3.10.1 \(\int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx\) [901]

Optimal. Leaf size=567 \[ \frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {a+b x+c x^2}}+\frac {4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e g) \left (c f^2-b f g+a g^2\right ) \sqrt {\frac {c (f+g x)}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g}\right )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}} \]

[Out]

2/15*e*(-4*b*e*g+7*c*d*g+c*e*f)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2/g+2/5*e*(e*x+d)*(g*x+f)^(1/2)*(c*x^2+b*x
+a)^(1/2)/c+1/15*(8*b^2*e^2*g^2-c*e*g*(9*a*e*g+20*b*d*g+3*b*e*f)-c^2*(-15*d^2*g^2-10*d*e*f*g+2*e^2*f^2))*Ellip
ticE(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^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)+4/15*e*(2*b*e*g-5*c*d*g+c*e*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^3/g^2/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2)

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Rubi [A]
time = 0.66, antiderivative size = 567, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {955, 1667, 857, 732, 435, 430} \begin {gather*} \frac {4 \sqrt {2} e \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 )}} (2 b e g-5 c d g+c e f) F\left (\text {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 )}{15 c^3 g^2 \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 (-c e g (9 a e g+20 b d g+3 b e f)+8 b^2 e^2 g^2-\left (c^2 \left (-15 d^2 g^2-10 d e f g+2 e^2 f^2\right )\right )\right ) E\left (\text {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 )}{15 c^3 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 e \sqrt {f+g x} \sqrt {a+b x+c x^2} (-4 b e g+7 c d g+c e f)}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e*(c*e*f + 7*c*d*g - 4*b*e*g)*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2])/(15*c^2*g) + (2*e*(d + e*x)*Sqrt[f + g*x
]*Sqrt[a + b*x + c*x^2])/(5*c) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(8*b^2*e^2*g^2 - c*e*g*(3*b*e*f + 20*b*d*g + 9*a*e
*g) - c^2*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*Sqrt[f + g*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*El
lipticE[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^3*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]) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*(c*e*f - 5*c*d*g + 2*b*e*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[Sq
rt[(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^3*g^2*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 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]

Rule 955

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

Rule 1667

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq
, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m
+ q + 2*p + 1))), x] + Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^
q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q
- 1) - c*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p +
1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2
, 0] &&  !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rubi steps

\begin {align*} \int \frac {(d+e x)^2 \sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx &=\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}-\frac {\int \frac {-5 c d^2 f+e (b d f+2 a e f+a d g)-(c d (8 e f+5 d g)-e (3 b e f+2 b d g+3 a e g)) x-e (c e f+7 c d g-4 b e g) x^2}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{5 c}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}-\frac {2 \int \frac {-\frac {1}{2} g \left (4 b^2 e^2 f g+b e \left (4 a e g^2-c f (e f+10 d g)\right )+c g \left (15 c d^2 f-a e (7 e f+10 d g)\right )\right )-\frac {1}{2} g \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (2 e (c e f-5 c d g+2 b e 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^2 g^2}+\frac {\left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+b x+c x^2}} \, dx}{15 c^2 g^2}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\left (\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\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 )}{15 c^3 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 (4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e 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 ) \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 )}{15 c^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ &=\frac {2 e (c e f+7 c d g-4 b e g) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{15 c^2 g}+\frac {2 e (d+e x) \sqrt {f+g x} \sqrt {a+b x+c x^2}}{5 c}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)-c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\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^3 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 {4 \sqrt {2} \sqrt {b^2-4 a c} e (c e f-5 c d g+2 b e 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^3 g^2 \sqrt {f+g x} \sqrt {a+b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 29.28, size = 1002, normalized size = 1.77 \begin {gather*} \frac {\left (-\frac {2 e (-c e f-10 c d g+4 b e g)}{15 c^2 g}+\frac {2 e^2 x}{5 c}\right ) \sqrt {f+g x} \left (a+b x+c x^2\right )}{\sqrt {a+x (b+c x)}}-\frac {2 (f+g x)^{3/2} \sqrt {a+b x+c x^2} \left (\left (-8 b^2 e^2 g^2+c e g (3 b e f+20 b d g+9 a e g)+c^2 \left (2 e^2 f^2-10 d e f g-15 d^2 g^2\right )\right ) \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )+\frac {i \sqrt {1-\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \sqrt {1+\frac {2 \left (c f^2+g (-b f+a g)\right )}{\left (-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} \left (\left (2 c f-b g+\sqrt {\left (b^2-4 a c\right ) g^2}\right ) \left (8 b^2 e^2 g^2-c e g (3 b e f+20 b d g+9 a e g)+c^2 \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+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 (-30 c^3 d^2 f g^2+8 b^2 e^2 g^2 \left (b g-\sqrt {\left (b^2-4 a c\right ) g^2}\right )+c e g \left (-17 a b e g^2+9 a e g \sqrt {\left (b^2-4 a c\right ) g^2}+b \sqrt {\left (b^2-4 a c\right ) g^2} (3 e f+20 d g)-b^2 g (11 e f+20 d g)\right )-c^2 \left (-15 b d g^2 (2 e f+d g)-2 a e g^2 (7 e f+10 d g)+\sqrt {\left (b^2-4 a c\right ) g^2} \left (-2 e^2 f^2+10 d e f g+15 d^2 g^2\right )\right )\right ) F\left (i \sinh ^{-1}\left (\frac {\sqrt {2} \sqrt {\frac {c f^2-b f g+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 )}{2 \sqrt {2} \sqrt {\frac {c f^2+g (-b f+a g)}{-2 c f+b g+\sqrt {\left (b^2-4 a c\right ) g^2}}} \sqrt {f+g x}}\right )}{15 c^3 g^3 \sqrt {a+x (b+c x)} \sqrt {\frac {(f+g x)^2 \left (c \left (-1+\frac {f}{f+g x}\right )^2+\frac {g \left (b-\frac {b f}{f+g x}+\frac {a g}{f+g x}\right )}{f+g x}\right )}{g^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(((-2*e*(-(c*e*f) - 10*c*d*g + 4*b*e*g))/(15*c^2*g) + (2*e^2*x)/(5*c))*Sqrt[f + g*x]*(a + b*x + c*x^2))/Sqrt[a
+ x*(b + c*x)] - (2*(f + g*x)^(3/2)*Sqrt[a + b*x + c*x^2]*((-8*b^2*e^2*g^2 + c*e*g*(3*b*e*f + 20*b*d*g + 9*a*
e*g) + c^2*(2*e^2*f^2 - 10*d*e*f*g - 15*d^2*g^2))*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f
+ g*x)))/(f + g*x)) + ((I/2)*Sqrt[1 - (2*(c*f^2 + g*(-(b*f) + a*g)))/((2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])
*(f + g*x))]*Sqrt[1 + (2*(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])*(8*b^2*e^2*g^2 - c*e*g*(3*b*e*f + 20*b*d*g + 9*a*e*g) + c^2*(-2*e^2*f^2
+ 10*d*e*f*g + 15*d^2*g^2))*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]))] + (-30*c^3*d^2*f*g^2 + 8*b^2*e^2*g^2*(b*g - Sqrt[(b^2 - 4*a*c)*g^2]) + c*e*g*(-17*a*b*e*g^2 + 9*a*e*
g*Sqrt[(b^2 - 4*a*c)*g^2] + b*Sqrt[(b^2 - 4*a*c)*g^2]*(3*e*f + 20*d*g) - b^2*g*(11*e*f + 20*d*g)) - c^2*(-15*b
*d*g^2*(2*e*f + d*g) - 2*a*e*g^2*(7*e*f + 10*d*g) + Sqrt[(b^2 - 4*a*c)*g^2]*(-2*e^2*f^2 + 10*d*e*f*g + 15*d^2*
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])])/Sq
rt[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[2]*
Sqrt[(c*f^2 + g*(-(b*f) + a*g))/(-2*c*f + b*g + Sqrt[(b^2 - 4*a*c)*g^2])]*Sqrt[f + g*x])))/(15*c^3*g^3*Sqrt[a
+ x*(b + c*x)]*Sqrt[((f + g*x)^2*(c*(-1 + f/(f + g*x))^2 + (g*(b - (b*f)/(f + g*x) + (a*g)/(f + g*x)))/(f + g*
x)))/g^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(8247\) vs. \(2(503)=1006\).
time = 0.15, size = 8248, normalized size = 14.55

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+b x +a \right )}\, \left (\frac {2 e^{2} x \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{5 c}+\frac {2 \left (2 g d e +e^{2} f -\frac {2 \left (2 b g +2 c f \right ) e^{2}}{5 c}\right ) \sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}{3 c g}+\frac {2 \left (f \,d^{2}-\frac {2 f a \,e^{2}}{5 c}-\frac {2 \left (2 g d e +e^{2} f -\frac {2 \left (2 b g +2 c f \right ) e^{2}}{5 c}\right ) \left (\frac {a g}{2}+\frac {b f}{2}\right )}{3 c g}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}+\frac {2 \left (d^{2} g +2 d e f -\frac {2 e^{2} \left (\frac {3 a g}{2}+\frac {3 b f}{2}\right )}{5 c}-\frac {2 \left (2 g d e +e^{2} f -\frac {2 \left (2 b g +2 c f \right ) e^{2}}{5 c}\right ) \left (b g +c f \right )}{3 c g}\right ) \left (-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}\right ) \sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}\, \sqrt {\frac {x -\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \sqrt {\frac {x +\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}}\, \left (\left (-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )+\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \EllipticF \left (\sqrt {\frac {x +\frac {f}{g}}{-\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}+\frac {f}{g}}}, \sqrt {\frac {-\frac {f}{g}+\frac {b +\sqrt {-4 a c +b^{2}}}{2 c}}{-\frac {f}{g}-\frac {-b +\sqrt {-4 a c +b^{2}}}{2 c}}}\right )}{2 c}\right )}{\sqrt {c g \,x^{3}+b g \,x^{2}+c f \,x^{2}+a g x +b f x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}}\)	\(985\)
risch	\(\text {Expression too large to display}\)	\(2623\)
default	\(\text {Expression too large to display}\)	\(8248\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.55, size = 613, normalized size = 1.08 \begin {gather*} \frac {2 \, {\left ({\left (30 \, c^{3} d^{2} f g^{2} - 15 \, b c^{2} d^{2} g^{3} + {\left (2 \, c^{3} f^{3} + 2 \, b c^{2} f^{2} g + {\left (7 \, b^{2} c - 12 \, a c^{2}\right )} f g^{2} - {\left (8 \, b^{3} - 21 \, a b c\right )} g^{3}\right )} e^{2} - 10 \, {\left (c^{3} d f^{2} g + 2 \, b c^{2} d f g^{2} - {\left (2 \, b^{2} c - 3 \, a c^{2}\right )} d g^{3}\right )} e\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 (15 \, c^{3} d^{2} g^{3} - {\left (2 \, c^{3} f^{2} g + 3 \, b c^{2} f g^{2} - {\left (8 \, b^{2} c - 9 \, a c^{2}\right )} g^{3}\right )} e^{2} + 10 \, {\left (c^{3} d f g^{2} - 2 \, b c^{2} d g^{3}\right )} e\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 (10 \, c^{3} d g^{3} e + {\left (3 \, c^{3} g^{3} x + c^{3} f g^{2} - 4 \, b c^{2} g^{3}\right )} e^{2}\right )} \sqrt {c x^{2} + b x + a} \sqrt {g x + f}\right )}}{45 \, c^{4} g^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/45*((30*c^3*d^2*f*g^2 - 15*b*c^2*d^2*g^3 + (2*c^3*f^3 + 2*b*c^2*f^2*g + (7*b^2*c - 12*a*c^2)*f*g^2 - (8*b^3
- 21*a*b*c)*g^3)*e^2 - 10*(c^3*d*f^2*g + 2*b*c^2*d*f*g^2 - (2*b^2*c - 3*a*c^2)*d*g^3)*e)*sqrt(c*g)*weierstrass
PInverse(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^3*d^2*g^3 - (2*c
^3*f^2*g + 3*b*c^2*f*g^2 - (8*b^2*c - 9*a*c^2)*g^3)*e^2 + 10*(c^3*d*f*g^2 - 2*b*c^2*d*g^3)*e)*sqrt(c*g)*weiers
trassZeta(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*(10*c^3*d*g^3*e + (3*c^3*g^3*x + c^3*f*g^2 - 4*b*c^2*g^3)*e^2)*sqrt(
c*x^2 + b*x + a)*sqrt(g*x + f))/(c^4*g^3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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