3.1029 \(\int \sqrt{x} (A+B x) \sqrt{a+b x+c x^2} \, dx\)
Optimal. Leaf size=454 \[ -\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-\sqrt{a} \sqrt{c} \left (-10 a B c-7 A b c+4 b^2 B\right )-2 \left (b^2-3 a c\right ) (4 b B-7 A c)+5 a b B c\right ) \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right ),\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{105 c^2}-\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right )}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x\right )}+\frac{2 \sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x+c x^2}}+\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
[Out]
(-2*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b*B - 7*A*c))*Sqrt[x]*Sqrt[a + b*x + c*x^2])/(105*c^(5/2)*(Sqrt[a] + Sqrt[
c]*x)) - (2*Sqrt[x]*(4*b^2*B - 7*A*b*c + 5*a*B*c + 3*c*(4*b*B - 7*A*c)*x)*Sqrt[a + b*x + c*x^2])/(105*c^2) + (
2*B*Sqrt[x]*(a + b*x + c*x^2)^(3/2))/(7*c) + (2*a^(1/4)*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b*B - 7*A*c))*(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])/(105*c^(11/4)*Sqrt[a + b*x + c*x^2]) - (a^(1/4)*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b
*B - 7*A*c) - Sqrt[a]*Sqrt[c]*(4*b^2*B - 7*A*b*c - 10*a*B*c))*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(105*c^(11/
4)*Sqrt[a + b*x + c*x^2])
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Rubi [A] time = 0.582343, antiderivative size = 454, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used =
{832, 814, 839, 1197, 1103, 1195} \[ -\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} \left (5 a B c+3 c x (4 b B-7 A c)-7 A b c+4 b^2 B\right )}{105 c^2}-\frac{2 \sqrt{x} \sqrt{a+b x+c x^2} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right )}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{\sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (-\sqrt{a} \sqrt{c} \left (-10 a B c-7 A b c+4 b^2 B\right )-2 \left (b^2-3 a c\right ) (4 b B-7 A c)+5 a b B c\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x+c x^2}}+\frac{2 \sqrt [4]{a} \left (\sqrt{a}+\sqrt{c} x\right ) \sqrt{\frac{a+b x+c x^2}{\left (\sqrt{a}+\sqrt{c} x\right )^2}} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{a}}\right )|\frac{1}{4} \left (2-\frac{b}{\sqrt{a} \sqrt{c}}\right )\right )}{105 c^{11/4} \sqrt{a+b x+c x^2}}+\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[x]*(A + B*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
(-2*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b*B - 7*A*c))*Sqrt[x]*Sqrt[a + b*x + c*x^2])/(105*c^(5/2)*(Sqrt[a] + Sqrt[
c]*x)) - (2*Sqrt[x]*(4*b^2*B - 7*A*b*c + 5*a*B*c + 3*c*(4*b*B - 7*A*c)*x)*Sqrt[a + b*x + c*x^2])/(105*c^2) + (
2*B*Sqrt[x]*(a + b*x + c*x^2)^(3/2))/(7*c) + (2*a^(1/4)*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b*B - 7*A*c))*(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])/(105*c^(11/4)*Sqrt[a + b*x + c*x^2]) - (a^(1/4)*(5*a*b*B*c - 2*(b^2 - 3*a*c)*(4*b
*B - 7*A*c) - Sqrt[a]*Sqrt[c]*(4*b^2*B - 7*A*b*c - 10*a*B*c))*(Sqrt[a] + Sqrt[c]*x)*Sqrt[(a + b*x + c*x^2)/(Sq
rt[a] + Sqrt[c]*x)^2]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(105*c^(11/
4)*Sqrt[a + b*x + c*x^2])
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 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{x} (A+B x) \sqrt{a+b x+c x^2} \, dx &=\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{2 \int \frac{\left (-\frac{a B}{2}-\frac{1}{2} (4 b B-7 A c) x\right ) \sqrt{a+b x+c x^2}}{\sqrt{x}} \, dx}{7 c}\\ &=-\frac{2 \sqrt{x} \left (4 b^2 B-7 A b c+5 a B c+3 c (4 b B-7 A c) x\right ) \sqrt{a+b x+c x^2}}{105 c^2}+\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{4 \int \frac{-\frac{1}{4} a \left (4 b^2 B-7 A b c-10 a B c\right )+\frac{1}{4} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) x}{\sqrt{x} \sqrt{a+b x+c x^2}} \, dx}{105 c^2}\\ &=-\frac{2 \sqrt{x} \left (4 b^2 B-7 A b c+5 a B c+3 c (4 b B-7 A c) x\right ) \sqrt{a+b x+c x^2}}{105 c^2}+\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac{8 \operatorname{Subst}\left (\int \frac{-\frac{1}{4} a \left (4 b^2 B-7 A b c-10 a B c\right )+\frac{1}{4} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) x^2}{\sqrt{a+b x^2+c x^4}} \, dx,x,\sqrt{x}\right )}{105 c^2}\\ &=-\frac{2 \sqrt{x} \left (4 b^2 B-7 A b c+5 a B c+3 c (4 b B-7 A c) x\right ) \sqrt{a+b x+c x^2}}{105 c^2}+\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{\left (2 \sqrt{a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right )\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 )}{105 c^{5/2}}-\frac{\left (2 \sqrt{a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)-\sqrt{a} \sqrt{c} \left (4 b^2 B-7 A b c-10 a B c\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2+c x^4}} \, dx,x,\sqrt{x}\right )}{105 c^{5/2}}\\ &=-\frac{2 \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \sqrt{x} \sqrt{a+b x+c x^2}}{105 c^{5/2} \left (\sqrt{a}+\sqrt{c} x\right )}-\frac{2 \sqrt{x} \left (4 b^2 B-7 A b c+5 a B c+3 c (4 b B-7 A c) x\right ) \sqrt{a+b x+c x^2}}{105 c^2}+\frac{2 B \sqrt{x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac{2 \sqrt [4]{a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\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 )}{105 c^{11/4} \sqrt{a+b x+c x^2}}-\frac{\sqrt [4]{a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)-\sqrt{a} \sqrt{c} \left (4 b^2 B-7 A b c-10 a B c\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 )}{105 c^{11/4} \sqrt{a+b x+c x^2}}\\ \end{align*}
Mathematica [C] time = 4.96995, size = 638, normalized size = 1.41 \[ \frac{2 \sqrt{x} \sqrt{a+x (b+c x)} \left (c (10 a B+3 c x (7 A+5 B x))+b c (7 A+3 B x)-4 b^2 B\right )}{105 c^2}-\frac{i x^{3/2} \sqrt{\frac{2 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+1} \sqrt{\frac{-2 x \sqrt{b^2-4 a c}+4 a+2 b x}{b x-x \sqrt{b^2-4 a c}}} \left (2 a c^2 \left (10 a B-21 A \sqrt{b^2-4 a c}\right )-2 b^3 \left (4 B \sqrt{b^2-4 a c}+7 A c\right )+b^2 \left (14 A c \sqrt{b^2-4 a c}-37 a B c\right )+a b c \left (29 B \sqrt{b^2-4 a c}+56 A c\right )+8 b^4 B\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}}}{\sqrt{x}}\right ),\frac{\sqrt{b^2-4 a c}+b}{b-\sqrt{b^2-4 a c}}\right )+i x^{3/2} \left (\sqrt{b^2-4 a c}-b\right ) \sqrt{\frac{2 a}{x \left (\sqrt{b^2-4 a c}+b\right )}+1} \sqrt{\frac{-2 x \sqrt{b^2-4 a c}+4 a+2 b x}{b x-x \sqrt{b^2-4 a c}}} \left (42 a A c^2-29 a b B c-14 A b^2 c+8 b^3 B\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a}{b+\sqrt{b^2-4 a c}}}}{\sqrt{x}}\right )|\frac{b+\sqrt{b^2-4 a c}}{b-\sqrt{b^2-4 a c}}\right )-4 \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} (a+x (b+c x)) \left (42 a A c^2-29 a b B c-14 A b^2 c+8 b^3 B\right )}{210 c^3 \sqrt{x} \sqrt{\frac{a}{\sqrt{b^2-4 a c}+b}} \sqrt{a+x (b+c x)}} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[x]*(A + B*x)*Sqrt[a + b*x + c*x^2],x]
[Out]
(2*Sqrt[x]*Sqrt[a + x*(b + c*x)]*(-4*b^2*B + b*c*(7*A + 3*B*x) + c*(10*a*B + 3*c*x*(7*A + 5*B*x))))/(105*c^2)
- (-4*(8*b^3*B - 14*A*b^2*c - 29*a*b*B*c + 42*a*A*c^2)*Sqrt[a/(b + Sqrt[b^2 - 4*a*c])]*(a + x*(b + c*x)) + I*(
8*b^3*B - 14*A*b^2*c - 29*a*b*B*c + 42*a*A*c^2)*(-b + Sqrt[b^2 - 4*a*c])*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c
])*x)]*x^(3/2)*Sqrt[(4*a + 2*b*x - 2*Sqrt[b^2 - 4*a*c]*x)/(b*x - Sqrt[b^2 - 4*a*c]*x)]*EllipticE[I*ArcSinh[(Sq
rt[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])] + I*(8*b^4*B
+ 2*a*c^2*(10*a*B - 21*A*Sqrt[b^2 - 4*a*c]) - 2*b^3*(7*A*c + 4*B*Sqrt[b^2 - 4*a*c]) + a*b*c*(56*A*c + 29*B*Sq
rt[b^2 - 4*a*c]) + b^2*(-37*a*B*c + 14*A*c*Sqrt[b^2 - 4*a*c]))*Sqrt[1 + (2*a)/((b + Sqrt[b^2 - 4*a*c])*x)]*x^(
3/2)*Sqrt[(4*a + 2*b*x - 2*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])])/(210*c^3*Sqrt[a/(b +
Sqrt[b^2 - 4*a*c])]*Sqrt[x]*Sqrt[a + x*(b + c*x)])
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Maple [B] time = 0.089, size = 2884, normalized size = 6.4 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)*x^(1/2)*(c*x^2+b*x+a)^(1/2),x)
[Out]
1/105*(-8*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*Ellip
ticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*b^5-8*B*((-2*c*x+(-4*a*c+b^2)^(1
/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(
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))*((2*c*x+(-4*a*c+b^
2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*b^4-84*A*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b
^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(
-4*a*c+b^2)^(1/2)))^(1/2)*a^2*c^3+168*A*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*
a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^2*
c^3+14*A*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*Elliptic
E(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*b^4*c+26*B*x^2*a*b*c^3+36*B*x^4*b*c
^4+56*A*x^3*b*c^4+50*B*x^3*a*c^4-2*B*x^3*b^2*c^3+42*A*x^2*a*c^4+14*A*x^2*b^2*c^3-8*B*x^2*b^3*c^2+20*B*x*a^2*c^
3+14*A*x*a*b*c^3-8*B*x*a*b^2*c^2-42*A*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*
c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*
c+b^2)^(1/2)*a*b*c^2+4*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2))
)^(1/2)*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*
a*b^2*c+29*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*Elli
pticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a*b^2*c-7*A*
((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((2*c*x
+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*a*b*c^2+30*B*x^5*c^5+42*A
*x^4*c^5-116*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*El
lipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^2*b*c^2+61*B*((-2*c*x+(-4*a
*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^
(1/2)+b)/(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))*((2*c*x+
(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^3*c+14*A*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(
1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*
c+b^2)^(1/2)))^(1/2)*(-4*a*c+b^2)^(1/2)*b^3*c-10*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-
c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))
^(1/2)*(-4*a*c+b^2)^(1/2)*a^2*c^2+48*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a
*c+b^2)^(1/2)))^(1/2)*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a^2*b
*c^2-12*B*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*Ellipti
cF(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^3*c+21*A*((-2*c*x+(-4*a*c+b^2)
^(1/2)-b)/(-4*a*c+b^2)^(1/2))^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticF(((2*c*x+(-4*a*c+b^2)^(1/2)+b
)/(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))*((2*c*x+(-4*a*c
+b^2)^(1/2)+b)/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*a*b^2*c^2-98*A*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)/(-4*a*c+b^2)^(1/2))
^(1/2)*(-c*x/(b+(-4*a*c+b^2)^(1/2)))^(1/2)*EllipticE(((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(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))*((2*c*x+(-4*a*c+b^2)^(1/2)+b)/(b+(-4*a*c+b^2
)^(1/2)))^(1/2)*a*b^2*c^2)/x^(1/2)/(c*x^2+b*x+a)^(1/2)/c^4
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{c x^{2} + b x + a}{\left (B x + A\right )} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*x^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
[Out]
integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)*sqrt(x), 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}{\left (B x + A\right )} \sqrt{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*x^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
[Out]
integral(sqrt(c*x^2 + b*x + a)*(B*x + A)*sqrt(x), x)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x} \left (A + B x\right ) \sqrt{a + b x + c x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*x**(1/2)*(c*x**2+b*x+a)**(1/2),x)
[Out]
Integral(sqrt(x)*(A + B*x)*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 (B x + A\right )} \sqrt{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)*x^(1/2)*(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)*sqrt(x), x)