\(\int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx\) [888]

Optimal result

Integrand size = 29, antiderivative size = 652 \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

[Out]

2/7*e*(c*x^2+b*x+a)^(3/2)*(g*x+f)^(1/2)/c-2/105*(4*b^2*e*g^2+c^2*f*(-7*d*g+4*e*f)-c*g*(-5*a*e*g+7*b*d*g+2*b*e*
f)-3*c*g*(-4*b*e*g+7*c*d*g+c*e*f)*x)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/g^2+1/105*((-4*b*e*g+7*c*d*g+c*e*f)
*(8*c^2*f^2-2*b^2*g^2-3*c*g*(-2*a*g+b*f))-5*c*g*(-b*g+2*c*f)*(7*c*d*f-e*(a*g+3*b*f)))*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*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))
))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(g*x+f)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)/c^3/g^3/(c*x^2+b*x+a)
^(1/2)/(c*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)+2/105*(a*g^2-b*f*g+c*f^2)*(4*b^2*e*g^2-2*c^2*f*(-7*d
*g+4*e*f)+c*g*(-10*a*e*g-7*b*d*g+b*e*f))*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*g*(-4*a*c+b^2)^(1/2)/(2*c*f-g*(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*(g*x+f)/(2*c*f-g*(b+(-4*a*c+b^2)^(1/2))))^(1/2)/c^3/g^3/(g*x+f)^(1/2)/(c*x^2+
b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {846, 828, 857, 732, 435, 430} \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} \left (c g (-10 a e g-7 b d g+b e f)+4 b^2 e g^2-2 c^2 f (4 e f-7 d g)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (\left (-3 c g (b f-2 a g)-2 b^2 g^2+8 c^2 f^2\right ) (-4 b e g+7 c d g+c e f)-5 c g (2 c f-b g) (7 c d f-e (a g+3 b f))\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} \left (-c g (-5 a e g+7 b d g+2 b e f)+4 b^2 e g^2-3 c g x (-4 b e g+7 c d g+c e f)+c^2 f (4 e f-7 d g)\right )}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c} \]

[In]

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

[Out]

(-2*Sqrt[f + g*x]*(4*b^2*e*g^2 + c^2*f*(4*e*f - 7*d*g) - c*g*(2*b*e*f + 7*b*d*g - 5*a*e*g) - 3*c*g*(c*e*f + 7*
c*d*g - 4*b*e*g)*x)*Sqrt[a + b*x + c*x^2])/(105*c^2*g^2) + (2*e*Sqrt[f + g*x]*(a + b*x + c*x^2)^(3/2))/(7*c) +
(Sqrt[2]*Sqrt[b^2 - 4*a*c]*((c*e*f + 7*c*d*g - 4*b*e*g)*(8*c^2*f^2 - 2*b^2*g^2 - 3*c*g*(b*f - 2*a*g)) - 5*c*g
*(2*c*f - b*g)*(7*c*d*f - e*(3*b*f + a*g)))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti
cE[ArcSin[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]*g)/(2*c*f -
(b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^3*g^3*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[a + b*x
+ c*x^2]) + (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*f^2 - b*f*g + a*g^2)*(4*b^2*e*g^2 - 2*c^2*f*(4*e*f - 7*d*g) + c*g
*(b*e*f - 7*b*d*g - 10*a*e*g))*Sqrt[(c*(f + g*x))/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)]*Sqrt[-((c*(a + b*x + c*
x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*S
qrt[b^2 - 4*a*c]*g)/(2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)])/(105*c^3*g^3*Sqrt[f + g*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 {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {2 \int \frac {\left (\frac {1}{2} (7 c d f-3 b e f-a e g)+\frac {1}{2} (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{\sqrt {f+g x}} \, dx}{7 c} \\ & = -\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}-\frac {4 \int \frac {\frac {1}{4} \left (5 c g (b f-2 a g) (7 c d f-e (3 b f+a g))-2 (c e f+7 c d g-4 b e g) \left (\frac {1}{2} b f (4 c f-b g)-a g \left (c f+\frac {b g}{2}\right )\right )\right )-\frac {1}{4} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^2} \\ & = -\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (\left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^3}+\frac {\left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{105 c^2 g^3} \\ & = -\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \sqrt {f+g 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} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}}{\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^3 g^3 \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \sqrt {a+b x+c x^2}}+\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right ) \sqrt {\frac {c (f+g x)}{2 c f-b g-\sqrt {b^2-4 a c} g}} \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} g x^2}{2 c f-b g-\sqrt {b^2-4 a c} g}}} \, 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 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ & = -\frac {2 \sqrt {f+g x} \left (4 b^2 e g^2+c^2 f (4 e f-7 d g)-c g (2 b e f+7 b d g-5 a e g)-3 c g (c e f+7 c d g-4 b e g) x\right ) \sqrt {a+b x+c x^2}}{105 c^2 g^2}+\frac {2 e \sqrt {f+g x} \left (a+b x+c x^2\right )^{3/2}}{7 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left ((c e f+7 c d g-4 b e g) \left (8 c^2 f^2-2 b^2 g^2-3 c g (b f-2 a g)\right )-5 c g (2 c f-b g) (7 c d f-e (3 b f+a g))\right ) \sqrt {f+g 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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c f^2-b f g+a g^2\right ) \left (4 b^2 e g^2-2 c^2 f (4 e f-7 d g)+c g (b e f-7 b d g-10 a e g)\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \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} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{105 c^3 g^3 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.21 (sec) , antiderivative size = 8432, normalized size of antiderivative = 12.93 \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1228\) vs. \(2(588)=1176\).

Time = 1.95 (sec) , antiderivative size = 1229, normalized size of antiderivative = 1.88

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1229\)
risch	\(\text {Expression too large to display}\)	\(3893\)
default	\(\text {Expression too large to display}\)	\(10711\)

[In]

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

[Out]

((g*x+f)*(c*x^2+b*x+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/7*e*x^2*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*
x+a*f)^(1/2)+2/5*(b*e*g+c*d*g+c*e*f-2/7*e*(3*b*g+3*c*f))/c/g*x*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)
+2/3*(a*e*g+b*d*g+b*e*f+c*d*f-2/7*e*(5/2*a*g+5/2*b*f)-2/5*(b*e*g+c*d*g+c*e*f-2/7*e*(3*b*g+3*c*f))/c/g*(2*b*g+2
*c*f))/c/g*(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)+2*(a*d*f-2/5*(b*e*g+c*d*g+c*e*f-2/7*e*(3*b*g+3*c*f)
)/c/g*f*a-2/3*(a*e*g+b*d*g+b*e*f+c*d*f-2/7*e*(5/2*a*g+5/2*b*f)-2/5*(b*e*g+c*d*g+c*e*f-2/7*e*(3*b*g+3*c*f))/c/g
*(2*b*g+2*c*f))/c/g*(1/2*a*g+1/2*b*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-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)))/(-f/g-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)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)*E
llipticF(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-1/2/c*(
-b+(-4*a*c+b^2)^(1/2))))^(1/2))+2*(a*d*g+3/7*a*e*f+b*d*f-2/5*(b*e*g+c*d*g+c*e*f-2/7*e*(3*b*g+3*c*f))/c/g*(3/2*
a*g+3/2*b*f)-2/3*(a*e*g+b*d*g+b*e*f+c*d*f-2/7*e*(5/2*a*g+5/2*b*f)-2/5*(b*e*g+c*d*g+c*e*f-2/7*e*(3*b*g+3*c*f))/
c/g*(2*b*g+2*c*f))/c/g*(b*g+c*f))*(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+f/g)/(f/g-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)))/(-f/g-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)/(-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)^(1/2)*((-f/g
-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+f/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4
*a*c+b^2)^(1/2))/c)/(-f/g-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+f
/g)/(f/g-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-f/g+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-f/g-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 = 0.12 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.11 \[ \int (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{4} e f^{4} - {\left (14 \, c^{4} d + 9 \, b c^{3} e\right )} f^{3} g + {\left (21 \, b c^{3} d - 2 \, {\left (2 \, b^{2} c^{2} - 11 \, a c^{3}\right )} e\right )} f^{2} g^{2} + {\left (21 \, {\left (b^{2} c^{2} - 6 \, a c^{3}\right )} d - {\left (9 \, b^{3} c - 41 \, a b c^{2}\right )} e\right )} f g^{3} - {\left (7 \, {\left (2 \, b^{3} c - 9 \, a b c^{2}\right )} d - {\left (8 \, b^{4} - 41 \, a b^{2} c + 30 \, a^{2} c^{2}\right )} e\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right ) + 3 \, {\left (8 \, c^{4} e f^{3} g - {\left (14 \, c^{4} d + 5 \, b c^{3} e\right )} f^{2} g^{2} + {\left (14 \, b c^{3} d - {\left (5 \, b^{2} c^{2} - 16 \, a c^{3}\right )} e\right )} f g^{3} - {\left (14 \, {\left (b^{2} c^{2} - 3 \, a c^{3}\right )} d - {\left (8 \, b^{3} c - 29 \, a b c^{2}\right )} e\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} f^{2} - b c f g + {\left (b^{2} - 3 \, a c\right )} g^{2}\right )}}{3 \, c^{2} g^{2}}, -\frac {4 \, {\left (2 \, c^{3} f^{3} - 3 \, b c^{2} f^{2} g - 3 \, {\left (b^{2} c - 6 \, a c^{2}\right )} f g^{2} + {\left (2 \, b^{3} - 9 \, a b c\right )} g^{3}\right )}}{27 \, c^{3} g^{3}}, \frac {3 \, c g x + c f + b g}{3 \, c g}\right )\right ) - 3 \, {\left (15 \, c^{4} e g^{4} x^{2} - 4 \, c^{4} e f^{2} g^{2} + {\left (7 \, c^{4} d + 2 \, b c^{3} e\right )} f g^{3} + {\left (7 \, b c^{3} d - 2 \, {\left (2 \, b^{2} c^{2} - 5 \, a c^{3}\right )} e\right )} g^{4} + 3 \, {\left (c^{4} e f g^{3} + {\left (7 \, c^{4} d + b c^{3} e\right )} g^{4}\right )} x\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{315 \, c^{4} g^{4}} \]

[In]

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

[Out]

-2/315*((8*c^4*e*f^4 - (14*c^4*d + 9*b*c^3*e)*f^3*g + (21*b*c^3*d - 2*(2*b^2*c^2 - 11*a*c^3)*e)*f^2*g^2 + (21*
(b^2*c^2 - 6*a*c^3)*d - (9*b^3*c - 41*a*b*c^2)*e)*f*g^3 - (7*(2*b^3*c - 9*a*b*c^2)*d - (8*b^4 - 41*a*b^2*c + 3
0*a^2*c^2)*e)*g^4)*sqrt(c*g)*weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*
(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f
+ b*g)/(c*g)) + 3*(8*c^4*e*f^3*g - (14*c^4*d + 5*b*c^3*e)*f^2*g^2 + (14*b*c^3*d - (5*b^2*c^2 - 16*a*c^3)*e)*f*
g^3 - (14*(b^2*c^2 - 3*a*c^3)*d - (8*b^3*c - 29*a*b*c^2)*e)*g^4)*sqrt(c*g)*weierstrassZeta(4/3*(c^2*f^2 - b*c*
f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*
a*b*c)*g^3)/(c^3*g^3), weierstrassPInverse(4/3*(c^2*f^2 - b*c*f*g + (b^2 - 3*a*c)*g^2)/(c^2*g^2), -4/27*(2*c^3
*f^3 - 3*b*c^2*f^2*g - 3*(b^2*c - 6*a*c^2)*f*g^2 + (2*b^3 - 9*a*b*c)*g^3)/(c^3*g^3), 1/3*(3*c*g*x + c*f + b*g)
/(c*g))) - 3*(15*c^4*e*g^4*x^2 - 4*c^4*e*f^2*g^2 + (7*c^4*d + 2*b*c^3*e)*f*g^3 + (7*b*c^3*d - 2*(2*b^2*c^2 - 5
*a*c^3)*e)*g^4 + 3*(c^4*e*f*g^3 + (7*c^4*d + b*c^3*e)*g^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(g*x + f))/(c^4*g^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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