3.889 \(\int \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx\)

Optimal. Leaf size=513 \[ \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c f-b g) \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}-\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (2 c f-b g)}{15 c g} \]

[Out]

2/5*(g*x+f)^(3/2)*(c*x^2+b*x+a)^(1/2)/g-2/15*(-b*g+2*c*f)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/g-2/15*(c^2*f^2+
b^2*g^2-c*g*(3*a*g+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^2/g^2/(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)+2/1
5*(-b*g+2*c*f)*(a*g^2-b*f*g+c*f^2)*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^2/g^2/(g*x+f)^(1/2)/(c*x^2+b*x+a)
^(1/2)

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Rubi [A]  time = 0.54, antiderivative size = 513, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {734, 832, 843, 718, 424, 419} \[ \frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c f-b g) \left (a g^2-b f g+c f^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^2 g^2 \sqrt {a+b x+c x^2} \sqrt {\frac {c (f+g x)}{2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}}+\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}-\frac {2 \sqrt {f+g x} \sqrt {a+b x+c x^2} (2 c f-b g)}{15 c g} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]

[Out]

(-2*(2*c*f - b*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(15*c*g) + (2*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2])/(5
*g) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c^2*f^2 + b^2*g^2 - c*g*(b*f + 3*a*g))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x +
c*x^2))/(b^2 - 4*a*c))]*EllipticE[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)])/(15*c^2*g^2*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]*(2*c*f - b*g)*(c*f^2 - b*f*g + a*g^2)*
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*Sqrt[b^2 - 4*a*c]*g)/(2*c*f - (b
+ Sqrt[b^2 - 4*a*c])*g)])/(15*c^2*g^2*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])

Rule 419

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

Rule 424

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

Rule 718

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 734

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*x + c*x^2)^p)/(e*(m + 2*p + 1)), x] - Dist[p/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[b*d - 2*a*e + (2*c*
d - b*e)*x, x]*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, m}, 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] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !RationalQ[m] || Lt
Q[m, 1]) &&  !ILtQ[m + 2*p, 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 843

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*} \int \sqrt {f+g x} \sqrt {a+b x+c x^2} \, dx &=\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}-\frac {\int \frac {\sqrt {f+g x} (b f-2 a g+(2 c f-b g) x)}{\sqrt {a+b x+c x^2}} \, dx}{5 g}\\ &=-\frac {2 (2 c f-b g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c g}+\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}-\frac {2 \int \frac {\frac {1}{2} \left (b c f^2+b^2 f g-8 a c f g+a b g^2\right )+\left (c^2 f^2+b^2 g^2-c g (b f+3 a g)\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c g}\\ &=-\frac {2 (2 c f-b g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c g}+\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}+\frac {\left ((2 c f-b g) \left (c f^2-b f g+a g^2\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c g^2}-\frac {\left (2 \left (c^2 f^2+b^2 g^2-c g (b f+3 a g)\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c g^2}\\ &=-\frac {2 (2 c f-b g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c g}+\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}-\frac {\left (2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 f^2+b^2 g^2-c g (b f+3 a g)\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}}\right ) \operatorname {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 )}{15 c^2 g^2 \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} (2 c f-b g) \left (c f^2-b f g+a g^2\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 ) \operatorname {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 )}{15 c^2 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=-\frac {2 (2 c f-b g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c g}+\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2}}{5 g}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c^2 f^2+b^2 g^2-c g (b f+3 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 )}{15 c^2 g^2 \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} (2 c f-b g) \left (c f^2-b f g+a g^2\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 )}{15 c^2 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C]  time = 10.69, size = 697, normalized size = 1.36 \[ \frac {-\frac {4 g^2 (a+x (b+c x)) \left (-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right )}{\sqrt {f+g x}}+\frac {i (f+g x) \sqrt {1-\frac {2 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt {g^2 \left (b^2-4 a c\right )}-b g+2 c f\right )}} \sqrt {\frac {4 \left (g (a g-b f)+c f^2\right )}{(f+g x) \left (\sqrt {g^2 \left (b^2-4 a c\right )}+b g-2 c f\right )}+2} \left (\left (\sqrt {g^2 \left (b^2-4 a c\right )}-b g+2 c f\right ) \left (-c g (3 a g+b f)+b^2 g^2+c^2 f^2\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )+\left (c \left (a g^2 \left (3 \sqrt {g^2 \left (b^2-4 a c\right )}+8 c f\right )-c f^2 \sqrt {g^2 \left (b^2-4 a c\right )}\right )-b^2 g^2 \left (\sqrt {g^2 \left (b^2-4 a c\right )}+2 c f\right )+b c g \left (f \sqrt {g^2 \left (b^2-4 a c\right )}-4 a g^2\right )+b^3 g^3\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b g f+a g^2}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}}}{\sqrt {f+g x}}\right )|-\frac {-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}}\right )\right )}{\sqrt {\frac {g (a g-b f)+c f^2}{\sqrt {g^2 \left (b^2-4 a c\right )}+b g-2 c f}}}+2 c g^2 \sqrt {f+g x} (a+x (b+c x)) (b g+c (f+3 g x))}{15 c^2 g^3 \sqrt {a+x (b+c x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2],x]

[Out]

((-4*g^2*(c^2*f^2 + b^2*g^2 - c*g*(b*f + 3*a*g))*(a + x*(b + c*x)))/Sqrt[f + g*x] + 2*c*g^2*Sqrt[f + g*x]*(a +
x*(b + c*x))*(b*g + c*(f + 3*g*x)) + (I*(f + g*x)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqr
t[(b^2 - 4*a*c)*g^2])*(f + g*x))]*Sqrt[2 + (4*(c*f^2 + g*(-(b*f) + a*g)))/((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*
g^2])*(f + g*x))]*((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])*(c^2*f^2 + b^2*g^2 - c*g*(b*f + 3*a*g))*EllipticE[I
*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-
2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]))] + (b^3*g^3 - b^2*g^2*(2*c*f +
Sqrt[(b^2 - 4*a*c)*g^2]) + b*c*g*(-4*a*g^2 + f*Sqrt[(b^2 - 4*a*c)*g^2]) + c*(-(c*f^2*Sqrt[(b^2 - 4*a*c)*g^2])
+ a*g^2*(8*c*f + 3*Sqrt[(b^2 - 4*a*c)*g^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[(c*f^2 - b*f*g + a*g^2)/(-2*c
*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/Sqrt[f + g*x]], -((-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])/(2*c*f - b*g
+ Sqrt[(b^2 - 4*a*c)*g^2]))]))/Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])])/(15*
c^2*g^3*Sqrt[a + x*(b + c*x)])

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {c x^{2} + b x + a} \sqrt {g x + f}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)*sqrt(g*x + f), x)

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maple [B]  time = 0.04, size = 4356, normalized size = 8.49 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2*(8*x*a*c^2*f*g^3+8*x^3*b*c^2*g^4+8*x^3*c^3*f*g^3+6*x^2*a*c^2*g^4+2*
x^2*b^2*c*g^4+2*x^2*c^3*f^2*g^2+2*a*c^2*f^2*g^2+6*x^4*c^3*g^4+2*a*b*c*f*g^3-3*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b
^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b
+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(
1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*g^4+3
*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4
*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1
/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*
c+b^2)^(1/2)))^(1/2))*b^3*f*g^3-4*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c
+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)
+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/
2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b^3*f*g^3+12*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+
b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b
^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2
*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a^2*c*g^4-12*2^(1/2)*
(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)
^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*
x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1
/2)))^(1/2))*a^2*c*g^4+4*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/
2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*
f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*
c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b^2*g^4-8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)
)^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))
*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/
2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*c^2*f^2*g^2+12*2^(1/2)*(-(g*x
+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)
))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c
/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^
(1/2))*a*c^2*f^2*g^2-12*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2
)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f
))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c
*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*f*g^3+3*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f
))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2)
)*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1
/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*b*c*f^2*g^2
-2*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(
-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^
(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*
a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*c*f*g^3+8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(
1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/
(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),
(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*a*b*c*f*g^3+2*x*b^2*c*f*g^3+2*x*b*
c^2*f^2*g^2+10*x^2*b*c^2*f*g^3+2*x*a*b*c*g^4+8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-
2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a
*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4
*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*f^2*g^2+2^(1/2)*(-(g*x+f)*c/(g*(-4*a
*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c
*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^
2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c
+b^2)^(1/2)*a*b*g^4-8*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-
b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))
^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f
)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b*c^2*f^3*g-2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(
1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/
(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),
(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*b^2*f*g^3-2*2^(
1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c
+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*
(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^
2)^(1/2)))^(1/2))*(-4*a*c+b^2)^(1/2)*c^2*f^3*g-3*2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*(
(-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4
*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticF(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(
-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*b^2*c*f^2*g^2+4*2^(1/2)*(-(g*x+f)*c/(g*(
-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*g/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2)*(
(2*c*x+b+(-4*a*c+b^2)^(1/2))*g/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f))^(1/2)*EllipticE(2^(1/2)*(-(g*x+f)*c/(g*(-4*a*
c+b^2)^(1/2)+b*g-2*c*f))^(1/2),(-(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2)))^(1/2))*c^3
*f^4)/(c*g*x^3+b*g*x^2+c*f*x^2+a*g*x+b*f*x+a*f)/g^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)*sqrt(g*x + f), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {f + g x} \sqrt {a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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