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

Optimal. Leaf size=851 \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+a} (d+e x)^4}{11 e}+\frac{4 \sqrt{-a} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{4 \sqrt{-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{c x^2+a}}{99 g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{c x^2+a}}{693 c g^4}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{c x^2+a}}{3465 c g^4}-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{c x^2+a}}{3465 c^2 e g^4} \]

[Out]

(-2*(150*a^2*e^4*g^4 - 6*a*c*e^2*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^2*(187*e^4*f^4 - 732*d*e^3*f^3
*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^4))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3465*c^2*e*g^4) +
(2*(d + e*x)^4*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(11*e) - (2*(2*a*e^2*g^2*(74*e*f - 231*d*g) - c*(233*e^3*f^3 - 8
43*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(3465*c*g^4) + (2*e*(18*a*e
^2*g^2 - c*(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(693*c*g^4) + (2*e^2*(e*f
- 3*d*g)*(f + g*x)^(7/2)*Sqrt[a + c*x^2])/(99*g^4) + (4*Sqrt[-a]*(3*a^2*e^2*g^4*(26*e*f + 231*d*g) - c^2*f^2*(
64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) - 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2
*e*f*g^2 + 77*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)])/(3465*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)*(75*a^2*e^3*g^4 - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*
d^2*g^2) - c^2*f*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqr
t[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)])/(3465*c^(5/2)*g^5*Sqrt[f + g*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 2.69679, antiderivative size = 851, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{2 \sqrt{f+g x} \sqrt{c x^2+a} (d+e x)^4}{11 e}+\frac{4 \sqrt{-a} \left (3 a^2 e^2 (26 e f+231 d g) g^4-9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2-c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{\frac{c x^2}{a}+1} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{3/2} g^5 \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{c x^2+a}}-\frac{4 \sqrt{-a} \left (c f^2+a g^2\right ) \left (75 a^2 e^3 g^4-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2-c^2 f \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{c} f+\sqrt{-a} g}} \sqrt{\frac{c x^2}{a}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{3465 c^{5/2} g^5 \sqrt{f+g x} \sqrt{c x^2+a}}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{c x^2+a}}{99 g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e g f+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{c x^2+a}}{693 c g^4}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 g f^2+1107 d^2 e g^2 f-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{c x^2+a}}{3465 c g^4}-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+c^2 \left (187 e^4 f^4-732 d e^3 g f^3+1098 d^2 e^2 g^2 f^2-798 d^3 e g^3 f+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{c x^2+a}}{3465 c^2 e g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(150*a^2*e^4*g^4 - 6*a*c*e^2*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2*g^2) + c^2*(187*e^4*f^4 - 732*d*e^3*f^3
*g + 1098*d^2*e^2*f^2*g^2 - 798*d^3*e*f*g^3 + 315*d^4*g^4))*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(3465*c^2*e*g^4) +
(2*(d + e*x)^4*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(11*e) - (2*(2*a*e^2*g^2*(74*e*f - 231*d*g) - c*(233*e^3*f^3 - 8
43*d*e^2*f^2*g + 1107*d^2*e*f*g^2 - 567*d^3*g^3))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(3465*c*g^4) + (2*e*(18*a*e
^2*g^2 - c*(29*e^2*f^2 - 96*d*e*f*g + 81*d^2*g^2))*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(693*c*g^4) + (2*e^2*(e*f
- 3*d*g)*(f + g*x)^(7/2)*Sqrt[a + c*x^2])/(99*g^4) + (4*Sqrt[-a]*(3*a^2*e^2*g^4*(26*e*f + 231*d*g) - c^2*f^2*(
64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) - 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2
*e*f*g^2 + 77*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)])/(3465*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)*(75*a^2*e^3*g^4 - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*
d^2*g^2) - c^2*f*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3))*Sqrt[(Sqrt[c]*(f + g*x))/(Sqr
t[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)])/(3465*c^(5/2)*g^5*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)^3 \sqrt{f+g x} \sqrt{a+c x^2} \, dx &=\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}+\frac{\int \frac{(d+e x)^3 \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}{11 e}\\ &=\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}+\frac{2 \int \frac{-\frac{1}{2} a c g^2 \left (7 e^4 f^4-21 d e^3 f^3 g-27 d^3 e f g^3+9 d^4 g^4\right )-\frac{1}{2} c g \left (3 a e g^2 \left (7 e^3 f^3-21 d e^2 f^2 g-27 d^2 e f g^2+3 d^3 g^3\right )+2 c \left (e^4 f^5-3 d e^3 f^4 g+9 d^4 f g^4\right )\right ) x-\frac{3}{2} c g^2 \left (a e^2 g^2 \left (7 e^2 f^2-48 d e f g-9 d^2 g^2\right )+c \left (5 e^4 f^4-15 d e^3 f^3 g+15 d^3 e f g^3+9 d^4 g^4\right )\right ) x^2+\frac{1}{2} c e g^3 \left (2 a e^2 g^2 (10 e f+33 d g)-3 c \left (11 e^3 f^3-33 d e^2 f^2 g+9 d^2 e f g^2+27 d^3 g^3\right )\right ) x^3+\frac{1}{2} c e^2 g^4 \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) x^4}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{99 c e g^5}\\ &=\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{a+c x^2}}{693 c g^4}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}+\frac{4 \int \frac{-\frac{3}{4} a c g^6 \left (30 a e^4 f^2 g^2-c \left (32 e^4 f^4-111 d e^3 f^3 g+135 d^2 e^2 f^2 g^2+63 d^3 e f g^3-21 d^4 g^4\right )\right )-\frac{1}{4} c g^5 \left (180 a^2 e^4 f g^4-a c e g^2 \left (107 e^3 f^3-519 d e^2 f^2 g+1377 d^2 e f g^2-63 d^3 g^3\right )-2 c^2 \left (22 e^4 f^5-75 d e^3 f^4 g+81 d^2 e^2 f^3 g^2-63 d^4 f g^4\right )\right ) x-\frac{1}{4} c g^6 \left (90 a^2 e^4 g^4+2 a c e^2 g^2 \left (100 e^2 f^2-264 d e f g-297 d^2 g^2\right )-c^2 \left (214 e^4 f^4-741 d e^3 f^3 g+891 d^2 e^2 f^2 g^2-315 d^3 e f g^3-189 d^4 g^4\right )\right ) x^2-\frac{1}{4} c^2 e g^7 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) x^3}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{693 c^2 e g^9}\\ &=\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{3465 c g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{a+c x^2}}{693 c g^4}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}+\frac{8 \int \frac{-\frac{3}{8} a c^2 g^9 \left (2 a e^3 f g^2 (e f+231 d g)+c \left (73 e^4 f^4-288 d e^3 f^3 g+432 d^2 e^2 f^2 g^2-882 d^3 e f g^3+105 d^4 g^4\right )\right )-\frac{3}{4} c^2 g^8 \left (a^2 e^3 g^4 (76 e f+231 d g)-11 a c e g^2 \left (2 e^3 f^3-15 d e^2 f^2 g+54 d^2 e f g^2+21 d^3 g^3\right )+c^2 f \left (41 e^4 f^4-156 d e^3 f^3 g+234 d^2 e^2 f^2 g^2-189 d^3 e f g^3+105 d^4 g^4\right )\right ) x-\frac{3}{8} c^2 g^9 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) x^2}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{3465 c^3 e g^{12}}\\ &=-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{3465 c^2 e g^4}+\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{3465 c g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{a+c x^2}}{693 c g^4}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}+\frac{16 \int \frac{\frac{3}{8} a c^2 e g^{11} \left (75 a^2 e^3 g^4-9 a c e g^2 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-c^2 f \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )\right )-\frac{3}{8} c^3 e g^{10} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right ) x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{10395 c^4 e g^{14}}\\ &=-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{3465 c^2 e g^4}+\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{3465 c g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{a+c x^2}}{693 c g^4}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}-\frac{\left (2 \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{3465 c g^5}+\frac{\left (16 \left (\frac{3}{8} a c^2 e g^{12} \left (75 a^2 e^3 g^4-9 a c e g^2 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-c^2 f \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )\right )+\frac{3}{8} c^3 e f g^{10} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\right )\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{10395 c^4 e g^{15}}\\ &=-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{3465 c^2 e g^4}+\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{3465 c g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{a+c x^2}}{693 c g^4}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}-\frac{\left (4 a \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\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 )}{3465 \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 (32 a \left (\frac{3}{8} a c^2 e g^{12} \left (75 a^2 e^3 g^4-9 a c e g^2 \left (e^2 f^2+66 d e f g+55 d^2 g^2\right )-c^2 f \left (16 e^3 f^3-66 d e^2 f^2 g+99 d^2 e f g^2-924 d^3 g^3\right )\right )+\frac{3}{8} c^3 e f g^{10} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 d^3 g^3\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 )}{10395 \sqrt{-a} c^{9/2} e g^{15} \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{2 \left (150 a^2 e^4 g^4-6 a c e^2 g^2 \left (2 e^2 f^2-33 d e f g+165 d^2 g^2\right )+c^2 \left (187 e^4 f^4-732 d e^3 f^3 g+1098 d^2 e^2 f^2 g^2-798 d^3 e f g^3+315 d^4 g^4\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{3465 c^2 e g^4}+\frac{2 (d+e x)^4 \sqrt{f+g x} \sqrt{a+c x^2}}{11 e}-\frac{2 \left (2 a e^2 g^2 (74 e f-231 d g)-c \left (233 e^3 f^3-843 d e^2 f^2 g+1107 d^2 e f g^2-567 d^3 g^3\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{3465 c g^4}+\frac{2 e \left (18 a e^2 g^2-c \left (29 e^2 f^2-96 d e f g+81 d^2 g^2\right )\right ) (f+g x)^{5/2} \sqrt{a+c x^2}}{693 c g^4}+\frac{2 e^2 (e f-3 d g) (f+g x)^{7/2} \sqrt{a+c x^2}}{99 g^4}+\frac{4 \sqrt{-a} \left (3 a^2 e^2 g^4 (26 e f+231 d g)-c^2 f^2 \left (64 e^3 f^3-264 d e^2 f^2 g+396 d^2 e f g^2-231 d^3 g^3\right )-9 a c g^2 \left (6 e^3 f^3-33 d e^2 f^2 g+88 d^2 e f g^2+77 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 )}{3465 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 (64 c^2 e^3 f^4-264 c^2 d e^2 f^3 g+396 c^2 d^2 e f^2 g^2+6 a c e^3 f^2 g^2-231 c^2 d^3 f g^3-99 a c d e^2 f g^3+495 a c d^2 e g^4-75 a^2 e^3 g^4\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 )}{3465 c^{5/2} g^5 \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 10.5846, size = 1034, normalized size = 1.22 \[ \frac{2 \sqrt{f+g x} \left (\frac{2 \left (-3 a^2 e^2 (26 e f+231 d g) g^4+9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2+c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\right )\right ) \left (c x^2+a\right ) g^2}{f+g x}-\left (c x^2+a\right ) \left (150 a^2 e^3 g^4-2 a c e \left (\left (-23 f^2+16 g x f+45 g^2 x^2\right ) e^2+33 d g (4 f+7 g x) e+495 d^2 g^2\right ) g^2+c^2 \left (\left (64 f^4-48 g x f^3+40 g^2 x^2 f^2-35 g^3 x^3 f-315 g^4 x^4\right ) e^3-33 d g \left (8 f^3-6 g x f^2+5 g^2 x^2 f+35 g^3 x^3\right ) e^2-99 d^2 g^2 \left (-4 f^2+3 g x f+15 g^2 x^2\right ) e-231 d^3 g^3 (f+3 g x)\right )\right ) g^2+\frac{2 \sqrt{a} \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (75 a^2 e^3 g^4-3 i a^{3/2} \sqrt{c} e^2 (e f+231 d g) g^3-3 a c e \left (2 e^2 f^2-33 d e g f+165 d^2 g^2\right ) g^2+3 i \sqrt{a} c^{3/2} \left (16 e^3 f^3-66 d e^2 g f^2+99 d^2 e g^2 f+231 d^3 g^3\right ) g+c^2 f \left (-64 e^3 f^3+264 d e^2 g f^2-396 d^2 e g^2 f+231 d^3 g^3\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}} \sqrt{f+g x} \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}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}+\frac{2 \sqrt{c} \left (\sqrt{a} g-i \sqrt{c} f\right ) \left (-3 a^2 e^2 (26 e f+231 d g) g^4+9 a c \left (6 e^3 f^3-33 d e^2 g f^2+88 d^2 e g^2 f+77 d^3 g^3\right ) g^2+c^2 f^2 \left (64 e^3 f^3-264 d e^2 g f^2+396 d^2 e g^2 f-231 d^3 g^3\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}} \sqrt{f+g x} 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 )}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}\right )}{3465 c^2 g^6 \sqrt{c x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*((2*g^2*(-3*a^2*e^2*g^4*(26*e*f + 231*d*g) + c^2*f^2*(64*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*
e*f*g^2 - 231*d^3*g^3) + 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*e*f*g^2 + 77*d^3*g^3))*(a + c*x^2))/(f
+ g*x) - g^2*(a + c*x^2)*(150*a^2*e^3*g^4 - 2*a*c*e*g^2*(495*d^2*g^2 + 33*d*e*g*(4*f + 7*g*x) + e^2*(-23*f^2
+ 16*f*g*x + 45*g^2*x^2)) + c^2*(-231*d^3*g^3*(f + 3*g*x) - 99*d^2*e*g^2*(-4*f^2 + 3*f*g*x + 15*g^2*x^2) - 33*
d*e^2*g*(8*f^3 - 6*f^2*g*x + 5*f*g^2*x^2 + 35*g^3*x^3) + e^3*(64*f^4 - 48*f^3*g*x + 40*f^2*g^2*x^2 - 35*f*g^3*
x^3 - 315*g^4*x^4))) + (2*Sqrt[c]*((-I)*Sqrt[c]*f + Sqrt[a]*g)*(-3*a^2*e^2*g^4*(26*e*f + 231*d*g) + c^2*f^2*(6
4*e^3*f^3 - 264*d*e^2*f^2*g + 396*d^2*e*f*g^2 - 231*d^3*g^3) + 9*a*c*g^2*(6*e^3*f^3 - 33*d*e^2*f^2*g + 88*d^2*
e*f*g^2 + 77*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))]*Sqrt[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 - (I*Sqrt[a]*g)/Sqrt[c]] + (2*Sqrt[a]*g*(Sqrt[c]*f + I*Sqrt[a]*g)*(
75*a^2*e^3*g^4 - (3*I)*a^(3/2)*Sqrt[c]*e^2*g^3*(e*f + 231*d*g) - 3*a*c*e*g^2*(2*e^2*f^2 - 33*d*e*f*g + 165*d^2
*g^2) + c^2*f*(-64*e^3*f^3 + 264*d*e^2*f^2*g - 396*d^2*e*f*g^2 + 231*d^3*g^3) + (3*I)*Sqrt[a]*c^(3/2)*g*(16*e^
3*f^3 - 66*d*e^2*f^2*g + 99*d^2*e*f*g^2 + 231*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))]*Sqrt[f + g*x]*EllipticF[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/Sq
rt[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]]))/(3465*c
^2*g^6*Sqrt[a + c*x^2])

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Maple [B]  time = 0.479, size = 6457, normalized size = 7.6 \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)^3*(g*x+f)^(1/2)*(c*x^2+a)^(1/2),x)

[Out]

result too large to display

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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 )}^{3} \sqrt{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integral((e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*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 )^{3} \sqrt{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + c*x**2)*(d + e*x)**3*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 )}^{3} \sqrt{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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