3.11.29 \(\int \sqrt {x} (A+B x) \sqrt {a+b x+c x^2} \, dx\) [1029]

Optimal. Leaf size=454 \[ -\frac {2 \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \sqrt {x} \sqrt {a+b x+c x^2}}{105 c^{5/2} \left (\sqrt {a}+\sqrt {c} x\right )}-\frac {2 \sqrt {x} \left (4 b^2 B-7 A b c+5 a B c+3 c (4 b B-7 A c) x\right ) \sqrt {a+b x+c x^2}}{105 c^2}+\frac {2 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {2 \sqrt [4]{a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{105 c^{11/4} \sqrt {a+b x+c x^2}}-\frac {\sqrt [4]{a} \left (5 a b B c-2 \left (b^2-3 a c\right ) (4 b B-7 A c)-\sqrt {a} \sqrt {c} \left (4 b^2 B-7 A b c-10 a B c\right )\right ) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {a+b x+c x^2}{\left (\sqrt {a}+\sqrt {c} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{105 c^{11/4} \sqrt {a+b x+c x^2}} \]

[Out]

2/7*B*(c*x^2+b*x+a)^(3/2)*x^(1/2)/c-2/105*(4*b^2*B-7*A*b*c+5*a*B*c+3*c*(-7*A*c+4*B*b)*x)*x^(1/2)*(c*x^2+b*x+a)
^(1/2)/c^2-2/105*(5*a*b*B*c-2*(-3*a*c+b^2)*(-7*A*c+4*B*b))*x^(1/2)*(c*x^2+b*x+a)^(1/2)/c^(5/2)/(a^(1/2)+x*c^(1
/2))+2/105*a^(1/4)*(5*a*b*B*c-2*(-3*a*c+b^2)*(-7*A*c+4*B*b))*(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))*((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^(11/4)/(c*x^2+b*x+a)^(1/2)-1/105*
a^(1/4)*(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))*(5*a*b*B*c-2*(-3*a*c+b
^2)*(-7*A*c+4*B*b)-(-7*A*b*c-10*B*a*c+4*B*b^2)*a^(1/2)*c^(1/2))*((c*x^2+b*x+a)/(a^(1/2)+x*c^(1/2))^2)^(1/2)/c^
(11/4)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.37, antiderivative size = 454, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {846, 828, 853, 1211, 1117, 1209} \begin {gather*} -\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 \text {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 )}{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 \text {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 )}{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 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 {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 B \sqrt {x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \end {gather*}

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 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 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*} \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 \text {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 ) \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 )}{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 ) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 23.25, size = 638, normalized size = 1.41 \begin {gather*} \frac {2 \sqrt {x} \sqrt {a+x (b+c x)} \left (-4 b^2 B+b c (7 A+3 B x)+c (10 a B+3 c x (7 A+5 B x))\right )}{105 c^2}-\frac {-4 \left (8 b^3 B-14 A b^2 c-29 a b B c+42 a A c^2\right ) \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} (a+x (b+c x))+i \left (8 b^3 B-14 A b^2 c-29 a b B c+42 a A c^2\right ) \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} 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 )+i \left (8 b^4 B+2 a c^2 \left (10 a B-21 A \sqrt {b^2-4 a c}\right )-2 b^3 \left (7 A c+4 B \sqrt {b^2-4 a c}\right )+a b c \left (56 A c+29 B \sqrt {b^2-4 a c}\right )+b^2 \left (-37 a B c+14 A c \sqrt {b^2-4 a c}\right )\right ) \sqrt {1+\frac {2 a}{\left (b+\sqrt {b^2-4 a c}\right ) x}} x^{3/2} \sqrt {\frac {4 a+2 b x-2 \sqrt {b^2-4 a c} x}{b x-\sqrt {b^2-4 a c} x}} F\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 )}{210 c^3 \sqrt {\frac {a}{b+\sqrt {b^2-4 a c}}} \sqrt {x} \sqrt {a+x (b+c x)}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(2883\) vs. \(2(440)=880\).
time = 0.98, size = 2884, normalized size = 6.35 Too large to display

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,method=_RETURNVERBOSE)

[Out]

-1/105*(-4*B*(-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^2*c-29*B
*(-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^2*c+98*A*((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^2*c^2+116*B*((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*c^2-14*A*(-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)*Ellipt
icE(((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^3*c-21*A*((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^2*c^2+10*B*(
-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*c^2-48*B*((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*c^2+12*B*((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^3*c-61*B*((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^3*c
+7*A*(-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*c^2+42*A*(-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*c^2-14*A*((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^4*c+8*B*(-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^4-56*A*b*c^4*x^3-14*A*a*b*c^3*x+8*B*a*b^2*c^2*x-42*A*c^5*x^4-30*B*c^5*x^5+2*
B*b^2*c^3*x^3-42*A*a*c^4*x^2-14*A*b^2*c^3*x^2+8*B*b^3*c^2*x^2-36*B*b*c^4*x^4-50*B*a*c^4*x^3+8*B*((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^5-20*B*a^2*c^3*x-168*A*((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...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.88, size = 265, normalized size = 0.58 \begin {gather*} -\frac {2 \, {\left ({\left (8 \, B b^{4} + 3 \, {\left (10 \, B a^{2} + 21 \, A a b\right )} c^{2} - {\left (41 \, B a b^{2} + 14 \, A b^{3}\right )} c\right )} \sqrt {c} {\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 ) + 3 \, {\left (8 \, B b^{3} c + 42 \, A a c^{3} - {\left (29 \, B a b + 14 \, A b^{2}\right )} c^{2}\right )} \sqrt {c} {\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 (15 \, B c^{4} x^{2} - 4 \, B b^{2} c^{2} + {\left (10 \, B a + 7 \, A b\right )} c^{3} + 3 \, {\left (B b c^{3} + 7 \, A c^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {x}\right )}}{315 \, c^{4}} \end {gather*}

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]

-2/315*((8*B*b^4 + 3*(10*B*a^2 + 21*A*a*b)*c^2 - (41*B*a*b^2 + 14*A*b^3)*c)*sqrt(c)*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*(8*B*b^3*c + 42*A*a*c^3 - (29*B*a*b + 14*
A*b^2)*c^2)*sqrt(c)*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*(15*B*c^4*x^2 - 4*B*b^2*c^2 + (10*B*
a + 7*A*b)*c^3 + 3*(B*b*c^3 + 7*A*c^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(x))/c^4

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x} \left (A + B x\right ) \sqrt {a + b x + c x^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {x}\,\left (A+B\,x\right )\,\sqrt {c\,x^2+b\,x+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)*(A + B*x)*(a + b*x + c*x^2)^(1/2),x)

[Out]

int(x^(1/2)*(A + B*x)*(a + b*x + c*x^2)^(1/2), x)

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