\(\int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx\) [636]

Optimal result

Integrand size = 28, antiderivative size = 531 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}+\frac {2 \sqrt {-a} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

2/35*e^2*(11*d*g+e*f)*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/c/g^2-2/105*e*(25*a*e^2*g^2+c*(-90*d^2*g^2+12*d*e*f*g+7*e^
2*f^2))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c^2/g^2+2/7*e*(e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c+2/105*(a*e^2*g^2
*(189*d*g+19*e*f)-c*(105*d^3*g^3+105*d^2*e*f*g^2-42*d*e^2*f^2*g+8*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^3/(c*x^2+a)^(1/2)/((g*x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)-2/105*e*(a*g^2+c*f^2)*(25*a*e^2*g^2-c
*(105*d^2*g^2-42*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^(5/2)/
g^3/(g*x+f)^(1/2)/(c*x^2+a)^(1/2)

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {956, 1668, 858, 733, 435, 430} \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \sqrt {-a} e \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (25 a e^2 g^2-c \left (105 d^2 g^2-42 d e f g+8 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{5/2} g^3 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {2 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (a e^2 g^2 (189 d g+19 e f)-c \left (105 d^3 g^3+105 d^2 e f g^2-42 d e^2 f^2 g+8 e^3 f^3\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^3 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {2 e \sqrt {a+c x^2} \sqrt {f+g x} \left (-e^2 \left (\frac {25 a}{c}+\frac {7 f^2}{g^2}\right )+90 d^2-\frac {12 d e f}{g}\right )}{105 c}+\frac {2 e^2 \sqrt {a+c x^2} (f+g x)^{3/2} (11 d g+e f)}{35 c g^2}+\frac {2 e \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 c} \]

[In]

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

[Out]

(2*e*(90*d^2 - e^2*((25*a)/c + (7*f^2)/g^2) - (12*d*e*f)/g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(105*c) + (2*e*(d +
e*x)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*c) + (2*e^2*(e*f + 11*d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(35*c*g^
2) + (2*Sqrt[-a]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*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)])/(105*c^(3/2)*g^3*Sqrt[(Sqrt[c]*(f + g*x))/(Sqrt[c]*f + Sqrt[-a]*g)]*Sqrt[a + c*x^2]) -
(2*Sqrt[-a]*e*(c*f^2 + a*g^2)*(25*a*e^2*g^2 - c*(8*e^2*f^2 - 42*d*e*f*g + 105*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)])/(105*c^(5/2)*g^3*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 956

Int[(((d_.) + (e_.)*(x_))^(m_)*Sqrt[(f_.) + (g_.)*(x_)])/Sqrt[(a_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2*e*(d +
e*x)^(m - 1)*Sqrt[f + g*x]*(Sqrt[a + c*x^2]/(c*(2*m + 1))), x] - Dist[1/(c*(2*m + 1)), Int[((d + e*x)^(m - 2)
/(Sqrt[f + g*x]*Sqrt[a + c*x^2]))*Simp[a*e*(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)))*x - c*e*(e*f + d*g*(4*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, 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 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}-\frac {\int \frac {(d+e x) \left (-7 c d^2 f+a e (4 e f+d g)+\left (5 a e^2 g-c d (12 e f+7 d g)\right ) x-c e (e f+11 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{7 c} \\ & = \frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {2 \int \frac {-\frac {1}{2} c g^2 \left (35 c d^3 f g-a e \left (3 e^2 f^2+53 d e f g+5 d^2 g^2\right )\right )+\frac {1}{2} c g \left (a e^2 g^2 (23 e f+63 d g)+c \left (2 e^3 f^3+22 d e^2 f^2 g-95 d^2 e f g^2-35 d^3 g^3\right )\right ) x+\frac {1}{2} c e g^2 \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{35 c^2 g^3} \\ & = -\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {4 \int \frac {-\frac {1}{4} c g^4 \left (105 c^2 d^3 f g+25 a^2 e^3 g^2-a c e \left (2 e^2 f^2+147 d e f g+105 d^2 g^2\right )\right )+\frac {1}{4} c^2 g^3 \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c^3 g^5} \\ & = -\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}+\frac {\left (e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e f g+105 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c^2 g^3}-\frac {\left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{105 c g^3} \\ & = -\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}-\frac {\left (2 a \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 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 )}{105 \sqrt {-a} c^{3/2} g^3 \sqrt {\frac {c (f+g x)}{c f-\frac {a \sqrt {c} g}{\sqrt {-a}}}} \sqrt {a+c x^2}}+\frac {\left (2 a e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e f g+105 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 )}{105 \sqrt {-a} c^{5/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = -\frac {2 e \left (25 a e^2 g^2+c \left (7 e^2 f^2+12 d e f g-90 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c^2 g^2}+\frac {2 e (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 c}+\frac {2 e^2 (e f+11 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 c g^2}+\frac {2 \sqrt {-a} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 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 )}{105 c^{3/2} g^3 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}-\frac {2 \sqrt {-a} e \left (c f^2+a g^2\right ) \left (25 a e^2 g^2-c \left (8 e^2 f^2-42 d e f g+105 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 )}{105 c^{5/2} g^3 \sqrt {f+g x} \sqrt {a+c x^2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 25.52 (sec) , antiderivative size = 777, normalized size of antiderivative = 1.46 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (-25 a e^3 g^2+c e \left (105 d^2 g^2+21 d e g (f+3 g x)+e^2 \left (-4 f^2+3 f g x+15 g^2 x^2\right )\right )\right )}{c^2 g^2}-\frac {2 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a e^2 g^2 (19 e f+189 d g)-c \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \left (a+c x^2\right )-\sqrt {c} \left (i a \sqrt {c} e^2 f g^2 (19 e f+189 d g)-a^{3/2} e^2 g^3 (19 e f+189 d g)-i c^{3/2} f \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )+\sqrt {a} c g \left (8 e^3 f^3-42 d e^2 f^2 g+105 d^2 e f g^2+105 d^3 g^3\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )-g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (105 i c^{3/2} d^3 g^2+25 a^{3/2} e^3 g^2+3 i a \sqrt {c} e^2 g (2 e f-63 d g)+\sqrt {a} c e \left (-8 e^2 f^2+42 d e f g-105 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c^2 g^4 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{105 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(Sqrt[f + g*x]*((2*(a + c*x^2)*(-25*a*e^3*g^2 + c*e*(105*d^2*g^2 + 21*d*e*g*(f + 3*g*x) + e^2*(-4*f^2 + 3*f*g*
x + 15*g^2*x^2))))/(c^2*g^2) - (2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a*e^2*g^2*(19*e*f + 189*d*g) - c*(8*e
^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 + 105*d^3*g^3))*(a + c*x^2) - Sqrt[c]*(I*a*Sqrt[c]*e^2*f*g^2*(19*e*f
+ 189*d*g) - a^(3/2)*e^2*g^3*(19*e*f + 189*d*g) - I*c^(3/2)*f*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*d^2*e*f*g^2 +
105*d^3*g^3) + Sqrt[a]*c*g*(8*e^3*f^3 - 42*d*e^2*f^2*g + 105*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))]*(f + g*x)^(3/2)*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)] - g*(Sq
rt[c]*f + I*Sqrt[a]*g)*((105*I)*c^(3/2)*d^3*g^2 + 25*a^(3/2)*e^3*g^2 + (3*I)*a*Sqrt[c]*e^2*g*(2*e*f - 63*d*g)
+ Sqrt[a]*c*e*(-8*e^2*f^2 + 42*d*e*f*g - 105*d^2*g^2))*Sqrt[(g*((I*Sqrt[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*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/S
qrt[f + g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(c^2*g^4*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]
]*(f + g*x))))/(105*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 3.37 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.66

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{3} x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{7 c}+\frac {2 \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right ) x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5 c g}+\frac {2 \left (3 d^{2} e g +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 g}-\frac {5 g a \,e^{3}}{7 c}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (d^{3} f -\frac {2 a f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 c g}-\frac {a \left (3 d^{2} e g +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 g}-\frac {5 g a \,e^{3}}{7 c}\right )}{3 c}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}+\frac {2 \left (d^{3} g +3 d^{2} e f -\frac {4 a f \,e^{3}}{7 c}-\frac {3 a \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 c}-\frac {2 f \left (3 d^{2} e g +3 d \,e^{2} f -\frac {4 f \left (3 d \,e^{2} g +\frac {1}{7} e^{3} f \right )}{5 g}-\frac {5 g a \,e^{3}}{7 c}\right )}{3 g}\right ) \left (\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}}\, \left (\left (-\frac {f}{g}-\frac {\sqrt {-a c}}{c}\right ) E\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )+\frac {\sqrt {-a c}\, F\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}, \sqrt {\frac {-\frac {f}{g}+\frac {\sqrt {-a c}}{c}}{-\frac {f}{g}-\frac {\sqrt {-a c}}{c}}}\right )}{c}\right )}{\sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+a}}\)	\(882\)
risch	\(\text {Expression too large to display}\)	\(1610\)
default	\(\text {Expression too large to display}\)	\(3924\)

[In]

int((e*x+d)^3*(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)*(2/7*e^3/c*x^2*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(
3*d*e^2*g+1/7*e^3*f)/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(3*d^2*e*g+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*e
^3*f)-5/7/c*g*a*e^3)/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(d^3*f-2/5*a/c*f/g*(3*d*e^2*g+1/7*e^3*f)-1/3/c*a*
(3*d^2*e*g+3*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*e^3*f)-5/7/c*g*a*e^3))*(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*(d^3*g+3*d^2*e*f-4/7*a/c*f*e^3-3/5/c*a*(3*d*e^2*g+1/7*e^3*f)-2/3*f/g*(3*d^2*e*g+3
*d*e^2*f-4/5*f/g*(3*d*e^2*g+1/7*e^3*f)-5/7/c*g*a*e^3))*(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 = 0.11 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3 \sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx=-\frac {2 \, {\left ({\left (8 \, c^{2} e^{3} f^{4} - 42 \, c^{2} d e^{2} f^{3} g + {\left (105 \, c^{2} d^{2} e - 13 \, a c e^{3}\right )} f^{2} g^{2} - 42 \, {\left (5 \, c^{2} d^{3} - 6 \, a c d e^{2}\right )} f g^{3} + 15 \, {\left (21 \, a c d^{2} e - 5 \, a^{2} e^{3}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 3 \, {\left (8 \, c^{2} e^{3} f^{3} g - 42 \, c^{2} d e^{2} f^{2} g^{2} + {\left (105 \, c^{2} d^{2} e - 19 \, a c e^{3}\right )} f g^{3} + 21 \, {\left (5 \, c^{2} d^{3} - 9 \, a c d e^{2}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) - 3 \, {\left (15 \, c^{2} e^{3} g^{4} x^{2} - 4 \, c^{2} e^{3} f^{2} g^{2} + 21 \, c^{2} d e^{2} f g^{3} + 5 \, {\left (21 \, c^{2} d^{2} e - 5 \, a c e^{3}\right )} g^{4} + 3 \, {\left (c^{2} e^{3} f g^{3} + 21 \, c^{2} d e^{2} g^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{3} g^{4}} \]

[In]

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

[Out]

-2/315*((8*c^2*e^3*f^4 - 42*c^2*d*e^2*f^3*g + (105*c^2*d^2*e - 13*a*c*e^3)*f^2*g^2 - 42*(5*c^2*d^3 - 6*a*c*d*e
^2)*f*g^3 + 15*(21*a*c*d^2*e - 5*a^2*e^3)*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) + 3*(8*c^2*e^3*f^3*g - 42*c^2*d*e^2*f^2*g^2 + (105*c^2*d^2
*e - 19*a*c*e^3)*f*g^3 + 21*(5*c^2*d^3 - 9*a*c*d*e^2)*g^4)*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*(15*c^2*e^3*g^4*x^2 - 4*c^2*e^3*f^2*g^2 + 21*c^2*d*e^2*f*g^3 + 5*(21*
c^2*d^2*e - 5*a*c*e^3)*g^4 + 3*(c^2*e^3*f*g^3 + 21*c^2*d*e^2*g^4)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^3*g^4)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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