3.2440 \(\int (d x)^{5/2} \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=504 \[ -\frac{\sqrt [4]{a} d^3 \sqrt{x} \left (42 a^2 c^2-72 a b^2 c+\sqrt{a} b \sqrt{c} \left (8 b^2-27 a c\right )+16 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}-\frac{4 d^3 x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \sqrt{a+b x+c x^2}}{315 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{4 \sqrt [4]{a} d^3 \sqrt{x} \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{2 d^2 \sqrt{d x} \left (3 c x \left (8 b^2-7 a c\right )+b \left (3 a c+8 b^2\right )\right ) \sqrt{a+b x+c x^2}}{315 c^3}-\frac{4 b d^2 \sqrt{d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.645228, antiderivative size = 504, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {742, 832, 814, 841, 839, 1197, 1103, 1195} \[ -\frac{4 d^3 x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \sqrt{a+b x+c x^2}}{315 c^{7/2} \sqrt{d x} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} d^3 \sqrt{x} \left (42 a^2 c^2-72 a b^2 c+\sqrt{a} b \sqrt{c} \left (8 b^2-27 a c\right )+16 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{4 \sqrt [4]{a} d^3 \sqrt{x} \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{315 c^{15/4} \sqrt{d x} \sqrt{a+b x+c x^2}}+\frac{2 d^2 \sqrt{d x} \left (3 c x \left (8 b^2-7 a c\right )+b \left (3 a c+8 b^2\right )\right ) \sqrt{a+b x+c x^2}}{315 c^3}-\frac{4 b d^2 \sqrt{d x} \left (a+b x+c x^2\right )^{3/2}}{21 c^2}+\frac{2 d (d x)^{3/2} \left (a+b x+c x^2\right )^{3/2}}{9 c} \]

Antiderivative was successfully verified.

[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 742

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 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 (d x)^{5/2} \sqrt{a+b x+c x^2} \, dx &=\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 ) \operatorname{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 ) \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 )}{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 ) \operatorname{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]  time = 3.53395, size = 594, normalized size = 1.18 \[ \frac{(d x)^{5/2} \left (-\frac{i x \left (21 a^2 c^2 \sqrt{b^2-4 a c}-48 a^2 b c^2+8 b^4 \sqrt{b^2-4 a c}+44 a b^3 c-36 a b^2 c \sqrt{b^2-4 a c}-8 b^5\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}{\sqrt{x}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}-\frac{4 \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) (a+x (b+c x))}{\sqrt{x}}+\frac{i x \left (21 a^2 c^2-36 a b^2 c+8 b^4\right ) \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{4 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+2} \sqrt{\frac{-x \sqrt{b^2-4 a c}+2 a+b x}{b x-x \sqrt{b^2-4 a c}}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )}{\sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}+2 c \sqrt{x} \left (b c \left (5 c x^2-27 a\right )+7 c^2 x \left (2 a+5 c x^2\right )-6 b^2 c x+8 b^3\right ) (a+x (b+c x))\right )}{315 c^4 x^{5/2} \sqrt{a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d*x)^(5/2)*((-4*(8*b^4 - 36*a*b^2*c + 21*a^2*c^2)*(a + x*(b + c*x)))/Sqrt[x] + 2*c*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)) + (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)]*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)]*x*Sqrt[(2*a + b*x - Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticF[I*A
rcSinh[(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])])/S
qrt[a/(b + Sqrt[b^2 - 4*a*c])]))/(315*c^4*x^(5/2)*Sqrt[a + x*(b + c*x)])

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Maple [B]  time = 0.26, size = 2062, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/315*d^2*(d*x)^(1/2)*(70*x^6*c^6+80*x^5*b*c^5+84*((-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))*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^
(1/2)*a^3*c^3-117*((-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))*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^2*b^2*c^2+27*((-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))*((b+2
*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a^2*b*c^2+24*((-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))*((b+2*c*x+(-4*a*c
+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^4*c-8*((-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))*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)
))^(1/2)*(-4*a*c+b^2)^(1/2)*a*b^3*c-168*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^3*
c^3+330*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^2*b^2*c^2+42*((-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+2*c*x+(-4*a
*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a^2*b*c^2-136*((-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+2*c*x+(-4*a*c+b^2)^(1/
2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^4*c-72*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)
*(-4*a*c+b^2)^(1/2)*a*b^3*c+16*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*b^6+16*((-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+2*c*x+(-4*a*c+b^2)^(1/2))/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*b^5+98*x^4*a*c^5-2*x^4*b^2*c^4-
16*x^3*a*b*c^4+4*x^3*b^3*c^3+28*x^2*a^2*c^4-66*x^2*a*b^2*c^3+16*x^2*b^4*c^2-54*x*a^2*b*c^3+16*x*a*b^3*c^2)/x/(
c*x^2+b*x+a)^(1/2)/c^5

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)