\(\int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx\) [623]

Optimal result

Integrand size = 28, antiderivative size = 635 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=-\frac {2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {4 \sqrt {-a} \left (21 a^2 e^2 g^4+3 a c g^2 \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (3 a e g^2 (e f-10 d g)+c f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

4/315*(7*a*e^2*g^2-c*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/c/g^3+2/63*e*(-3*d*g+e*f
)*(g*x+f)^(5/2)*(c*x^2+a)^(1/2)/g^3-2/315*(6*a*e^2*g^2*(-10*d*g+e*f)-c*(-35*d^3*g^3+63*d^2*e*f*g^2-57*d*e^2*f^
2*g+19*e^3*f^3))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/e/g^3+2/9*(e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/e+4/315*(21
*a^2*e^2*g^4+3*a*c*g^2*(-21*d^2*g^2-16*d*e*f*g+3*e^2*f^2)+c^2*f^2*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*EllipticE
(1/2*(1-x*c^(1/2)/(-a)^(1/2))^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(g*x+f)^(1/
2)*(1+c*x^2/a)^(1/2)/c^(3/2)/g^4/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-4/315*(a*g^2
+c*f^2)*(3*a*e*g^2*(-10*d*g+e*f)+c*f*(21*d^2*g^2-24*d*e*f*g+8*e^2*f^2))*EllipticF(1/2*(1-x*c^(1/2)/(-a)^(1/2))
^(1/2)*2^(1/2),(-2*a*g/(-a*g+f*(-a)^(1/2)*c^(1/2)))^(1/2))*(-a)^(1/2)*(1+c*x^2/a)^(1/2)*((g*x+f)*c^(1/2)/(g*(-
a)^(1/2)+f*c^(1/2)))^(1/2)/c^(3/2)/g^4/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {933, 1668, 858, 733, 435, 430} \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (3 a e g^2 (e f-10 d g)+c f \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{315 c g^3}-\frac {2 \sqrt {a+c x^2} \sqrt {f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (-35 d^3 g^3+63 d^2 e f g^2-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{315 c e g^3}+\frac {2 e \sqrt {a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{63 g^3}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 e} \]

[In]

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

[Out]

(-2*(6*a*e^2*g^2*(e*f - 10*d*g) - c*(19*e^3*f^3 - 57*d*e^2*f^2*g + 63*d^2*e*f*g^2 - 35*d^3*g^3))*Sqrt[f + g*x]
*Sqrt[a + c*x^2])/(315*c*e*g^3) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*e) + (4*(7*a*e^2*g^2 - c*(8
*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^3) + (2*e*(e*f - 3*d*g)*(f + g*
x)^(5/2)*Sqrt[a + c*x^2])/(63*g^3) + (4*Sqrt[-a]*(21*a^2*e^2*g^4 + 3*a*c*g^2*(3*e^2*f^2 - 16*d*e*f*g - 21*d^2*
g^2) + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*Sqrt[f + g*x]*Sqrt[1 + (c*x^2)/a]*EllipticE[ArcSin[Sqrt[
1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(315*c^(3/2)*g^4*Sqrt[(Sqrt[c]*(f +
g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) - (4*Sqrt[-a]*(c*f^2 + a*g^2)*(3*a*e*g^2*(e*f - 10*d*g) + c*f
*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[1 + (c*x^2)/a]
*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(315*c^(3/2)*
g^4*Sqrt[f + g*x]*Sqrt[a + 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 733

Int[((d_) + (e_.)*(x_))^(m_)/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2*a*Rt[-c/a, 2]*(d + e*x)^m*(Sqrt[1
+ c*(x^2/a)]/(c*Sqrt[a + c*x^2]*(c*((d + e*x)/(c*d - a*e*Rt[-c/a, 2])))^m)), Subst[Int[(1 + 2*a*e*Rt[-c/a, 2]*
(x^2/(c*d - a*e*Rt[-c/a, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(1 - Rt[-c/a, 2]*x)/2]], x] /; FreeQ[{a, c, d, e},
x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m^2, 1/4]

Rule 858

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

Rule 933

Int[((d_.) + (e_.)*(x_))^(m_.)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*(d + e*
x)^(m + 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(e*(2*m + 5))), x] + Dist[1/(e*(2*m + 5)), Int[((d + e*x)^m/(Sqrt[f
+ g*x]*Sqrt[a + c*x^2]))*Simp[3*a*e*f - a*d*g - 2*(c*d*f - a*e*g)*x + (c*e*f - 3*c*d*g)*x^2, x], x], x] /; Fre
eQ[{a, c, d, e, f, g, m}, x] && NeQ[e*f - d*g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (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 + 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 + 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)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps \begin{align*} \text {integral}& = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {\int \frac {(d+e x)^2 \left (a (3 e f-d g)-2 (c d f-a e g) x+c (e f-3 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{9 e} \\ & = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {2 \int \frac {-\frac {1}{2} a c g^2 \left (5 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+7 d^3 g^3\right )-c f g \left (a e^2 g^2 (5 e f-36 d g)+c \left (e^3 f^3-3 d e^2 f^2 g+7 d^3 g^3\right )\right ) x+\frac {1}{2} c g^2 \left (4 a e^2 g^2 (4 e f+9 d g)-c \left (11 e^3 f^3-33 d e^2 f^2 g+21 d^2 e f g^2+21 d^3 g^3\right )\right ) x^2+c e g^3 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{63 c e g^4} \\ & = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {4 \int \frac {-\frac {1}{4} a c g^5 \left (42 a e^3 f g^2-c \left (23 e^3 f^3-69 d e^2 f^2 g+231 d^2 e f g^2-35 d^3 g^3\right )\right )-\frac {1}{2} c g^4 \left (21 a^2 e^3 g^4+3 a c e g^2 \left (5 e^2 f^2-36 d e f g-21 d^2 g^2\right )-c^2 f \left (11 e^3 f^3-33 d e^2 f^2 g+42 d^2 e f g^2-35 d^3 g^3\right )\right ) x-\frac {3}{4} c^2 g^5 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{315 c^2 e g^7} \\ & = -\frac {2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {8 \int \frac {-\frac {3}{2} a c^2 e g^7 \left (3 a e g^2 (3 e f+5 d g)-c f \left (e^2 f^2-3 d e f g+42 d^2 g^2\right )\right )-\frac {3}{4} c^2 e g^6 \left (21 a^2 e^2 g^4+3 a c g^2 \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{945 c^3 e g^9} \\ & = -\frac {2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {\left (2 \left (c f^2+a g^2\right ) \left (3 a e g^2 (e f-10 d g)+c f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{315 c g^4}-\frac {\left (2 \left (21 a^2 e^2 g^4+3 a c g^2 \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{315 c g^4} \\ & = -\frac {2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}-\frac {\left (4 a \left (21 a^2 e^2 g^4+3 a c g^2 \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}}{\sqrt {1-x^2}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{315 \sqrt {-a} c^{3/2} g^4 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (4 a \left (c f^2+a g^2\right ) \left (3 a e g^2 (e f-10 d g)+c f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {1+\frac {c x^2}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {2 a \sqrt {c} g x^2}{\sqrt {-a} \left (c f-\frac {a \sqrt {c} g}{\sqrt {-a}}\right )}}} \, dx,x,\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )}{315 \sqrt {-a} c^{3/2} g^4 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {2 \left (6 a e^2 g^2 (e f-10 d g)-c \left (19 e^3 f^3-57 d e^2 f^2 g+63 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c e g^3}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 e}+\frac {4 \left (7 a e^2 g^2-c \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^3}+\frac {2 e (e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^3}+\frac {4 \sqrt {-a} \left (21 a^2 e^2 g^4+3 a c g^2 \left (3 e^2 f^2-16 d e f g-21 d^2 g^2\right )+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (3 a e g^2 (e f-10 d g)+c f \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.89 (sec) , antiderivative size = 809, normalized size of antiderivative = 1.27 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (2 a e g^2 (4 e f+30 d g+7 e g x)+c \left (21 d^2 g^2 (f+3 g x)+6 d e g \left (-4 f^2+3 f g x+15 g^2 x^2\right )+e^2 \left (8 f^3-6 f^2 g x+5 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{c g^3}-\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (21 a^2 e^2 g^4+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 a c g^2 \left (-3 e^2 f^2+16 d e f g+21 d^2 g^2\right )\right ) \left (a+c x^2\right )-i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 a^2 e^2 g^4+c^2 f^2 \left (8 e^2 f^2-24 d e f g+21 d^2 g^2\right )-3 a c g^2 \left (-3 e^2 f^2+16 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 i a^{3/2} e^2 g^3-3 a \sqrt {c} e g^2 (e f-10 d g)+c^{3/2} f \left (-8 e^2 f^2+24 d e f g-21 d^2 g^2\right )-3 i \sqrt {a} c g \left (-2 e^2 f^2+6 d e f g+21 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c^2 g^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{315 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(Sqrt[f + g*x]*((2*(a + c*x^2)*(2*a*e*g^2*(4*e*f + 30*d*g + 7*e*g*x) + c*(21*d^2*g^2*(f + 3*g*x) + 6*d*e*g*(-4
*f^2 + 3*f*g*x + 15*g^2*x^2) + e^2*(8*f^3 - 6*f^2*g*x + 5*f*g^2*x^2 + 35*g^3*x^3))))/(c*g^3) - (4*(g^2*Sqrt[-f
- (I*Sqrt[a]*g)/Sqrt[c]]*(21*a^2*e^2*g^4 + c^2*f^2*(8*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*
f^2 + 16*d*e*f*g + 21*d^2*g^2))*(a + c*x^2) - I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(21*a^2*e^2*g^4 + c^2*f^2*(8
*e^2*f^2 - 24*d*e*f*g + 21*d^2*g^2) - 3*a*c*g^2*(-3*e^2*f^2 + 16*d*e*f*g + 21*d^2*g^2))*Sqrt[(g*((I*Sqrt[a])/S
qrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[S
qrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)] + Sqrt[a]
*Sqrt[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((21*I)*a^(3/2)*e^2*g^3 - 3*a*Sqrt[c]*e*g^2*(e*f - 10*d*g) + c^(3/2)*f*(-
8*e^2*f^2 + 24*d*e*f*g - 21*d^2*g^2) - (3*I)*Sqrt[a]*c*g*(-2*e^2*f^2 + 6*d*e*f*g + 21*d^2*g^2))*Sqrt[(g*((I*Sq
rt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*A
rcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sqrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))
/(c^2*g^5*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(315*Sqrt[a + c*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1141\) vs. \(2(551)=1102\).

Time = 1.40 (sec) , antiderivative size = 1142, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1142\)
risch	\(\text {Expression too large to display}\)	\(1677\)
default	\(\text {Expression too large to display}\)	\(4352\)

[In]

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

[Out]

((g*x+f)*(c*x^2+a))^(1/2)/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)*(2/9*e^2*x^3*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/7*(2*
c*d*e*g+1/9*c*e^2*f)/c/g*x^2*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c
*d*e*g+1/9*c*e^2*f))/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(2*a*d*e*g+1/3*a*e^2*f+c*d^2*f-4/5*f/g*(2/9*a
*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9*c*e^2*f))-5/7*a/c*(2*c*d*e*g+1/9*c*e^2*f))/c/g*(c*g*x^3+c*f*x^
2+a*g*x+a*f)^(1/2)+2*(a*d^2*f-2/5*f*a/c/g*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9*c*e^2*f))-1/3*
a/c*(2*a*d*e*g+1/3*a*e^2*f+c*d^2*f-4/5*f/g*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9*c*e^2*f))-5/7
*a/c*(2*c*d*e*g+1/9*c*e^2*f)))*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a*c)^(1/2)/c)/(
-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)
*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))+2*(a*d^2*
g+2*a*d*e*f-4/7*f*a/c/g*(2*c*d*e*g+1/9*c*e^2*f)-3/5*a/c*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9*
c*e^2*f))-2/3*f/g*(2*a*d*e*g+1/3*a*e^2*f+c*d^2*f-4/5*f/g*(2/9*a*e^2*g+c*d^2*g+2*c*d*e*f-6/7*f/g*(2*c*d*e*g+1/9
*c*e^2*f))-5/7*a/c*(2*c*d*e*g+1/9*c*e^2*f)))*(f/g-(-a*c)^(1/2)/c)*((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2)*((x-(-a
*c)^(1/2)/c)/(-f/g-(-a*c)^(1/2)/c))^(1/2)*((x+(-a*c)^(1/2)/c)/(-f/g+(-a*c)^(1/2)/c))^(1/2)/(c*g*x^3+c*f*x^2+a*
g*x+a*f)^(1/2)*((-f/g-(-a*c)^(1/2)/c)*EllipticE(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/c)/(-
f/g-(-a*c)^(1/2)/c))^(1/2))+(-a*c)^(1/2)/c*EllipticF(((x+f/g)/(f/g-(-a*c)^(1/2)/c))^(1/2),((-f/g+(-a*c)^(1/2)/
c)/(-f/g-(-a*c)^(1/2)/c))^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 510, normalized size of antiderivative = 0.80 \[ \int (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2} \, dx=\frac {2 \, {\left (2 \, {\left (8 \, c^{2} e^{2} f^{5} - 24 \, c^{2} d e f^{4} g - 66 \, a c d e f^{2} g^{3} - 90 \, a^{2} d e g^{5} + 3 \, {\left (7 \, c^{2} d^{2} + 5 \, a c e^{2}\right )} f^{3} g^{2} + 3 \, {\left (63 \, a c d^{2} - 11 \, a^{2} e^{2}\right )} f g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (8 \, c^{2} e^{2} f^{4} g - 24 \, c^{2} d e f^{3} g^{2} - 48 \, a c d e f g^{4} + 3 \, {\left (7 \, c^{2} d^{2} + 3 \, a c e^{2}\right )} f^{2} g^{3} - 21 \, {\left (3 \, a c d^{2} - a^{2} e^{2}\right )} g^{5}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (35 \, c^{2} e^{2} g^{5} x^{3} + 8 \, c^{2} e^{2} f^{3} g^{2} - 24 \, c^{2} d e f^{2} g^{3} + 60 \, a c d e g^{5} + {\left (21 \, c^{2} d^{2} + 8 \, a c e^{2}\right )} f g^{4} + 5 \, {\left (c^{2} e^{2} f g^{4} + 18 \, c^{2} d e g^{5}\right )} x^{2} - {\left (6 \, c^{2} e^{2} f^{2} g^{3} - 18 \, c^{2} d e f g^{4} - 7 \, {\left (9 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{945 \, c^{2} g^{5}} \]

[In]

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

[Out]

2/945*(2*(8*c^2*e^2*f^5 - 24*c^2*d*e*f^4*g - 66*a*c*d*e*f^2*g^3 - 90*a^2*d*e*g^5 + 3*(7*c^2*d^2 + 5*a*c*e^2)*f
^3*g^2 + 3*(63*a*c*d^2 - 11*a^2*e^2)*f*g^4)*sqrt(c*g)*weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27
*(c*f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g) + 6*(8*c^2*e^2*f^4*g - 24*c^2*d*e*f^3*g^2 - 48*a*c*d*e*f*g^4
+ 3*(7*c^2*d^2 + 3*a*c*e^2)*f^2*g^3 - 21*(3*a*c*d^2 - a^2*e^2)*g^5)*sqrt(c*g)*weierstrassZeta(4/3*(c*f^2 - 3*a
*g^2)/(c*g^2), -8/27*(c*f^3 + 9*a*f*g^2)/(c*g^3), weierstrassPInverse(4/3*(c*f^2 - 3*a*g^2)/(c*g^2), -8/27*(c*
f^3 + 9*a*f*g^2)/(c*g^3), 1/3*(3*g*x + f)/g)) + 3*(35*c^2*e^2*g^5*x^3 + 8*c^2*e^2*f^3*g^2 - 24*c^2*d*e*f^2*g^3
+ 60*a*c*d*e*g^5 + (21*c^2*d^2 + 8*a*c*e^2)*f*g^4 + 5*(c^2*e^2*f*g^4 + 18*c^2*d*e*g^5)*x^2 - (6*c^2*e^2*f^2*g
^3 - 18*c^2*d*e*f*g^4 - 7*(9*c^2*d^2 + 2*a*c*e^2)*g^5)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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