\(\int \sqrt {d x} (a+b x+c x^2)^{5/2} \, dx\) [2454]

Optimal result

Integrand size = 22, antiderivative size = 616 \[ \int \sqrt {d x} \left (a+b x+c x^2\right )^{5/2} \, dx=-\frac {4 \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d x \sqrt {a+b x+c x^2}}{9009 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 \sqrt {d x} \left (b \left (24 b^4-151 a b^2 c+108 a^2 c^2\right )+3 c \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac {4 \sqrt [4]{a} \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{9009 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (\sqrt {a} b \sqrt {c} \left (24 b^4-241 a b^2 c+708 a^2 c^2\right )+2 \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right )\right ) d \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{9009 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}} \]

[Out]

2/13*(d*x)^(3/2)*(c*x^2+b*x+a)^(5/2)/d-10/9009*(3*b*(-19*a*c+6*b^2)+14*c*(-11*a*c+3*b^2)*x)*(c*x^2+b*x+a)^(3/2
)*(d*x)^(1/2)/c^2+10/143*b*(c*x^2+b*x+a)^(5/2)*(d*x)^(1/2)/c-4/9009*(-924*a^3*c^3+951*a^2*b^2*c^2-268*a*b^4*c+
24*b^6)*d*x*(c*x^2+b*x+a)^(1/2)/c^(7/2)/(a^(1/2)+x*c^(1/2))/(d*x)^(1/2)+2/9009*(b*(108*a^2*c^2-151*a*b^2*c+24*
b^4)+3*c*(308*a^2*c^2-181*a*b^2*c+24*b^4)*x)*(d*x)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3+4/9009*a^(1/4)*(-924*a^3*c^3+
951*a^2*b^2*c^2-268*a*b^4*c+24*b^6)*d*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^
(1/2)/a^(1/4)))*EllipticE(sin(2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x*c
^(1/2))*x^(1/2)*((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(15/4)/(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)-1/9009*a^
(1/4)*d*(cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))^2)^(1/2)/cos(2*arctan(c^(1/4)*x^(1/2)/a^(1/4)))*EllipticF(sin(
2*arctan(c^(1/4)*x^(1/2)/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))*(a^(1/2)+x*c^(1/2))*(-1848*a^3*c^3+1902*a^
2*b^2*c^2-536*a*b^4*c+48*b^6+b*(708*a^2*c^2-241*a*b^2*c+24*b^4)*a^(1/2)*c^(1/2))*x^(1/2)*((c*x^2+b*x+a)/(a^(1/
2)+x*c^(1/2))^2)^(1/2)/c^(15/4)/(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 616, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {748, 846, 828, 855, 853, 1211, 1117, 1209} \[ \int \sqrt {d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 \sqrt {d x} \left (3 c x \left (308 a^2 c^2-181 a b^2 c+24 b^4\right )+b \left (108 a^2 c^2-151 a b^2 c+24 b^4\right )\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {\sqrt [4]{a} d \sqrt {x} \left (\sqrt {a} b \sqrt {c} \left (708 a^2 c^2-241 a b^2 c+24 b^4\right )+2 \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right )\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{9009 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt [4]{a} d \sqrt {x} \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{9009 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {4 d x \left (-924 a^3 c^3+951 a^2 b^2 c^2-268 a b^4 c+24 b^6\right ) \sqrt {a+b x+c x^2}}{9009 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {10 \sqrt {d x} \left (14 c x \left (3 b^2-11 a c\right )+3 b \left (6 b^2-19 a c\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c} \]

[In]

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

[Out]

(-4*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*d*x*Sqrt[a + b*x + c*x^2])/(9009*c^(7/2)*Sqrt[d*x]*
(Sqrt[a] + Sqrt[c]*x)) + (2*Sqrt[d*x]*(b*(24*b^4 - 151*a*b^2*c + 108*a^2*c^2) + 3*c*(24*b^4 - 181*a*b^2*c + 30
8*a^2*c^2)*x)*Sqrt[a + b*x + c*x^2])/(9009*c^3) - (10*Sqrt[d*x]*(3*b*(6*b^2 - 19*a*c) + 14*c*(3*b^2 - 11*a*c)*
x)*(a + b*x + c*x^2)^(3/2))/(9009*c^2) + (10*b*Sqrt[d*x]*(a + b*x + c*x^2)^(5/2))/(143*c) + (2*(d*x)^(3/2)*(a
+ b*x + c*x^2)^(5/2))/(13*d) + (4*a^(1/4)*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*d*Sqrt[x]*(Sq
rt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticE[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4
)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(9009*c^(15/4)*Sqrt[d*x]*Sqrt[a + b*x + c*x^2]) - (a^(1/4)*(Sqrt[a]*b*Sqrt[c
]*(24*b^4 - 241*a*b^2*c + 708*a^2*c^2) + 2*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3))*d*Sqrt[x]*(
Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sqrt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1
/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(9009*c^(15/4)*Sqrt[d*x]*Sqrt[a + b*x + c*x^2])

Rule 748

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 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 846

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 853

Int[((f_) + (g_.)*(x_))/(Sqrt[x_]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[2, Subst[Int[(f +
g*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 855

Int[((f_) + (g_.)*(x_))/(Sqrt[(e_)*(x_)]*Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[x]/Sq
rt[e*x], Int[(f + g*x)/(Sqrt[x]*Sqrt[a + b*x + c*x^2]), x], x] /; FreeQ[{a, b, c, e, f, g}, x] && NeQ[b^2 - 4*
a*c, 0]

Rule 1117

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(
a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(
4*c))], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1209

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(
-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 +
q^2*x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c))], x] /; EqQ[e + d*q^2,
0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1211

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Dist[(
e + d*q)/q, Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Dist[e/q, Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]
/; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}-\frac {5 \int \sqrt {d x} (-2 a d-b d x) \left (a+b x+c x^2\right )^{3/2} \, dx}{13 d} \\ & = \frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}-\frac {10 \int \frac {\left (\frac {1}{2} a b d^2+\left (3 b^2-11 a c\right ) d^2 x\right ) \left (a+b x+c x^2\right )^{3/2}}{\sqrt {d x}} \, dx}{143 c d} \\ & = -\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac {20 \int \frac {\left (\frac {1}{2} a b \left (3 b^2-20 a c\right ) d^4+\frac {1}{4} \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) d^4 x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d x}} \, dx}{3003 c^2 d^3} \\ & = \frac {2 \sqrt {d x} \left (b \left (24 b^4-151 a b^2 c+108 a^2 c^2\right )+3 c \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}-\frac {8 \int \frac {\frac {1}{8} a b \left (24 b^4-241 a b^2 c+708 a^2 c^2\right ) d^6+\frac {1}{4} \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d^6 x}{\sqrt {d x} \sqrt {a+b x+c x^2}} \, dx}{9009 c^3 d^5} \\ & = \frac {2 \sqrt {d x} \left (b \left (24 b^4-151 a b^2 c+108 a^2 c^2\right )+3 c \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}-\frac {\left (8 \sqrt {x}\right ) \int \frac {\frac {1}{8} a b \left (24 b^4-241 a b^2 c+708 a^2 c^2\right ) d^6+\frac {1}{4} \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d^6 x}{\sqrt {x} \sqrt {a+b x+c x^2}} \, dx}{9009 c^3 d^5 \sqrt {d x}} \\ & = \frac {2 \sqrt {d x} \left (b \left (24 b^4-151 a b^2 c+108 a^2 c^2\right )+3 c \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}-\frac {\left (16 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {1}{8} a b \left (24 b^4-241 a b^2 c+708 a^2 c^2\right ) d^6+\frac {1}{4} \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d^6 x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{9009 c^3 d^5 \sqrt {d x}} \\ & = \frac {2 \sqrt {d x} \left (b \left (24 b^4-151 a b^2 c+108 a^2 c^2\right )+3 c \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac {\left (4 \sqrt {a} \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d \sqrt {x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {c} x^2}{\sqrt {a}}}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{9009 c^{7/2} \sqrt {d x}}-\frac {\left (2 \sqrt {a} \left (\sqrt {a} b \left (24 b^4-241 a b^2 c+708 a^2 c^2\right )+\frac {2 \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right )}{\sqrt {c}}\right ) d \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{9009 c^3 \sqrt {d x}} \\ & = -\frac {4 \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d x \sqrt {a+b x+c x^2}}{9009 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 \sqrt {d x} \left (b \left (24 b^4-151 a b^2 c+108 a^2 c^2\right )+3 c \left (24 b^4-181 a b^2 c+308 a^2 c^2\right ) x\right ) \sqrt {a+b x+c x^2}}{9009 c^3}-\frac {10 \sqrt {d x} \left (3 b \left (6 b^2-19 a c\right )+14 c \left (3 b^2-11 a c\right ) x\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac {10 b \sqrt {d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}+\frac {2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac {4 \sqrt [4]{a} \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) d \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{9009 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (\sqrt {a} b \left (24 b^4-241 a b^2 c+708 a^2 c^2\right )+\frac {2 \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right )}{\sqrt {c}}\right ) d \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{9009 c^{13/4} \sqrt {d x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.86 (sec) , antiderivative size = 708, normalized size of antiderivative = 1.15 \[ \int \sqrt {d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {\sqrt {d x} \left (-\frac {4 \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) (a+x (b+c x))}{\sqrt {x}}+2 c \sqrt {x} (a+x (b+c x)) \left (24 b^5-18 b^4 c x+b^3 c \left (-241 a+15 c x^2\right )+3 b^2 c^2 x \left (54 a+371 c x^2\right )+77 c^3 x \left (31 a^2+28 a c x^2+9 c^2 x^4\right )+b c^2 \left (708 a^2+3071 a c x^2+1701 c^2 x^4\right )\right )+\frac {i \left (24 b^6-268 a b^4 c+951 a^2 b^2 c^2-924 a^3 c^3\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}+\frac {i \left (24 b^7-292 a b^5 c+1192 a^2 b^3 c^2-1632 a^3 b c^3-24 b^6 \sqrt {b^2-4 a c}+268 a b^4 c \sqrt {b^2-4 a c}-951 a^2 b^2 c^2 \sqrt {b^2-4 a c}+924 a^3 c^3 \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{9009 c^4 \sqrt {x} \sqrt {a+x (b+c x)}} \]

[In]

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

[Out]

(Sqrt[d*x]*((-4*(24*b^6 - 268*a*b^4*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*(a + x*(b + c*x)))/Sqrt[x] + 2*c*Sqrt[x
]*(a + x*(b + c*x))*(24*b^5 - 18*b^4*c*x + b^3*c*(-241*a + 15*c*x^2) + 3*b^2*c^2*x*(54*a + 371*c*x^2) + 77*c^3
*x*(31*a^2 + 28*a*c*x^2 + 9*c^2*x^4) + b*c^2*(708*a^2 + 3071*a*c*x^2 + 1701*c^2*x^4)) + (I*(24*b^6 - 268*a*b^4
*c + 951*a^2*b^2*c^2 - 924*a^3*c^3)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x*Sqr
t[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[a/(b + Sqrt
[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sqrt[b^2 - 4*a*c])] +
(I*(24*b^7 - 292*a*b^5*c + 1192*a^2*b^3*c^2 - 1632*a^3*b*c^3 - 24*b^6*Sqrt[b^2 - 4*a*c] + 268*a*b^4*c*Sqrt[b^
2 - 4*a*c] - 951*a^2*b^2*c^2*Sqrt[b^2 - 4*a*c] + 924*a^3*c^3*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2
- 4*a*c])*x)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*ArcSinh[(Sqrt[
2]*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])])/Sqrt[x]], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/Sqrt[a/(b + Sq
rt[b^2 - 4*a*c])]))/(9009*c^4*Sqrt[x]*Sqrt[a + x*(b + c*x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1415\) vs. \(2(579)=1158\).

Time = 0.87 (sec) , antiderivative size = 1416, normalized size of antiderivative = 2.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(1416\)
elliptic	\(\text {Expression too large to display}\)	\(1695\)
default	\(\text {Expression too large to display}\)	\(2810\)

[In]

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

[Out]

2/9009/c^3*(693*c^5*x^5+1701*b*c^4*x^4+2156*a*c^4*x^3+1113*b^2*c^3*x^3+3071*a*b*c^3*x^2+15*b^3*c^2*x^2+2387*a^
2*c^3*x+162*a*b^2*c^2*x-18*b^4*c*x+708*a^2*b*c^2-241*a*b^3*c+24*b^5)*x*(c*x^2+b*x+a)^(1/2)*d/(d*x)^(1/2)-1/900
9/c^3*(24*a*b^5*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(
1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(
-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*EllipticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(
1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c
*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))+708*a^3*b*c*(b+(-4*a*c+b^2)^(1/2))*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)
/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(
-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*EllipticF(2^(1/
2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b
+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-241*a^2*b^3*(b+(-4*a*c+b^2)^(1/2))*2^(1/2)*((x+1
/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a
*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+
a*d*x)^(1/2)*EllipticF(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^(1/2),1/2*(-2*(b+(-
4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-(1848*a^3*c^3-1902*a
^2*b^2*c^2+536*a*b^4*c-48*b^6)*(b+(-4*a*c+b^2)^(1/2))/c*2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b
^2)^(1/2))*c)^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(
1/2))))^(1/2)*(-2*c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)/(c*d*x^3+b*d*x^2+a*d*x)^(1/2)*((-1/2*(b+(-4*a*c+b^2)^(1/2)
)/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*
c)^(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-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(2^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(b+(-4*a*c+b^2)^(1/2))*c)^
(1/2),1/2*(-2*(b+(-4*a*c+b^2)^(1/2))/c/(-1/2*(b+(-4*a*c+b^2)^(1/2))/c-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))))
*d*(d*x*(c*x^2+b*x+a))^(1/2)/(d*x)^(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 = 0.11 (sec) , antiderivative size = 331, normalized size of antiderivative = 0.54 \[ \int \sqrt {d x} \left (a+b x+c x^2\right )^{5/2} \, dx=\frac {2 \, {\left ({\left (48 \, b^{7} - 608 \, a b^{5} c + 2625 \, a^{2} b^{3} c^{2} - 3972 \, a^{3} b c^{3}\right )} \sqrt {c d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (24 \, b^{6} c - 268 \, a b^{4} c^{2} + 951 \, a^{2} b^{2} c^{3} - 924 \, a^{3} c^{4}\right )} \sqrt {c d} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (693 \, c^{7} x^{5} + 1701 \, b c^{6} x^{4} + 24 \, b^{5} c^{2} - 241 \, a b^{3} c^{3} + 708 \, a^{2} b c^{4} + 7 \, {\left (159 \, b^{2} c^{5} + 308 \, a c^{6}\right )} x^{3} + {\left (15 \, b^{3} c^{4} + 3071 \, a b c^{5}\right )} x^{2} - {\left (18 \, b^{4} c^{3} - 162 \, a b^{2} c^{4} - 2387 \, a^{2} c^{5}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{27027 \, c^{5}} \]

[In]

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

[Out]

2/27027*((48*b^7 - 608*a*b^5*c + 2625*a^2*b^3*c^2 - 3972*a^3*b*c^3)*sqrt(c*d)*weierstrassPInverse(4/3*(b^2 - 3
*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c) + 6*(24*b^6*c - 268*a*b^4*c^2 + 951*a^2*b^2*c^3 - 9
24*a^3*c^4)*sqrt(c*d)*weierstrassZeta(4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, weierstrassPInverse(
4/3*(b^2 - 3*a*c)/c^2, -4/27*(2*b^3 - 9*a*b*c)/c^3, 1/3*(3*c*x + b)/c)) + 3*(693*c^7*x^5 + 1701*b*c^6*x^4 + 24
*b^5*c^2 - 241*a*b^3*c^3 + 708*a^2*b*c^4 + 7*(159*b^2*c^5 + 308*a*c^6)*x^3 + (15*b^3*c^4 + 3071*a*b*c^5)*x^2 -
(18*b^4*c^3 - 162*a*b^2*c^4 - 2387*a^2*c^5)*x)*sqrt(c*x^2 + b*x + a)*sqrt(d*x))/c^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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