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

Optimal. Leaf size=508 \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (5 a e^2 g^2-c \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (5 a e^2 g^2+c \left (10 d^2 g^2-34 d e f g+21 e^2 f^2\right )\right )}{105 c g^3}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 \sqrt{c} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{35 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^2 \sqrt{f+g x}}{7 g} \]

[Out]

(4*(5*a*e^2*g^2 + c*(21*e^2*f^2 - 34*d*e*f*g + 10*d^2*g^2))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(105*c*g^3) + (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*Sqrt[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])

________________________________________________________________________________________

Rubi [A]  time = 0.977467, antiderivative size = 503, normalized size of antiderivative = 0.99, number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {921, 1654, 844, 719, 424, 419} \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (5 a e^2 g^2-c \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (e^2 \left (\frac{5 a}{c}+\frac{21 f^2}{g^2}\right )+10 d^2-\frac{34 d e f}{g}\right )}{105 g}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{105 \sqrt{c} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} (3 e f-2 d g)}{35 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^2 \sqrt{f+g x}}{7 g} \]

Antiderivative was successfully verified.

[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 921

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)*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])/(Sqrt[f + g*x]*Sqrt[a
+ c*x^2]), 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 1654

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]))

Rule 844

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 719

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 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rule 419

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(1*EllipticF[ArcSin[Rt[-(d/c),
2]*x], (b*c)/(a*d)])/(Sqrt[a]*Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] &
& GtQ[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-(b/a), -(d/c)])

Rubi steps

\begin{align*} \int \frac{(d+e x)^2 \sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx &=\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 (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{\left (4 \left (-\frac{1}{2} c^2 f 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 )+\frac{1}{2} a c g^5 \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 )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{105 c^2 g^7}\\ &=\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 ) \operatorname{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 (8 a \left (-\frac{1}{2} c^2 f 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 )+\frac{1}{2} a c g^5 \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 )\right ) \sqrt{\frac{c (f+g x)}{c f-\frac{a \sqrt{c} g}{\sqrt{-a}}}} \sqrt{1+\frac{c x^2}{a}}\right ) \operatorname{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^7 \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 (24 c e^2 f^2-56 c d e f g+35 c d^2 g^2-5 a e^2 g^2\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]  time = 5.35991, size = 712, normalized size = 1.4 \[ \frac{2 \sqrt{f+g x} \left (g^2 \left (a+c x^2\right ) \left (10 a e^2 g^2+c \left (35 d^2 g^2+14 d e g (3 g x-4 f)+3 e^2 \left (8 f^2-6 f g x+5 g^2 x^2\right )\right )\right )-\frac{2 \left (\sqrt{a} g (f+g x)^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (6 i \sqrt{a} \sqrt{c} e g (3 e f-7 d g)+5 a e^2 g^2+c \left (-35 d^2 g^2+56 d e f g-24 e^2 f^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )+g^2 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (a^2 e g^2 (13 e f-42 d g)+a c \left (35 d^2 f g^2-14 d e g \left (4 f^2+3 g^2 x^2\right )+e^2 \left (24 f^3+13 f g^2 x^2\right )\right )+c^2 f x^2 \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right )-i \sqrt{c} (f+g x)^{3/2} \left (\sqrt{c} f+i \sqrt{a} g\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{-g x+\frac{i \sqrt{a} g}{\sqrt{c}}}{f+g x}} \left (a e g^2 (13 e f-42 d g)+c f \left (35 d^2 g^2-56 d e f g+24 e^2 f^2\right )\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{(f+g x) \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right )}{105 c g^5 \sqrt{a+c x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(g^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))) - (2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(a^2*e*g^2*(13*e*f - 42*d*g) + c^2*f*(24*e^2*f^
2 - 56*d*e*f*g + 35*d^2*g^2)*x^2 + a*c*(35*d^2*f*g^2 - 14*d*e*g*(4*f^2 + 3*g^2*x^2) + e^2*(24*f^3 + 13*f*g^2*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])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqr
t[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]]/Sqrt[f
+ g*x]], (Sqrt[c]*f - I*Sqrt[a]*g)/(Sqrt[c]*f + I*Sqrt[a]*g)]))/(Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x)))
)/(105*c*g^5*Sqrt[a + c*x^2])

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Maple [B]  time = 0.259, size = 3278, normalized size = 6.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(c*x^2+a)^(1/2)*(g*x+f)^(1/2)*(-84*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c
)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-
c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a^2*c*d*e*g^5+26*(-(g*x+f)*c/((-a*c)^(1/2)*g-c
*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2
)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a^2*c*
e^2*f*g^4+70*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+
(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*
g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c^2*d^2*f*g^4+74*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(
1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((
-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c^2*e^2*f^3*g^2-112*(-(g*x+f)*
c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)
^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+
c*f))^(1/2))*c^3*d*e*f^4*g+84*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c
*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)
,(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a^2*c*d*e*g^5-36*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*
((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF
((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a^2*c*e^2*f*g^4-3
6*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2
))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a
*c)^(1/2)*g+c*f))^(1/2))*a*c^2*e^2*f^3*g^2-70*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/(
(-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/
2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*a*c*d^2*g^5+112*(-(g*x+f)*c/
((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(
1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*
f))^(1/2))*(-a*c)^(1/2)*a*c*d*e*f*g^4+70*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c
)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-
c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*c^3*d^2*f^3*g^2-56*a*c^2*d*e*f^2*g^3+48*(-(g*x
+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-
a*c)^(1/2)*g-c*f))^(1/2)*EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2
)*g+c*f))^(1/2))*c^3*e^2*f^5-14*x*a*c^2*d*e*f*g^4+35*a*c^2*d^2*f*g^4+24*a*c^2*e^2*f^3*g^2+42*x^4*c^3*d*e*g^5-3
*x^4*c^3*e^2*f*g^4+25*x^3*a*c^2*e^2*g^5+6*x^3*c^3*e^2*f^2*g^3+35*x^2*c^3*d^2*f*g^4+24*x^2*c^3*e^2*f^3*g^2+10*x
*a^2*c*e^2*g^5+35*x*a*c^2*d^2*g^5+15*x^5*c^3*e^2*g^5+35*x^3*c^3*d^2*g^5+10*a^2*c*e^2*f*g^4+10*(-(g*x+f)*c/((-a
*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)
*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^
(1/2))*(-a*c)^(1/2)*a^2*e^2*g^5-70*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2
)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^
(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c^2*d^2*f^2*g^3-48*(-(g*x+f)*c/((-a*c)^
(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c
*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2
))*(-a*c)^(1/2)*c^2*e^2*f^4*g+84*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*
g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1
/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c^2*d*e*f^2*g^3-38*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))
^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*El
lipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/
2)*a*c*e^2*f^2*g^3+112*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1
/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*EllipticF((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a
*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*(-a*c)^(1/2)*c^2*d*e*f^3*g^2-196*(-(g*x+f)*c/((-a*c)^(1/2)*g-c*f
))^(1/2)*((-c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g+c*f))^(1/2)*((c*x+(-a*c)^(1/2))*g/((-a*c)^(1/2)*g-c*f))^(1/2)*
EllipticE((-(g*x+f)*c/((-a*c)^(1/2)*g-c*f))^(1/2),(-((-a*c)^(1/2)*g-c*f)/((-a*c)^(1/2)*g+c*f))^(1/2))*a*c^2*d*
e*f^2*g^3-14*x^3*c^3*d*e*f*g^4+42*x^2*a*c^2*d*e*g^5+7*x^2*a*c^2*e^2*f*g^4-56*x^2*c^3*d*e*f^2*g^3+6*x*a*c^2*e^2
*f^2*g^3)/g^5/(c*g*x^3+c*f*x^2+a*g*x+a*f)/c^2

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )} \sqrt{c x^{2} + a}}{\sqrt{g x + f}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)