\(\int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx\) [2440]

Optimal result

Integrand size = 22, antiderivative size = 504 \[ \int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx=-\frac {4 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 x \sqrt {a+b x+c x^2}}{315 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {4 \sqrt [4]{a} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (16 b^4-72 a b^2 c+42 a^2 c^2+\sqrt {a} b \sqrt {c} \left (8 b^2-27 a c\right )\right ) d^3 \sqrt {x} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}} \]

[Out]

2/9*d*(d*x)^(3/2)*(c*x^2+b*x+a)^(3/2)/c-4/21*b*d^2*(c*x^2+b*x+a)^(3/2)*(d*x)^(1/2)/c^2-4/315*(21*a^2*c^2-36*a*
b^2*c+8*b^4)*d^3*x*(c*x^2+b*x+a)^(1/2)/c^(7/2)/(a^(1/2)+x*c^(1/2))/(d*x)^(1/2)+2/315*d^2*(b*(3*a*c+8*b^2)+3*c*
(-7*a*c+8*b^2)*x)*(d*x)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3+4/315*a^(1/4)*(21*a^2*c^2-36*a*b^2*c+8*b^4)*d^3*(cos(2*a
rctan(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)-1/315*a^(1/4)*d^3*(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))*(16*b^4-72*a*b^2*c+42*a^2*c^2+b*(-27*a*c+8*b^2)*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 = 0.43 (sec) , antiderivative size = 504, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {756, 846, 828, 855, 853, 1211, 1117, 1209} \[ \int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx=-\frac {\sqrt [4]{a} d^3 \sqrt {x} \left (42 a^2 c^2-72 a b^2 c+\sqrt {a} b \sqrt {c} \left (8 b^2-27 a c\right )+16 b^4\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt [4]{a} d^3 \sqrt {x} \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {4 d^3 x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \sqrt {a+b x+c x^2}}{315 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 d^2 \sqrt {d x} \left (3 c x \left (8 b^2-7 a c\right )+b \left (3 a c+8 b^2\right )\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c} \]

[In]

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

[Out]

(-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*d^3*x*Sqrt[a + b*x + c*x^2])/(315*c^(7/2)*Sqrt[d*x]*(Sqrt[a] + Sqrt[c]*x
)) + (2*d^2*Sqrt[d*x]*(b*(8*b^2 + 3*a*c) + 3*c*(8*b^2 - 7*a*c)*x)*Sqrt[a + b*x + c*x^2])/(315*c^3) - (4*b*d^2*
Sqrt[d*x]*(a + b*x + c*x^2)^(3/2))/(21*c^2) + (2*d*(d*x)^(3/2)*(a + b*x + c*x^2)^(3/2))/(9*c) + (4*a^(1/4)*(8*
b^4 - 36*a*b^2*c + 21*a^2*c^2)*d^3*Sqrt[x]*(Sqrt[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])/(315*c^(15/4)*Sqrt[d*x]*Sqrt[a
+ b*x + c*x^2]) - (a^(1/4)*(16*b^4 - 72*a*b^2*c + 42*a^2*c^2 + Sqrt[a]*b*Sqrt[c]*(8*b^2 - 27*a*c))*d^3*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])/(315*c^(15/4)*Sqrt[d*x]*Sqrt[a + b*x + c*x^2])

Rule 756

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[e*(d + e*x)^(m - 1)*
((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, 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]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 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 (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {2 \int \sqrt {d x} \left (-\frac {3 a d^2}{2}-3 b d^2 x\right ) \sqrt {a+b x+c x^2} \, dx}{9 c} \\ & = -\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {4 \int \frac {\left (\frac {3}{2} a b d^3+\frac {3}{4} \left (8 b^2-7 a c\right ) d^3 x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d x}} \, dx}{63 c^2} \\ & = \frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}-\frac {8 \int \frac {\frac {3}{8} a b \left (8 b^2-27 a c\right ) d^5+\frac {3}{4} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^5 x}{\sqrt {d x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 d^2} \\ & = \frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (8 \sqrt {x}\right ) \int \frac {\frac {3}{8} a b \left (8 b^2-27 a c\right ) d^5+\frac {3}{4} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^5 x}{\sqrt {x} \sqrt {a+b x+c x^2}} \, dx}{945 c^3 d^2 \sqrt {d x}} \\ & = \frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (16 \sqrt {x}\right ) \text {Subst}\left (\int \frac {\frac {3}{8} a b \left (8 b^2-27 a c\right ) d^5+\frac {3}{4} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^5 x^2}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{945 c^3 d^2 \sqrt {d x}} \\ & = \frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {\left (4 \sqrt {a} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 \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 )}{315 c^{7/2} \sqrt {d x}}-\frac {\left (2 \sqrt {a} \left (\sqrt {a} b \left (8 b^2-27 a c\right )+\frac {2 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right )}{\sqrt {c}}\right ) d^3 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2+c x^4}} \, dx,x,\sqrt {x}\right )}{315 c^3 \sqrt {d x}} \\ & = -\frac {4 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 x \sqrt {a+b x+c x^2}}{315 c^{7/2} \sqrt {d x} \left (\sqrt {a}+\sqrt {c} x\right )}+\frac {2 d^2 \sqrt {d x} \left (b \left (8 b^2+3 a c\right )+3 c \left (8 b^2-7 a c\right ) x\right ) \sqrt {a+b x+c x^2}}{315 c^3}-\frac {4 b d^2 \sqrt {d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c}+\frac {4 \sqrt [4]{a} \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) d^3 \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 )}{315 c^{15/4} \sqrt {d x} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (\sqrt {a} b \left (8 b^2-27 a c\right )+\frac {2 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right )}{\sqrt {c}}\right ) d^3 \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 )}{315 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 = 24.58 (sec) , antiderivative size = 597, normalized size of antiderivative = 1.18 \[ \int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx=\frac {(d x)^{5/2} \left (\frac {2 \sqrt {x} (a+x (b+c x)) \left (8 b^3-6 b^2 c x+b c \left (-27 a+5 c x^2\right )+7 c^2 x \left (2 a+5 c x^2\right )\right )}{c^3}+\frac {x \left (-\frac {4 \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) (a+x (b+c x))}{x^{3/2}}+\frac {i \left (8 b^4-36 a b^2 c+21 a^2 c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}-\frac {i \left (-8 b^5+44 a b^3 c-48 a^2 b c^2+8 b^4 \sqrt {b^2-4 a c}-36 a b^2 c \sqrt {b^2-4 a c}+21 a^2 c^2 \sqrt {b^2-4 a c}\right ) \sqrt {2+\frac {4 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} \sqrt {\frac {2 a+b x-\sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}{\sqrt {x}}\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{\sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}}}\right )}{c^4}\right )}{315 x^{5/2} \sqrt {a+x (b+c x)}} \]

[In]

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

[Out]

((d*x)^(5/2)*((2*Sqrt[x]*(a + x*(b + c*x))*(8*b^3 - 6*b^2*c*x + b*c*(-27*a + 5*c*x^2) + 7*c^2*x*(2*a + 5*c*x^2
)))/c^3 + (x*((-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*(a + x*(b + c*x)))/x^(3/2) + (I*(8*b^4 - 36*a*b^2*c + 21*a
^2*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*Sqrt[(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*(-8*b^5 + 44*a*b^3*c - 48*
a^2*b*c^2 + 8*b^4*Sqrt[b^2 - 4*a*c] - 36*a*b^2*c*Sqrt[b^2 - 4*a*c] + 21*a^2*c^2*Sqrt[b^2 - 4*a*c])*Sqrt[2 + (4
*a)/((b + Sqrt[b^2 - 4*a*c])*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 + Sqrt[b^2 - 4*a*c])]))/c^4))/(315*x^(5/2)*Sqrt[a + x*(b + c*x)])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(989\) vs. \(2(476)=952\).

Time = 0.75 (sec) , antiderivative size = 990, normalized size of antiderivative = 1.96

method	result	size
elliptic	\(\frac {\sqrt {d x}\, \sqrt {d x \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 d^{2} x^{3} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{9}+\frac {2 b \,d^{2} x^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{63 c}+\frac {2 \left (\frac {2 a \,d^{3}}{9}-\frac {2 b^{2} d^{3}}{21 c}\right ) x \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{5 c d}+\frac {2 \left (-\frac {5 b \,d^{3} a}{63 c}-\frac {4 \left (\frac {2 a \,d^{3}}{9}-\frac {2 b^{2} d^{3}}{21 c}\right ) b}{5 c}\right ) \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}{3 c d}-\frac {\left (-\frac {5 b \,d^{3} a}{63 c}-\frac {4 \left (\frac {2 a \,d^{3}}{9}-\frac {2 b^{2} d^{3}}{21 c}\right ) b}{5 c}\right ) a \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, F\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{3 c^{2} \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}+\frac {\left (-\frac {3 \left (\frac {2 a \,d^{3}}{9}-\frac {2 b^{2} d^{3}}{21 c}\right ) a}{5 c}-\frac {2 \left (-\frac {5 b \,d^{3} a}{63 c}-\frac {4 \left (\frac {2 a \,d^{3}}{9}-\frac {2 b^{2} d^{3}}{21 c}\right ) b}{5 c}\right ) b}{3 c}\right ) \left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {-\frac {2 c x}{b +\sqrt {-4 a c +b^{2}}}}\, \left (\left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) E\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) F\left (\sqrt {2}\, \sqrt {\frac {\left (x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) c}{b +\sqrt {-4 a c +b^{2}}}}, \frac {\sqrt {-\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right )}{c \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right )}}}{2}\right )}{2 c}\right )}{c \sqrt {c d \,x^{3}+b d \,x^{2}+a d x}}\right )}{d x \sqrt {c \,x^{2}+b x +a}}\)	\(990\)
risch	\(\text {Expression too large to display}\)	\(1080\)
default	\(\text {Expression too large to display}\)	\(2062\)

[In]

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

[Out]

1/d/x*(d*x)^(1/2)/(c*x^2+b*x+a)^(1/2)*(d*x*(c*x^2+b*x+a))^(1/2)*(2/9*d^2*x^3*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)+2/6
3*b*d^2/c*x^2*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)+2/5*(2/9*a*d^3-2/21*b^2*d^3/c)/c/d*x*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)
+2/3*(-5/63*b*d^3/c*a-4/5*(2/9*a*d^3-2/21*b^2*d^3/c)/c*b)/c/d*(c*d*x^3+b*d*x^2+a*d*x)^(1/2)-1/3*(-5/63*b*d^3/c
*a-4/5*(2/9*a*d^3-2/21*b^2*d^3/c)/c*b)/c^2*a*(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))+(-3/5*(2/9*a*d^3-2/21*b^2*d^3/c)/c*a-2/3*(-5/63*b
*d^3/c*a-4/5*(2/9*a*d^3-2/21*b^2*d^3/c)/c*b)/c*b)*(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))))

Fricas [C] (verification not implemented)

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

Time = 0.11 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.53 \[ \int (d x)^{5/2} \sqrt {a+b x+c x^2} \, dx=\frac {2 \, {\left ({\left (16 \, b^{5} - 96 \, a b^{3} c + 123 \, a^{2} b c^{2}\right )} \sqrt {c d} d^{2} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right ) + 6 \, {\left (8 \, b^{4} c - 36 \, a b^{2} c^{2} + 21 \, a^{2} c^{3}\right )} \sqrt {c d} d^{2} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} - 3 \, a c\right )}}{3 \, c^{2}}, -\frac {4 \, {\left (2 \, b^{3} - 9 \, a b c\right )}}{27 \, c^{3}}, \frac {3 \, c x + b}{3 \, c}\right )\right ) + 3 \, {\left (35 \, c^{5} d^{2} x^{3} + 5 \, b c^{4} d^{2} x^{2} - 2 \, {\left (3 \, b^{2} c^{3} - 7 \, a c^{4}\right )} d^{2} x + {\left (8 \, b^{3} c^{2} - 27 \, a b c^{3}\right )} d^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {d x}\right )}}{945 \, c^{5}} \]

[In]

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

[Out]

2/945*((16*b^5 - 96*a*b^3*c + 123*a^2*b*c^2)*sqrt(c*d)*d^2*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*(8*b^4*c - 36*a*b^2*c^2 + 21*a^2*c^3)*sqrt(c*d)*d^2*weierstrassZet
a(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*(35*c^5*d^2*x^3 + 5*b*c^4*d^2*x^2 - 2*(3*b^2*c^3 - 7*a*c^4)*d^2*x + (8
*b^3*c^2 - 27*a*b*c^3)*d^2)*sqrt(c*x^2 + b*x + a)*sqrt(d*x))/c^5

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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