\(\int (b+2 c x) \sqrt {d+e x} \sqrt {a+b x+c x^2} \, dx\) [1628]

Optimal result

Integrand size = 30, antiderivative size = 576 \[ \int (b+2 c x) \sqrt {d+e x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c e^2}+\frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^2 e^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^2 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]

[Out]

4/7*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(3/2)-2/105*(8*c^2*d^2+b^2*e^2-c*e*(-10*a*e+11*b*d)-3*c*e*(-b*e+2*c*d)*x)*(e*x
+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e^2+2/105*(-b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))*EllipticE(1/2*((
b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2
)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^2/e^3/(c*x^
2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)-2/105*(a*e^2-b*d*e+c*d^2)*(16*c^2*d^2-b^2*e^
2-4*c*e*(-5*a*e+4*b*d))*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-
4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c
+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^2/e^3/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {846, 828, 857, 732, 435, 430} \[ \int (b+2 c x) \sqrt {d+e x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-4 c e (4 b d-5 a e)-b^2 e^2+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^2 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^2 e^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e (11 b d-10 a e)+b^2 e^2-3 c e x (2 c d-b e)+8 c^2 d^2\right )}{105 c e^2}+\frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2} \]

[In]

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

[Out]

(-2*Sqrt[d + e*x]*(8*c^2*d^2 + b^2*e^2 - c*e*(11*b*d - 10*a*e) - 3*c*e*(2*c*d - b*e)*x)*Sqrt[a + b*x + c*x^2])
/(105*c*e^2) + (4*Sqrt[d + e*x]*(a + b*x + c*x^2)^(3/2))/7 + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(4*c^2
*d^2 - b^2*e^2 - 4*c*e*(b*d - 2*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[Arc
Sin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + S
qrt[b^2 - 4*a*c])*e)])/(105*c^2*e^3*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x
^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(16*c^2*d^2 - b^2*e^2 - 4*c*e*(4*b*d - 5*a*e))*Sqr
t[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[Ar
cSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b +
Sqrt[b^2 - 4*a*c])*e)])/(105*c^2*e^3*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]
))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 732

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*Rt[b^2 - 4*a*c, 2]*
(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*
e - e*Rt[b^2 - 4*a*c, 2])))^m)), Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2*c*d - b*e - e*Rt[b^2 - 4*a*c, 2
])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b
, c, d, e}, 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] && EqQ[m^2, 1/4]

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 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + b*x + c*x^
2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]
&&  !IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}+\frac {2 \int \frac {\left (\frac {1}{2} c (b d-2 a e)+\frac {1}{2} c (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {d+e x}} \, dx}{7 c} \\ & = -\frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c e^2}+\frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}-\frac {4 \int \frac {\frac {1}{4} c \left (5 c e (b d-2 a e)^2-2 (2 c d-b e) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )\right )-\frac {1}{2} c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 e^2} \\ & = -\frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c e^2}+\frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}+\frac {\left (2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c e^3}-\frac {\left (4 \left (\frac {1}{2} c d (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+\frac {1}{4} c e \left (5 c e (b d-2 a e)^2-2 (2 c d-b e) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 e^3} \\ & = -\frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c e^2}+\frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{105 c^2 e^3 \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {a+b x+c x^2}}-\frac {\left (8 \sqrt {2} \sqrt {b^2-4 a c} \left (\frac {1}{2} c d (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right )+\frac {1}{4} c e \left (5 c e (b d-2 a e)^2-2 (2 c d-b e) \left (\frac {1}{2} b d (4 c d-b e)-a e \left (c d+\frac {b e}{2}\right )\right )\right )\right ) \sqrt {\frac {c (d+e x)}{2 c d-b e-\sqrt {b^2-4 a c} e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 \sqrt {b^2-4 a c} e x^2}{2 c d-b e-\sqrt {b^2-4 a c} e}}} \, dx,x,\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )}{105 c^3 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \sqrt {d+e x} \left (8 c^2 d^2+b^2 e^2-c e (11 b d-10 a e)-3 c e (2 c d-b e) x\right ) \sqrt {a+b x+c x^2}}{105 c e^2}+\frac {4}{7} \sqrt {d+e x} \left (a+b x+c x^2\right )^{3/2}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^2 e^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (16 c^2 d^2-16 b c d e-b^2 e^2+20 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{105 c^2 e^3 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 34.06 (sec) , antiderivative size = 1281, normalized size of antiderivative = 2.22 \[ \int (b+2 c x) \sqrt {d+e x} \sqrt {a+b x+c x^2} \, dx=\sqrt {d+e x} \left (-\frac {2 \left (8 c^2 d^2-11 b c d e+b^2 e^2-20 a c e^2\right )}{105 c e^2}+\frac {2 (2 c d+9 b e) x}{35 e}+\frac {4 c x^2}{7}\right ) \sqrt {a+x (b+c x)}+\frac {(d+e x)^{3/2} \sqrt {a+x (b+c x)} \left (4 (-2 c d+b e) \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )-\frac {i \sqrt {2} (-2 c d+b e) \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (-4 c^2 d^2+b^2 e^2+4 c e (b d-2 a e)\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}+\frac {i \sqrt {2} \left (-b^4 e^4+b^3 e^3 \left (-c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b^2 c e^2 \left (c d^2+9 a e^2+2 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )-4 b c e \left (3 c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (-c d+2 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )+4 c^2 \left (-5 a^2 e^4+2 c d^3 \sqrt {\left (b^2-4 a c\right ) e^2}+a d e^2 \left (-c d+4 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}-\frac {2 a e^2}{d+e x}-2 c d \left (-1+\frac {d}{d+e x}\right )+b e \left (-1+\frac {2 d}{d+e x}\right )}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {\frac {\sqrt {\left (b^2-4 a c\right ) e^2}+\frac {2 a e^2}{d+e x}+2 c d \left (-1+\frac {d}{d+e x}\right )+b \left (e-\frac {2 d e}{d+e x}\right )}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )}{\sqrt {d+e x}}\right )}{105 c^2 e^4 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {a+b x+c x^2} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]

[In]

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

[Out]

Sqrt[d + e*x]*((-2*(8*c^2*d^2 - 11*b*c*d*e + b^2*e^2 - 20*a*c*e^2))/(105*c*e^2) + (2*(2*c*d + 9*b*e)*x)/(35*e)
+ (4*c*x^2)/7)*Sqrt[a + x*(b + c*x)] + ((d + e*x)^(3/2)*Sqrt[a + x*(b + c*x)]*(4*(-2*c*d + b*e)*(-4*c^2*d^2 +
b^2*e^2 + 4*c*e*(b*d - 2*a*e))*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(c*(
-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) - (I*Sqrt[2]*(-2*c*d + b*e)*(2*c*
d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(-4*c^2*d^2 + b^2*e^2 + 4*c*e*(b*d - 2*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2]
- (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*
c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d +
e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-
2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b
*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(-(b^4*e^4) + b^3*e^3*(-(c*d) + Sqrt[(b^2 - 4*a*c)
*e^2]) + b^2*c*e^2*(c*d^2 + 9*a*e^2 + 2*d*Sqrt[(b^2 - 4*a*c)*e^2]) - 4*b*c*e*(3*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]
+ a*e^2*(-(c*d) + 2*Sqrt[(b^2 - 4*a*c)*e^2])) + 4*c^2*(-5*a^2*e^4 + 2*c*d^3*Sqrt[(b^2 - 4*a*c)*e^2] + a*d*e^2*
(-(c*d) + 4*Sqrt[(b^2 - 4*a*c)*e^2])))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a*e^2)/(d + e*x) - 2*c*d*(-1 + d/(d
+ e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] +
(2*a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c
)*e^2])]*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/
Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))])/Sqrt[d +
e*x]))/(105*c^2*e^4*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[a + b*x +
c*x^2]*Sqrt[((d + e*x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)))/e^2
])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1220\) vs. \(2(512)=1024\).

Time = 1.61 (sec) , antiderivative size = 1221, normalized size of antiderivative = 2.12

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1221\)
risch	\(\text {Expression too large to display}\)	\(2640\)
default	\(\text {Expression too large to display}\)	\(6516\)

[In]

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

[Out]

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(4/7*c*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*
x+a*d)^(1/2)+2/5*(3*b*c*e+2*c^2*d-4/7*c*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2
/3*(2*a*c*e+b^2*e+3*b*c*d-4/7*c*(5/2*a*e+5/2*b*d)-2/5*(3*b*c*e+2*c^2*d-4/7*c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))
/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(b*d*a-2/5*(3*b*c*e+2*c^2*d-4/7*c*(3*b*e+3*c*d))/c/e*a*
d-2/3*(2*a*c*e+b^2*e+3*b*c*d-4/7*c*(5/2*a*e+5/2*b*d)-2/5*(3*b*c*e+2*c^2*d-4/7*c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*
d))/c/e*(1/2*a*e+1/2*b*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/
2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2
))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x
+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b
^2)^(1/2))))^(1/2))+2*(a*b*e+6/7*a*c*d+b^2*d-2/5*(3*b*c*e+2*c^2*d-4/7*c*(3*b*e+3*c*d))/c/e*(3/2*a*e+3/2*b*d)-2
/3*(2*a*c*e+b^2*e+3*b*c*d-4/7*c*(5/2*a*e+5/2*b*d)-2/5*(3*b*c*e+2*c^2*d-4/7*c*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))
/c/e*(b*e+c*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/
c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e
+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c
+b^2)^(1/2)))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c
)/(-d/e-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(((x+d/e)/(d/e-1/2*(b+(-
4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-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 = 0.20 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.02 \[ \int (b+2 c x) \sqrt {d+e x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \, {\left ({\left (16 \, c^{4} d^{4} - 32 \, b c^{3} d^{3} e + {\left (13 \, b^{2} c^{2} + 44 \, a c^{3}\right )} d^{2} e^{2} + {\left (3 \, b^{3} c - 44 \, a b c^{2}\right )} d e^{3} + {\left (2 \, b^{4} - 19 \, a b^{2} c + 60 \, a^{2} c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 6 \, {\left (8 \, c^{4} d^{3} e - 12 \, b c^{3} d^{2} e^{2} + 2 \, {\left (b^{2} c^{2} + 8 \, a c^{3}\right )} d e^{3} + {\left (b^{3} c - 8 \, a b c^{2}\right )} e^{4}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + {\left (b^{2} - 3 \, a c\right )} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} d e^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (30 \, c^{4} e^{4} x^{2} - 8 \, c^{4} d^{2} e^{2} + 11 \, b c^{3} d e^{3} - {\left (b^{2} c^{2} - 20 \, a c^{3}\right )} e^{4} + 3 \, {\left (2 \, c^{4} d e^{3} + 9 \, b c^{3} e^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {e x + d}\right )}}{315 \, c^{3} e^{4}} \]

[In]

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

[Out]

-2/315*((16*c^4*d^4 - 32*b*c^3*d^3*e + (13*b^2*c^2 + 44*a*c^3)*d^2*e^2 + (3*b^3*c - 44*a*b*c^2)*d*e^3 + (2*b^4
- 19*a*b^2*c + 60*a^2*c^2)*e^4)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^
2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(
3*c*e*x + c*d + b*e)/(c*e)) + 6*(8*c^4*d^3*e - 12*b*c^3*d^2*e^2 + 2*(b^2*c^2 + 8*a*c^3)*d*e^3 + (b^3*c - 8*a*b
*c^2)*e^4)*sqrt(c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 -
3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^
2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*
b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e))) - 3*(30*c^4*e^4*x^2 - 8*c^4*d^2*e^2 + 11*b*c^
3*d*e^3 - (b^2*c^2 - 20*a*c^3)*e^4 + 3*(2*c^4*d*e^3 + 9*b*c^3*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^
3*e^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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