\(\int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx\) [630]

Optimal result

Integrand size = 28, antiderivative size = 508 \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {4 \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {4 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 g^3}+\frac {4 \sqrt {-a} \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 \sqrt {c} g^4 \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {a+c x^2}}+\frac {4 \sqrt {-a} \left (c f^2+a g^2\right ) \left (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {c} f+\sqrt {-a} g}} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^4 \sqrt {f+g x} \sqrt {a+c x^2}} \]

[Out]

-4/35*e*(-2*d*g+3*e*f)*(g*x+f)^(3/2)*(c*x^2+a)^(1/2)/g^3+4/105*(5*a*e^2*g^2+c*(10*d^2*g^2-34*d*e*f*g+21*e^2*f^
2))*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/c/g^3+2/7*(e*x+d)^2*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)/g+4/105*(a*e*g^2*(-42*d*g+
13*e*f)+c*f*(35*d^2*g^2-56*d*e*f*g+24*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)/g^4/c^(1/2)/(c*x^2+a)^(1/2)/((g*
x+f)*c^(1/2)/(g*(-a)^(1/2)+f*c^(1/2)))^(1/2)+4/105*(a*g^2+c*f^2)*(5*a*e^2*g^2-c*(35*d^2*g^2-56*d*e*f*g+24*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 = 0.86 (sec) , antiderivative size = 503, 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 = {935, 1668, 858, 733, 435, 430} \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}} \left (5 a e^2 g^2-c \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right ),-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 c^{3/2} g^4 \sqrt {a+c x^2} \sqrt {f+g x}}+\frac {4 \sqrt {-a} \sqrt {\frac {c x^2}{a}+1} \sqrt {f+g x} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {-a}}}}{\sqrt {2}}\right )|-\frac {2 a g}{\sqrt {-a} \sqrt {c} f-a g}\right )}{105 \sqrt {c} g^4 \sqrt {a+c x^2} \sqrt {\frac {\sqrt {c} (f+g x)}{\sqrt {-a} g+\sqrt {c} f}}}+\frac {4 \sqrt {a+c x^2} \sqrt {f+g x} \left (e^2 \left (\frac {5 a}{c}+\frac {21 f^2}{g^2}\right )+10 d^2-\frac {34 d e f}{g}\right )}{105 g}-\frac {4 e \sqrt {a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{35 g^3}+\frac {2 \sqrt {a+c x^2} (d+e x)^2 \sqrt {f+g x}}{7 g} \]

[In]

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

[Out]

(4*(10*d^2 + e^2*((5*a)/c + (21*f^2)/g^2) - (34*d*e*f)/g)*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(105*g) + (2*(d + e*x
)^2*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(7*g) - (4*e*(3*e*f - 2*d*g)*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(35*g^3) + (4
*Sqrt[-a]*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*f^2 - 56*d*e*f*g + 35*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)])/(105*Sq
rt[c]*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)*(5
*a*e^2*g^2 - c*(24*e^2*f^2 - 56*d*e*f*g + 35*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^(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 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)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {\int \frac {(d+e x) \left (2 a (2 e f-3 d g)+2 (c d f-a e g) x+2 c (3 e f-2 d g) x^2\right )}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{7 g} \\ & = \frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {4 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 g^3}-\frac {2 \int \frac {-a c g^2 \left (9 e^2 f^2-16 d e f g+15 d^2 g^2\right )+c g \left (a e g^2 (e f-14 d g)-c f \left (6 e^2 f^2-4 d e f g-5 d^2 g^2\right )\right ) x-c g^2 \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) x^2}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{35 c g^4} \\ & = \frac {4 \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {4 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 g^3}-\frac {4 \int \frac {\frac {1}{2} a c g^4 \left (5 a e^2 g^2-c \left (6 e^2 f^2-14 d e f g+35 d^2 g^2\right )\right )+\frac {1}{2} c^2 g^3 \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) x}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c^2 g^6} \\ & = \frac {4 \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {4 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 g^3}-\frac {\left (2 \left (c f^2+a g^2\right ) \left (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {1}{\sqrt {f+g x} \sqrt {a+c x^2}} \, dx}{105 c g^4}-\frac {\left (2 \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right )\right ) \int \frac {\sqrt {f+g x}}{\sqrt {a+c x^2}} \, dx}{105 g^4} \\ & = \frac {4 \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {4 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 g^3}-\frac {\left (4 a \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 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 )}{105 \sqrt {-a} \sqrt {c} 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 (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e f g+35 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^{3/2} g^4 \sqrt {f+g x} \sqrt {a+c x^2}} \\ & = \frac {4 \left (5 a e^2 g^2+c \left (21 e^2 f^2-34 d e f g+10 d^2 g^2\right )\right ) \sqrt {f+g x} \sqrt {a+c x^2}}{105 c g^3}+\frac {2 (d+e x)^2 \sqrt {f+g x} \sqrt {a+c x^2}}{7 g}-\frac {4 e (3 e f-2 d g) (f+g x)^{3/2} \sqrt {a+c x^2}}{35 g^3}+\frac {4 \sqrt {-a} \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 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 )}{105 \sqrt {c} 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 (5 a e^2 g^2-c \left (24 e^2 f^2-56 d e f g+35 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^{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 = 25.16 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.31 \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {\sqrt {f+g x} \left (\frac {2 \left (a+c x^2\right ) \left (10 a e^2 g^2+c \left (35 d^2 g^2+14 d e g (-4 f+3 g x)+3 e^2 \left (8 f^2-6 f g x+5 g^2 x^2\right )\right )\right )}{c g^3}-\frac {4 \left (g^2 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \left (a+c x^2\right )-i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (a e g^2 (13 e f-42 d g)+c f \left (24 e^2 f^2-56 d e f g+35 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right )|\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )+\sqrt {a} g \left (\sqrt {c} f+i \sqrt {a} g\right ) \left (5 a e^2 g^2+6 i \sqrt {a} \sqrt {c} e g (3 e f-7 d g)+c \left (-24 e^2 f^2+56 d e f g-35 d^2 g^2\right )\right ) \sqrt {\frac {g \left (\frac {i \sqrt {a}}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i \sqrt {a} g}{\sqrt {c}}-g x}{f+g x}} (f+g x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i \sqrt {a} g}{\sqrt {c} f+i \sqrt {a} g}\right )\right )}{c g^5 \sqrt {-f-\frac {i \sqrt {a} g}{\sqrt {c}}} (f+g x)}\right )}{105 \sqrt {a+c x^2}} \]

[In]

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

[Out]

(Sqrt[f + g*x]*((2*(a + c*x^2)*(10*a*e^2*g^2 + c*(35*d^2*g^2 + 14*d*e*g*(-4*f + 3*g*x) + 3*e^2*(8*f^2 - 6*f*g*
x + 5*g^2*x^2))))/(c*g^3) - (4*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a*e*g^2*(13*e*f - 42*d*g) + c*f*(24*e^2*
f^2 - 56*d*e*f*g + 35*d^2*g^2))*(a + c*x^2) - I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)*(a*e*g^2*(13*e*f - 42*d*g) +
c*f*(24*e^2*f^2 - 56*d*e*f*g + 35*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)*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[a]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*(5*a*e^2*g^2 +
(6*I)*Sqrt[a]*Sqrt[c]*e*g*(3*e*f - 7*d*g) + c*(-24*e^2*f^2 + 56*d*e*f*g - 35*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)*EllipticF[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)]))/(c*g^5*
Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(105*Sqrt[a + c*x^2])

Maple [A] (verified)

Time = 2.28 (sec) , antiderivative size = 848, normalized size of antiderivative = 1.67

method	result	size
elliptic	\(\frac {\sqrt {\left (g x +f \right ) \left (c \,x^{2}+a \right )}\, \left (\frac {2 e^{2} x^{2} \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{7 g}+\frac {2 \left (2 c d e -\frac {6 e^{2} c f}{7 g}\right ) x \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{5 c g}+\frac {2 \left (\frac {2 e^{2} a}{7}+c \,d^{2}-\frac {4 \left (2 c d e -\frac {6 e^{2} c f}{7 g}\right ) f}{5 g}\right ) \sqrt {c g \,x^{3}+c f \,x^{2}+a g x +f a}}{3 c g}+\frac {2 \left (a \,d^{2}-\frac {2 \left (2 c d e -\frac {6 e^{2} c f}{7 g}\right ) f a}{5 c g}-\frac {\left (\frac {2 e^{2} a}{7}+c \,d^{2}-\frac {4 \left (2 c d e -\frac {6 e^{2} c f}{7 g}\right ) f}{5 g}\right ) a}{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 (2 a d e -\frac {4 e^{2} f a}{7 g}-\frac {3 \left (2 c d e -\frac {6 e^{2} c f}{7 g}\right ) a}{5 c}-\frac {2 \left (\frac {2 e^{2} a}{7}+c \,d^{2}-\frac {4 \left (2 c d e -\frac {6 e^{2} c f}{7 g}\right ) f}{5 g}\right ) f}{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}}\)	\(848\)
risch	\(\text {Expression too large to display}\)	\(1370\)
default	\(\text {Expression too large to display}\)	\(3278\)

[In]

int((e*x+d)^2*(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)*(2/7*e^2/g*x^2*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/5*(
2*c*d*e-6/7*e^2/g*c*f)/c/g*x*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2/3*(2/7*e^2*a+c*d^2-4/5*(2*c*d*e-6/7*e^2/g*c*f
)/g*f)/c/g*(c*g*x^3+c*f*x^2+a*g*x+a*f)^(1/2)+2*(a*d^2-2/5*(2*c*d*e-6/7*e^2/g*c*f)/c/g*f*a-1/3*(2/7*e^2*a+c*d^2
-4/5*(2*c*d*e-6/7*e^2/g*c*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
*(2*a*d*e-4/7*e^2/g*f*a-3/5*(2*c*d*e-6/7*e^2/g*c*f)/c*a-2/3*(2/7*e^2*a+c*d^2-4/5*(2*c*d*e-6/7*e^2/g*c*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)*El
lipticE(((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 = 409, normalized size of antiderivative = 0.81 \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\frac {2 \, {\left (2 \, {\left (24 \, c^{2} e^{2} f^{4} - 56 \, c^{2} d e f^{3} g - 84 \, a c d e f g^{3} + {\left (35 \, c^{2} d^{2} + 31 \, a c e^{2}\right )} f^{2} g^{2} + 15 \, {\left (7 \, a c d^{2} - a^{2} e^{2}\right )} g^{4}\right )} \sqrt {c g} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right ) + 6 \, {\left (24 \, c^{2} e^{2} f^{3} g - 56 \, c^{2} d e f^{2} g^{2} - 42 \, a c d e g^{4} + {\left (35 \, c^{2} d^{2} + 13 \, a c e^{2}\right )} f g^{3}\right )} \sqrt {c g} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c f^{2} - 3 \, a g^{2}\right )}}{3 \, c g^{2}}, -\frac {8 \, {\left (c f^{3} + 9 \, a f g^{2}\right )}}{27 \, c g^{3}}, \frac {3 \, g x + f}{3 \, g}\right )\right ) + 3 \, {\left (15 \, c^{2} e^{2} g^{4} x^{2} + 24 \, c^{2} e^{2} f^{2} g^{2} - 56 \, c^{2} d e f g^{3} + 5 \, {\left (7 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} g^{4} - 6 \, {\left (3 \, c^{2} e^{2} f g^{3} - 7 \, c^{2} d e g^{4}\right )} x\right )} \sqrt {c x^{2} + a} \sqrt {g x + f}\right )}}{315 \, c^{2} g^{5}} \]

[In]

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

[Out]

2/315*(2*(24*c^2*e^2*f^4 - 56*c^2*d*e*f^3*g - 84*a*c*d*e*f*g^3 + (35*c^2*d^2 + 31*a*c*e^2)*f^2*g^2 + 15*(7*a*c
*d^2 - a^2*e^2)*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*(24*c^2*e^2*f^3*g - 56*c^2*d*e*f^2*g^2 - 42*a*c*d*e*g^4 + (35*c^2*d^2 + 13*a*c*e
^2)*f*g^3)*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), weierst
rassPInverse(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^2*g^4*x^2 + 24*c^2*e^2*f^2*g^2 - 56*c^2*d*e*f*g^3 + 5*(7*c^2*d^2 + 2*a*c*e^2)*g^4 - 6*(3*c^2*e^2*f*g^3 - 7*
c^2*d*e*g^4)*x)*sqrt(c*x^2 + a)*sqrt(g*x + f))/(c^2*g^5)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate((e*x+d)^2*(c*x^2+a)^(1/2)/(g*x+f)^(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. \[ \int \frac {(d+e x)^2 \sqrt {a+c x^2}}{\sqrt {f+g x}} \, dx=\int \frac {\sqrt {c\,x^2+a}\,{\left (d+e\,x\right )}^2}{\sqrt {f+g\,x}} \,d x \]

[In]

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

[Out]

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