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

Optimal. Leaf size=635 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (3 a e g^2 (e f-10 d g)+c f \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{4 \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{315 c g^3}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (63 d^2 e f g^2-35 d^3 g^3-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{315 c e g^3}+\frac{2 e \sqrt{a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{63 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 e} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 1.62228, antiderivative size = 635, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {919, 1654, 844, 719, 424, 419} \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^2 g^4+3 a c g^2 \left (-21 d^2 g^2-16 d e f g+3 e^2 f^2\right )+c^2 f^2 \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (3 a e g^2 (e f-10 d g)+c f \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^4 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2-c \left (21 d^2 g^2-24 d e f g+8 e^2 f^2\right )\right )}{315 c g^3}-\frac{2 \sqrt{a+c x^2} \sqrt{f+g x} \left (6 a e^2 g^2 (e f-10 d g)-c \left (63 d^2 e f g^2-35 d^3 g^3-57 d e^2 f^2 g+19 e^3 f^3\right )\right )}{315 c e g^3}+\frac{2 e \sqrt{a+c x^2} (f+g x)^{5/2} (e f-3 d g)}{63 g^3}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 e} \]

Antiderivative was successfully verified.

[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 919

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*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])/(Sqrt[f + g*x]*Sqrt[a + c*x^2]), 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 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 (d+e x)^2 \sqrt{f+g x} \sqrt{a+c x^2} \, dx &=\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 (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{\left (8 \left (\frac{3}{4} c^2 e f 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 )-\frac{3}{2} a c^2 e g^8 \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 )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{945 c^3 e g^{10}}\\ &=-\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 ) \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 )}{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 (16 a \left (\frac{3}{4} c^2 e f 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 )-\frac{3}{2} a c^2 e g^8 \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 )\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 )}{945 \sqrt{-a} c^{7/2} e g^{10} \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 (8 c e^2 f^3-24 c d e f^2 g+21 c d^2 f g^2+3 a e^2 f g^2-30 a d e g^3\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]  time = 7.41708, size = 809, normalized size = 1.27 \[ \frac{\sqrt{f+g x} \left (\frac{2 \left (c x^2+a\right ) \left (2 a e (4 e f+30 d g+7 e g x) g^2+c \left (\left (8 f^3-6 g x f^2+5 g^2 x^2 f+35 g^3 x^3\right ) e^2+6 d g \left (-4 f^2+3 g x f+15 g^2 x^2\right ) e+21 d^2 g^2 (f+3 g x)\right )\right )}{c g^3}-\frac{4 \left (\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (21 a^2 e^2 g^4-3 a c \left (-3 e^2 f^2+16 d e g f+21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) \left (c x^2+a\right ) g^2+\sqrt{a} \sqrt{c} \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (21 i a^{3/2} e^2 g^3-3 a \sqrt{c} e (e f-10 d g) g^2-3 i \sqrt{a} c \left (-2 e^2 f^2+6 d e g f+21 d^2 g^2\right ) g+c^{3/2} f \left (-8 e^2 f^2+24 d e g f-21 d^2 g^2\right )\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right ) g-i \sqrt{c} \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (21 a^2 e^2 g^4-3 a c \left (-3 e^2 f^2+16 d e g f+21 d^2 g^2\right ) g^2+c^2 f^2 \left (8 e^2 f^2-24 d e g f+21 d^2 g^2\right )\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{c^2 g^5 \sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)}\right )}{315 \sqrt{c x^2+a}} \]

Antiderivative was successfully verified.

[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]  time = 0.28, size = 4351, normalized size = 6.9 \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*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x)

[Out]

-2/315*(g*x+f)^(1/2)*(c*x^2+a)^(1/2)*(-108*(-(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^2*g^4-60*(-(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*d*e*g^6+54*(-(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^2*g^4-126*(-(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^2*f^2*g^4+12*(-(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^4*g^2-60*(-(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^2*g^4+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*c^2*d^2*f^2*g^4-34*(-(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^4*g^2+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
*d*e*f^5*g+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)^(1/2)*a^2*e^2*f*g^5+42*(-(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^3*g^3-14*x^2*a^2*c*e^2*g^6-8*a^2*c*e^2*f^2*g^4-21*a*c^2*d^2*f^2*g^4-8*a*c^2*e^2*f^4*g^2-42*(-(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^4*g^2-49*x^4*a*c^2*e^2*g^6+x^4*c^3*e^2*f^2*g^4-84*x^3*c^3*d^2*f*g^5-2*x^3*c^3*e^2*f^3*g^3-63*x
^2*a*c^2*d^2*g^6-21*x^2*c^3*d^2*f^2*g^4-8*x^2*c^3*e^2*f^4*g^2-90*x^5*c^3*d*e*g^6-40*x^5*c^3*e^2*f*g^5-35*x^6*c
^3*e^2*g^6-63*x^4*c^3*d^2*g^6-60*a^2*c*d*e*f*g^5+24*a*c^2*d*e*f^3*g^3-126*(-(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)*Ellip
ticF((-(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^2*g^6
+126*(-(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^2*g^6-168*x^2*a*c^2*d*e*f*g^5+6*x*a*c^2*d*e*f^2*g^4+42*(-(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^3*e^2*g^6-42*(-(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^3*e^2*g^6-16*(-(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^6+16*(-(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^5*g-108*x^4*c^3*d*e*f*g^5-150*x^3*a*c^2*d*e*g^6-62*x^3*a*c^2*e^2*f*g^5+6*x^3*c^
3*d*e*f^2*g^4-7*x^2*a*c^2*e^2*f^2*g^4+24*x^2*c^3*d*e*f^3*g^3-60*x*a^2*c*d*e*g^6-22*x*a^2*c*e^2*f*g^5-84*x*a*c^
2*d^2*f*g^5-2*x*a*c^2*e^2*f^3*g^3+42*(-(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*f*g^5+22*(-(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*e^2*f^3*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*d*e*f^4*g^2-36*(-(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*d*e*f*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*c^2*d*e*f^3*g^3+96*(-(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*f*g^5+144*(
-(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^3*g^3)/c^2/(c*g*x^3+c*f*x^2+a*g*x+a*f)/g^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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*(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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\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*(g*x+f)^(1/2)*(c*x^2+a)^(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 \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*(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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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*(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)