(-((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(-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))*Ellipt
icE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]],
-((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]) + (-(b^5*e^5) + b^4*e^4
*(-6*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + b^3*c*e^3*(24*c*d^2 + 16*a*e^2 + 7*d*Sqrt[(b^2 - 4*a*c)*e^2]) - b^2*c*e^
2*(16*c^2*d^3 + 135*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] + 15*a*e^2*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b*(-12*a^2*c^2*e^5 +
64*c^3*d^3*e*Sqrt[(b^2 - 4*a*c)*e^2] + 3*a*c^2*d*e^3*(-8*c*d + 19*Sqrt[(b^2 - 4*a*c)*e^2])) - 4*c^2*(32*c^2*d
^4*Sqrt[(b^2 - 4*a*c)*e^2] + 3*a^2*e^4*(-8*c*d + 7*Sqrt[(b^2 - 4*a*c)*e^2]) + a*c*d^2*e^2*(-16*c*d + 57*Sqrt[(
b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a
*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])
)]))/Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/(63*c^2*e^7*Sqrt[a + x*(b + c*
x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2071\) vs. \(2(648)=1296\).

Time = 3.33 (sec) , antiderivative size = 2072, normalized size of antiderivative = 2.89

method	result	size
risch	\(\text {Expression too large to display}\)	\(2072\)
elliptic	\(\text {Expression too large to display}\)	\(2528\)
default	\(\text {Expression too large to display}\)	\(9187\)

[In]

int((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/63/c*(7*c^3*e^3*x^3+19*b*c^2*e^3*x^2-17*c^3*d*e^2*x^2+28*a*c^2*e^3*x+15*b^2*c*e^3*x-50*b*c^2*d*e^2*x+33*c^3*
d^2*e*x+57*a*b*c*e^3-86*a*c^2*d*e^2+b^3*e^3-48*b^2*c*d*e^2+114*b*c^2*d^2*e-65*c^3*d^3)*(e*x+d)^(1/2)*(c*x^2+b*
x+a)^(1/2)/e^5+1/63/c/e^5*(2*(105*a^2*c^2*e^4+30*a*b^2*c*e^4-330*a*b*c^2*d*e^3+330*a*c^3*d^2*e^2-2*b^4*e^4-14*
b^3*c*d*e^3+207*b^2*c^2*d^2*e^2-386*b*c^3*d^3*e+193*c^4*d^4)*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-
1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^
(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*
e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/
c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+
b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/
(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))+63*(a^3*e^6-3*a^2*b*d*e^5+3*a^2*c*d^2*e^4+3*a*b^2*d^2*e^4-6*a*b*
c*d^3*e^3+3*a*c^2*d^4*e^2-b^3*d^3*e^3+3*b^2*c*d^4*e^2-3*b*c^2*d^5*e+c^3*d^6)*c/e*(-2*(c*e*x^2+b*e*x+a*e)/(a*e^
2-b*d*e+c*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2*(-(b*e-c*d)/(a*e^2-b*d*e+c*d^2)+b*e/(a*e^2-b*d*e+c*d^2))*
(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^
2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+
b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b
^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*c*e/(
a*e^2-b*d*e+c*d^2)*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1
/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-
d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*
a*c+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2)
)/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+1/2/c*(-b+(-4*a*c+b^2)^(1/2))*EllipticF(((x+d/e)/(d/e-1/2*(b
+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2
))))+2*(132*a^2*b*c*e^5-159*a^2*c^2*d*e^4-a*b^3*e^5-228*a*b^2*c*d*e^4+450*a*b*c^2*d^2*e^3-190*a*c^3*d^3*e^2-b^
4*d*e^4+111*b^3*c*d^2*e^3-303*b^2*c^2*d^3*e^2+254*b*c^3*d^4*e-63*c^4*d^5)/e*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)
*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c
+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e
*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b
+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)))*((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1
/2)/(c*x^2+b*x+a)^(1/2)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.29 (sec) , antiderivative size = 1057, normalized size of antiderivative = 1.48 \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

-2/189*((256*c^5*d^6 - 640*b*c^4*d^5*e + 2*(239*b^2*c^3 + 324*a*c^4)*d^4*e^2 - (77*b^3*c^2 + 972*a*b*c^3)*d^3*
e^3 - (13*b^4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d^2*e^4 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*d*e^5 + (256*c^5
*d^5*e - 640*b*c^4*d^4*e^2 + 2*(239*b^2*c^3 + 324*a*c^4)*d^3*e^3 - (77*b^3*c^2 + 972*a*b*c^3)*d^2*e^4 - (13*b^
4*c - 258*a*b^2*c^2 - 456*a^2*c^3)*d*e^5 - (2*b^5 - 33*a*b^3*c + 228*a^2*b*c^2)*e^6)*x)*sqrt(c*e)*weierstrassP
Inverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^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)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 6*(128*c^5*d^5*e - 256*b*
c^4*d^4*e^2 + 3*(45*b^2*c^3 + 76*a*c^4)*d^3*e^3 - (7*b^3*c^2 + 228*a*b*c^3)*d^2*e^4 - (b^4*c - 15*a*b^2*c^2 -
84*a^2*c^3)*d*e^5 + (128*c^5*d^4*e^2 - 256*b*c^4*d^3*e^3 + 3*(45*b^2*c^3 + 76*a*c^4)*d^2*e^4 - (7*b^3*c^2 + 22
8*a*b*c^3)*d*e^5 - (b^4*c - 15*a*b^2*c^2 - 84*a^2*c^3)*e^6)*x)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*
e + (b^2 - 3*a*c)*e^2)/(c^2*e^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)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^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)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(
c*e))) - 3*(7*c^5*e^6*x^4 - 128*c^5*d^4*e^2 + 240*b*c^4*d^3*e^3 - 63*a^2*c^3*e^6 - (111*b^2*c^3 + 212*a*c^4)*d
^2*e^4 + (b^3*c^2 + 183*a*b*c^3)*d*e^5 - (10*c^5*d*e^5 - 19*b*c^4*e^6)*x^3 + (16*c^5*d^2*e^4 - 31*b*c^4*d*e^5
+ (15*b^2*c^3 + 28*a*c^4)*e^6)*x^2 - (32*c^5*d^3*e^3 - 64*b*c^4*d^2*e^4 + (33*b^2*c^3 + 58*a*c^4)*d*e^5 - (b^3
*c^2 + 57*a*b*c^3)*e^6)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^3*e^8*x + c^3*d*e^7)

Sympy [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((c*x**2+b*x+a)**(5/2)/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*x + c*x**2)**(5/2)/(d + e*x)**(3/2), x)

Maxima [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(3/2), x)

Giac [F]

\[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((c*x^2+b*x+a)^(5/2)/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((c*x^2 + b*x + a)^(5/2)/(e*x + d)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x+c x^2\right )^{5/2}}{(d+e x)^{3/2}} \, dx=\int \frac {{\left (c\,x^2+b\,x+a\right )}^{5/2}}{{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2),x)

[Out]

int((a + b*x + c*x^2)^(5/2)/(d + e*x)^(3/2), x)