### 3.2454 $$\int \sqrt{d x} (a+b x+c x^2)^{5/2} \, dx$$

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

[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])

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Rubi [A]  time = 0.959471, antiderivative size = 616, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.364, Rules used = {734, 832, 814, 841, 839, 1197, 1103, 1195} $\frac{2 \sqrt{d x} \left (3 c x \left (308 a^2 c^2-181 a b^2 c+24 b^4\right )+b \left (108 a^2 c^2-151 a b^2 c+24 b^4\right )\right ) \sqrt{a+b x+c x^2}}{9009 c^3}-\frac{4 d x \left (951 a^2 b^2 c^2-924 a^3 c^3-268 a b^4 c+24 b^6\right ) \sqrt{a+b x+c x^2}}{9009 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} d \sqrt{x} \left (\sqrt{a} b \sqrt{c} \left (708 a^2 c^2-241 a b^2 c+24 b^4\right )+2 \left (951 a^2 b^2 c^2-924 a^3 c^3-268 a b^4 c+24 b^6\right )\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{a} d \sqrt{x} \left (951 a^2 b^2 c^2-924 a^3 c^3-268 a b^4 c+24 b^6\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{9009 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}-\frac{10 \sqrt{d x} \left (14 c x \left (3 b^2-11 a c\right )+3 b \left (6 b^2-19 a c\right )\right ) \left (a+b x+c x^2\right )^{3/2}}{9009 c^2}+\frac{2 (d x)^{3/2} \left (a+b x+c x^2\right )^{5/2}}{13 d}+\frac{10 b \sqrt{d x} \left (a+b x+c x^2\right )^{5/2}}{143 c}$

Antiderivative was successfully veriﬁed.

[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 734

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 832

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 814

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 841

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 839

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 1197

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]

Rule 1103

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)]*EllipticF[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(2*q*Sqrt[a + b*x^2 + c
*x^4]), x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rule 1195

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)]*EllipticE[2*ArcTan[q*x], 1/2 - (b*q^2)/(4*c)])/(q*Sqrt[a + b*x^2 + c*x^4]), x] /; EqQ[e + d*q^2, 0
]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]

Rubi steps

\begin{align*} \int \sqrt{d x} \left (a+b x+c x^2\right )^{5/2} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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]  time = 4.28974, size = 708, normalized size = 1.15 $\frac{\sqrt{d x} \left (\frac{i x \left (1192 a^2 b^3 c^2-951 a^2 b^2 c^2 \sqrt{b^2-4 a c}+924 a^3 c^3 \sqrt{b^2-4 a c}-1632 a^3 b c^3-24 b^6 \sqrt{b^2-4 a c}-292 a b^5 c+268 a b^4 c \sqrt{b^2-4 a c}+24 b^7\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}{\sqrt{x}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+2 c \sqrt{x} \left (b c^2 \left (708 a^2+3071 a c x^2+1701 c^2 x^4\right )+77 c^3 x \left (31 a^2+28 a c x^2+9 c^2 x^4\right )+3 b^2 c^2 x \left (54 a+371 c x^2\right )+b^3 c \left (15 c x^2-241 a\right )-18 b^4 c x+24 b^5\right ) (a+x (b+c x))-\frac{4 \left (951 a^2 b^2 c^2-924 a^3 c^3-268 a b^4 c+24 b^6\right ) (a+x (b+c x))}{\sqrt{x}}+\frac{i x \left (951 a^2 b^2 c^2-924 a^3 c^3-268 a b^4 c+24 b^6\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}\right )}{9009 c^4 \sqrt{x} \sqrt{a+x (b+c x)}}$

Antiderivative was successfully veriﬁed.

[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]  time = 0.286, size = 2810, normalized size = 4.6 \begin{align*} \text{output too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9009*(d*x)^(1/2)*(-1902*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b
-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(
-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)
)*a^2*b^3*c^2+536*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-
4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^
2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*b^5*c
+1848*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(
1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b
+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^3*b*c^3+6*x^4*b^4
*c^4-2256*x^5*b^3*c^5-5698*x^6*a*c^7-9086*x^4*a^2*c^6-4774*x^2*a^3*c^5-4788*x^7*b*c^7-12*x^3*b^5*c^3-48*x^2*b^
6*c^2-5628*x^6*b^2*c^6+3696*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2
)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))
/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^4*c^4-7392*((b
+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1
/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),
1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^4*c^4+831*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-
4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2))
)^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/
2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^2*b^4*c^2-4046*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((
-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x
+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/
2))*a^2*b^4*c^2+9456*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2)
)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4
*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^3*b^2*c^3-72*((b+2*c*
x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(
-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2
^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*b^6*c-3096*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*
c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1
/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2))*a^3*b^2*c^3+728*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*
c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*
a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a
*b^6*c+708*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b
^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2
))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^3*b*c^3-241*
(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/
(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a
*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a^2*b^3*c^2-48*(-4*a*c+b^
2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^
2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1
/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*b^7-1386*x^8*c^8-13856*x^5*a*b*c^6-
8692*x^4*a*b^2*c^5-12332*x^3*a^2*b*c^5+128*x^3*a*b^3*c^4-1740*x^2*a^2*b^2*c^4+518*x^2*a*b^4*c^3-1416*x*a^3*b*c
^4+482*x*a^2*b^3*c^3-48*x*a*b^5*c^2-48*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+
(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((b+2*c*x+(-4*a*c+
b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*b^8+2
4*(-4*a*c+b^2)^(1/2)*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*((-b-2*c*x+(-4*a*c+b^2)^(1/2)
)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4
*a*c+b^2)^(1/2)))^(1/2),1/2*2^(1/2)*((b+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2))*a*b^5*c)/(c*x^2+b*x+a)^
(1/2)/c^5/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}} \sqrt{d x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x +{\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )} \sqrt{c x^{2} + b x + a} \sqrt{d x}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^2*x^4 + 2*b*c*x^3 + 2*a*b*x + (b^2 + 2*a*c)*x^2 + a^2)*sqrt(c*x^2 + b*x + a)*sqrt(d*x), x)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c x^{2} + b x + a\right )}^{\frac{5}{2}} \sqrt{d x}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)