\(\int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx\) [629]

Optimal result

Integrand size = 28, antiderivative size = 666 \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}+\frac {4 \sqrt {-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

4/315*e*(7*a*e^2*g^2+c*(42*d^2*g^2-111*d*e*f*g+64*e^2*f^2))*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/c/g^4-4/63*e^2*(-3*d
*g+4*e*f)*(g*x+f)^(5/2)*(c*x^2+a)^(1/2)/g^4-4/315*(9*a*e^2*g^2*(-5*d*g+2*e*f)+c*(-35*d^3*g^3+168*d^2*e*f*g^2-2
04*d*e^2*f^2*g+76*e^3*f^3))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g^4+2/9*(e*x+d)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/g+
4/315*(21*a^2*e^3*g^4-3*a*c*e*g^2*(63*d^2*g^2-39*d*e*f*g+10*e^2*f^2)-c^2*f*(-105*d^3*g^3+252*d^2*e*f*g^2-216*d
*e^2*f^2*g+64*e^3*f^3))*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^5/(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)*(9*a*e^2*g^2*(-5*d*g+2*e*f)-c*(-105*d^3*g^3+252*d^2*e*f*g^2-216*d*e^2
*f^2*g+64*e^3*f^3))*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^5/(g*x+f)^(1/2)
/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 1.22 (sec) , antiderivative size = 666, 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 = {935, 1668, 858, 733, 435, 430} \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}-\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (-105 d^3 g^3+252 d^2 e f g^2-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 e \sqrt {a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac {4 \sqrt {a+c x^2} \sqrt {f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (-35 d^3 g^3+168 d^2 e f g^2-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac {4 e^2 \sqrt {a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac {2 \sqrt {a+c x^2} (d+e x)^3 \sqrt {f+g x}}{9 g} \]

[In]

Int[((d + e*x)^3*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]

[Out]

(-4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2*g + 168*d^2*e*f*g^2 - 35*d^3*g^3))*Sqrt[f + g
*x]*Sqrt[a + c*x^2])/(315*c*g^4) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*g) + (4*e*(7*a*e^2*g^2 + c
*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^4) - (4*e^2*(4*e*f - 3*d*g
)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^4) + (4*Sqrt[-a]*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f
*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*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^5*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)*(9*a*e^2*g^2*(2*e*f - 5*d*g) - c*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(Sqr
t[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^5*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 935

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

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 g}-\frac {\int \frac {(d+e x)^2 \left (2 a (3 e f-4 d g)+2 (c d f-a e g) x+2 c (4 e f-3 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{9 g} \\ & = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {2 \int \frac {-a c g^2 \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )-c g \left (a e g^2 \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+c \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )\right ) x+c g^2 \left (a e^2 g^2 (e f-27 d g)-c \left (44 e^3 f^3-33 d e^2 f^2 g-42 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 (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{63 c g^5} \\ & = \frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {4 \int \frac {\frac {1}{2} a c g^5 \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 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 (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+c^2 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right ) x+\frac {3}{2} c^2 g^5 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 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 g^8} \\ & = -\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {8 \int \frac {\frac {3}{4} a c^2 g^7 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )+\frac {3}{4} c^2 g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{945 c^3 g^{10}} \\ & = -\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}+\frac {\left (2 \left (c f^2+a g^2\right ) \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{315 c g^5}-\frac {\left (2 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{315 c g^5} \\ & = -\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}-\frac {\left (4 a \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\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^5 \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 (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\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^5 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{315 c g^4}+\frac {2 (d+e x)^3 \sqrt {f+g x} \sqrt {a+c x^2}}{9 g}+\frac {4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt {a+c x^2}}{315 c g^4}-\frac {4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt {a+c x^2}}{63 g^4}+\frac {4 \sqrt {-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\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^5 \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 (9 a e^2 g^2 (2 e f-5 d g)-c \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\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^5 \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 27.66 (sec) , antiderivative size = 872, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (2 a e^2 g^2 (-11 e f+45 d g+7 e g x)+c \left (105 d^3 g^3+63 d^2 e g^2 (-4 f+3 g x)+27 d e^2 g \left (8 f^2-6 f g x+5 g^2 x^2\right )+e^3 \left (-64 f^3+48 f^2 g x-40 f g^2 x^2+35 g^3 x^3\right )\right )\right )}{c g^4}+\frac {4 (f+g x) \left (\frac {g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (-21 a^2 e^3 g^4+3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )+c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \left (a+c x^2\right )}{(f+g x)^2}+\frac {i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )+c^2 f \left (-64 e^3 f^3+216 d e^2 f^2 g-252 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}+\frac {\sqrt {a} \sqrt {c} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (-21 i a^{3/2} e^3 g^3+9 a \sqrt {c} e^2 g^2 (2 e f-5 d g)+3 i \sqrt {a} c e g \left (16 e^2 f^2-54 d e f g+63 d^2 g^2\right )+c^{3/2} \left (-64 e^3 f^3+216 d e^2 f^2 g-252 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )}{\sqrt {f+g x}}\right )}{c^2 g^6 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}\right )}{315 \sqrt {a+c x^2}} \]

[In]

Integrate[((d + e*x)^3*Sqrt[a + c*x^2])/Sqrt[f + g*x],x]

[Out]

(Sqrt[f + g*x]*((2*(a + c*x^2)*(2*a*e^2*g^2*(-11*e*f + 45*d*g + 7*e*g*x) + c*(105*d^3*g^3 + 63*d^2*e*g^2*(-4*f
+ 3*g*x) + 27*d*e^2*g*(8*f^2 - 6*f*g*x + 5*g^2*x^2) + e^3*(-64*f^3 + 48*f^2*g*x - 40*f*g^2*x^2 + 35*g^3*x^3))
))/(c*g^4) + (4*(f + g*x)*((g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(-21*a^2*e^3*g^4 + 3*a*c*e*g^2*(10*e^2*f^2 -
39*d*e*f*g + 63*d^2*g^2) + c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*(a + c*x^2))/
(f + g*x)^2 + (I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63
*d^2*g^2) + c^2*f*(-64*e^3*f^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c
] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*EllipticE[I*ArcSinh[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)])/Sqrt[f + g*x] + (Sqrt[a]*Sqrt
[c]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*((-21*I)*a^(3/2)*e^3*g^3 + 9*a*Sqrt[c]*e^2*g^2*(2*e*f - 5*d*g) + (3*I)*Sqrt[a]
*c*e*g*(16*e^2*f^2 - 54*d*e*f*g + 63*d^2*g^2) + c^(3/2)*(-64*e^3*f^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105
*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*Elli
pticF[I*ArcSinh[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)])/Sqrt[f + g*x]))/(c^2*g^6*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]])))/(315*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 2.44 (sec) , antiderivative size = 1156, normalized size of antiderivative = 1.74

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1156\)
risch	\(\text {Expression too large to display}\)	\(1937\)
default	\(\text {Expression too large to display}\)	\(5079\)

[In]

int((e*x+d)^3*(c*x^2+a)^(1/2)/(g*x+f)^(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^3/g*x^3*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/7*(
3*c*d*e^2-8/9*e^3/g*c*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^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9
*e^3/g*c*f)/g*f)/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*a*d*e^2+c*d^3-2/3*e^3/g*f*a-5/7*(3*c*d*e^2-8/9
*e^3/g*c*f)/c*a-4/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/g*f)/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*
f)^(1/2)+2*(a*d^3-2/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/c/g*f*a-1/3*(3*a*d*e^2+c*d^3-2/3
*e^3/g*f*a-5/7*(3*c*d*e^2-8/9*e^3/g*c*f)/c*a-4/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/g*f)/
c*a)*(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*(3*a*d^2*e-4/7*(3*c*d*e^2-8/9*e^3
/g*c*f)/c/g*f*a-3/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/c*a-2/3*(3*a*d*e^2+c*d^3-2/3*e^3/g
*f*a-5/7*(3*c*d*e^2-8/9*e^3/g*c*f)/c*a-4/5*(2/9*a*e^3+3*c*d^2*e-6/7*(3*c*d*e^2-8/9*e^3/g*c*f)/g*f)/g*f)/g*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)*Elliptic
E(((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*Ell
ipticF(((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 = 0.18 (sec) , antiderivative size = 578, normalized size of antiderivative = 0.87 \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=-\frac {2 \, {\left (2 \, {\left (64 \, c^{2} e^{3} f^{5} - 216 \, c^{2} d e^{2} f^{4} g + 6 \, {\left (42 \, c^{2} d^{2} e + 13 \, a c e^{3}\right )} f^{3} g^{2} - 3 \, {\left (35 \, c^{2} d^{3} + 93 \, a c d e^{2}\right )} f^{2} g^{3} + 6 \, {\left (63 \, a c d^{2} e - 2 \, a^{2} e^{3}\right )} f g^{4} - 45 \, {\left (7 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} g^{5}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (64 \, c^{2} e^{3} f^{4} g - 216 \, c^{2} d e^{2} f^{3} g^{2} + 6 \, {\left (42 \, c^{2} d^{2} e + 5 \, a c e^{3}\right )} f^{2} g^{3} - 3 \, {\left (35 \, c^{2} d^{3} + 39 \, a c d e^{2}\right )} f g^{4} + 21 \, {\left (9 \, a c d^{2} e - a^{2} e^{3}\right )} g^{5}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (35 \, c^{2} e^{3} g^{5} x^{3} - 64 \, c^{2} e^{3} f^{3} g^{2} + 216 \, c^{2} d e^{2} f^{2} g^{3} - 2 \, {\left (126 \, c^{2} d^{2} e + 11 \, a c e^{3}\right )} f g^{4} + 15 \, {\left (7 \, c^{2} d^{3} + 6 \, a c d e^{2}\right )} g^{5} - 5 \, {\left (8 \, c^{2} e^{3} f g^{4} - 27 \, c^{2} d e^{2} g^{5}\right )} x^{2} + {\left (48 \, c^{2} e^{3} f^{2} g^{3} - 162 \, c^{2} d e^{2} f g^{4} + 7 \, {\left (27 \, c^{2} d^{2} e + 2 \, a c e^{3}\right )} g^{5}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{945 \, c^{2} g^{6}} \]

[In]

integrate((e*x+d)^3*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="fricas")

[Out]

-2/945*(2*(64*c^2*e^3*f^5 - 216*c^2*d*e^2*f^4*g + 6*(42*c^2*d^2*e + 13*a*c*e^3)*f^3*g^2 - 3*(35*c^2*d^3 + 93*a
*c*d*e^2)*f^2*g^3 + 6*(63*a*c*d^2*e - 2*a^2*e^3)*f*g^4 - 45*(7*a*c*d^3 - 3*a^2*d*e^2)*g^5)*sqrt(c*g)*weierstra
ssPInverse(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*(64*c^2*e^
3*f^4*g - 216*c^2*d*e^2*f^3*g^2 + 6*(42*c^2*d^2*e + 5*a*c*e^3)*f^2*g^3 - 3*(35*c^2*d^3 + 39*a*c*d*e^2)*f*g^4 +
21*(9*a*c*d^2*e - a^2*e^3)*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^3*g^5*x^3 - 64*c^2*e^3*f^3*g^2 + 216*c^2*d*e^2*f^2*g^3 - 2*(126*c^2*d^2*e + 11*a*c*e^
3)*f*g^4 + 15*(7*c^2*d^3 + 6*a*c*d*e^2)*g^5 - 5*(8*c^2*e^3*f*g^4 - 27*c^2*d*e^2*g^5)*x^2 + (48*c^2*e^3*f^2*g^3
- 162*c^2*d*e^2*f*g^4 + 7*(27*c^2*d^2*e + 2*a*c*e^3)*g^5)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^6)

Sympy [F]

\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {a + c x^{2}} \left (d + e x\right )^{3}}{\sqrt {f + g x}}\, dx \]

[In]

integrate((e*x+d)**3*(c*x**2+a)**(1/2)/(g*x+f)**(1/2),x)

[Out]

Integral(sqrt(a + c*x**2)*(d + e*x)**3/sqrt(f + g*x), x)

Maxima [F]

\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}} \,d x } \]

[In]

integrate((e*x+d)^3*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f), x)

Giac [F]

\[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int { \frac {\sqrt {c x^{2} + a} {\left (e x + d\right )}^{3}}{\sqrt {g x + f}} \,d x } \]

[In]

integrate((e*x+d)^3*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(c*x^2 + a)*(e*x + d)^3/sqrt(g*x + f), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+e x)^3 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^3}{\sqrt {f+g\,x}} \,d x \]

[In]

int(((a + c*x^2)^(1/2)*(d + e*x)^3)/(f + g*x)^(1/2),x)

[Out]

int(((a + c*x^2)^(1/2)*(d + e*x)^3)/(f + g*x)^(1/2), x)