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

Optimal. Leaf size=666 \[ -\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right ),-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (168 d^2 e f g^2-35 d^3 g^3-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]

[Out]

(-4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2*g + 168*d^2*e*f*g^2 - 35*d^3*g^3))*Sqrt[f + g
*x]*Sqrt[a + c*x^2])/(315*c*g^4) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*g) + (4*e*(7*a*e^2*g^2 + c
*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^4) - (4*e^2*(4*e*f - 3*d*g
)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^4) + (4*Sqrt[-a]*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f
*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[f + g*x]*Sqrt[1
+ (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(
315*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)*(9*a*e^2*g^2*(2*e*f - 5*d*g) - c*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(Sqr
t[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^5*Sqrt[f + g*x]*Sqrt[a + c*x^2])

________________________________________________________________________________________

Rubi [A]  time = 1.57141, antiderivative size = 666, 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 = {921, 1654, 844, 719, 424, 419} \[ \frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \sqrt{f+g x} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (63 d^2 g^2-39 d e f g+10 e^2 f^2\right )-c^2 f \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}}}-\frac{4 \sqrt{-a} \sqrt{\frac{c x^2}{a}+1} \left (a g^2+c f^2\right ) \sqrt{\frac{\sqrt{c} (f+g x)}{\sqrt{-a} g+\sqrt{c} f}} \left (9 a e^2 g^2 (2 e f-5 d g)-c \left (252 d^2 e f g^2-105 d^3 g^3-216 d e^2 f^2 g+64 e^3 f^3\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 c^{3/2} g^5 \sqrt{a+c x^2} \sqrt{f+g x}}+\frac{4 e \sqrt{a+c x^2} (f+g x)^{3/2} \left (7 a e^2 g^2+c \left (42 d^2 g^2-111 d e f g+64 e^2 f^2\right )\right )}{315 c g^4}-\frac{4 \sqrt{a+c x^2} \sqrt{f+g x} \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (168 d^2 e f g^2-35 d^3 g^3-204 d e^2 f^2 g+76 e^3 f^3\right )\right )}{315 c g^4}-\frac{4 e^2 \sqrt{a+c x^2} (f+g x)^{5/2} (4 e f-3 d g)}{63 g^4}+\frac{2 \sqrt{a+c x^2} (d+e x)^3 \sqrt{f+g x}}{9 g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(9*a*e^2*g^2*(2*e*f - 5*d*g) + c*(76*e^3*f^3 - 204*d*e^2*f^2*g + 168*d^2*e*f*g^2 - 35*d^3*g^3))*Sqrt[f + g
*x]*Sqrt[a + c*x^2])/(315*c*g^4) + (2*(d + e*x)^3*Sqrt[f + g*x]*Sqrt[a + c*x^2])/(9*g) + (4*e*(7*a*e^2*g^2 + c
*(64*e^2*f^2 - 111*d*e*f*g + 42*d^2*g^2))*(f + g*x)^(3/2)*Sqrt[a + c*x^2])/(315*c*g^4) - (4*e^2*(4*e*f - 3*d*g
)*(f + g*x)^(5/2)*Sqrt[a + c*x^2])/(63*g^4) + (4*Sqrt[-a]*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f
*g + 63*d^2*g^2) - c^2*f*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[f + g*x]*Sqrt[1
+ (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[-a]]/Sqrt[2]], (-2*a*g)/(Sqrt[-a]*Sqrt[c]*f - a*g)])/(
315*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)*(9*a*e^2*g^2*(2*e*f - 5*d*g) - c*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(Sqr
t[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^5*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)^3 \sqrt{a+c x^2}}{\sqrt{f+g x}} \, dx &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}-\frac{\int \frac{(d+e x)^2 \left (2 a (3 e f-4 d g)+2 (c d f-a e g) x+2 c (4 e f-3 d g) x^2\right )}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{9 g}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{2 \int \frac{-a c g^2 \left (20 e^3 f^3-15 d e^2 f^2 g-21 d^2 e f g^2+28 d^3 g^3\right )-c g \left (a e g^2 \left (40 e^2 f^2-72 d e f g+63 d^2 g^2\right )+c \left (8 e^3 f^4-6 d e^2 f^3 g-7 d^3 f g^3\right )\right ) x+c g^2 \left (a e^2 g^2 (e f-27 d g)-c \left (44 e^3 f^3-33 d e^2 f^2 g-42 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 (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) x^3}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{63 c g^5}\\ &=\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{4 \int \frac{\frac{1}{2} a c g^5 \left (21 a e^3 f g^2+c \left (92 e^3 f^3-258 d e^2 f^2 g+231 d^2 e f g^2-140 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 (2 e^2 f^2+9 d e f g-63 d^2 g^2\right )+c^2 f \left (88 e^3 f^3-192 d e^2 f^2 g+84 d^2 e f g^2+35 d^3 g^3\right )\right ) x+\frac{3}{2} c^2 g^5 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 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 g^8}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{8 \int \frac{\frac{3}{4} a c^2 g^7 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )+\frac{3}{4} c^2 g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) x}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{945 c^3 g^{10}}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{\left (2 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right ) \int \frac{\sqrt{f+g x}}{\sqrt{a+c x^2}} \, dx}{315 c g^5}-\frac{\left (8 \left (\frac{3}{4} a c^2 g^8 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-\frac{3}{4} c^2 f g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right )\right )\right ) \int \frac{1}{\sqrt{f+g x} \sqrt{a+c x^2}} \, dx}{945 c^3 g^{11}}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}-\frac{\left (4 a \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{1+\frac{c x^2}{a}}\right ) \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^5 \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} a c^2 g^8 \left (3 a e^2 g^2 (e f+15 d g)+c \left (16 e^3 f^3-54 d e^2 f^2 g+63 d^2 e f g^2-105 d^3 g^3\right )\right )-\frac{3}{4} c^2 f g^6 \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 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 )}{945 \sqrt{-a} c^{7/2} g^{11} \sqrt{f+g x} \sqrt{a+c x^2}}\\ &=-\frac{4 \left (9 a e^2 g^2 (2 e f-5 d g)+c \left (76 e^3 f^3-204 d e^2 f^2 g+168 d^2 e f g^2-35 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{a+c x^2}}{315 c g^4}+\frac{2 (d+e x)^3 \sqrt{f+g x} \sqrt{a+c x^2}}{9 g}+\frac{4 e \left (7 a e^2 g^2+c \left (64 e^2 f^2-111 d e f g+42 d^2 g^2\right )\right ) (f+g x)^{3/2} \sqrt{a+c x^2}}{315 c g^4}-\frac{4 e^2 (4 e f-3 d g) (f+g x)^{5/2} \sqrt{a+c x^2}}{63 g^4}+\frac{4 \sqrt{-a} \left (21 a^2 e^3 g^4-3 a c e g^2 \left (10 e^2 f^2-39 d e f g+63 d^2 g^2\right )-c^2 f \left (64 e^3 f^3-216 d e^2 f^2 g+252 d^2 e f g^2-105 d^3 g^3\right )\right ) \sqrt{f+g x} \sqrt{1+\frac{c x^2}{a}} E\left (\sin ^{-1}\left (\frac{\sqrt{1-\frac{\sqrt{c} x}{\sqrt{-a}}}}{\sqrt{2}}\right )|-\frac{2 a g}{\sqrt{-a} \sqrt{c} f-a g}\right )}{315 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 e^3 f^3-216 c d e^2 f^2 g+252 c d^2 e f g^2-18 a e^3 f g^2-105 c d^3 g^3+45 a d e^2 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^5 \sqrt{f+g x} \sqrt{a+c x^2}}\\ \end{align*}

Mathematica [C]  time = 8.1031, size = 864, normalized size = 1.3 \[ \frac{2 \sqrt{f+g x} \left (-c \left (c x^2+a\right ) \left (c \left (\left (64 f^3-48 g x f^2+40 g^2 x^2 f-35 g^3 x^3\right ) e^3-27 d g \left (8 f^2-6 g x f+5 g^2 x^2\right ) e^2+63 d^2 g^2 (4 f-3 g x) e-105 d^3 g^3\right )-2 a e^2 g^2 (-11 e f+45 d g+7 e g x)\right ) g^2-\frac{2 \left (\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2+c^2 f \left (-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right )\right ) \left (c x^2+a\right ) g^2+\sqrt{a} \sqrt{c} \left (\sqrt{c} f+i \sqrt{a} g\right ) \left (21 i a^{3/2} e^3 g^3-9 a \sqrt{c} e^2 (2 e f-5 d g) g^2-3 i \sqrt{a} c e \left (16 e^2 f^2-54 d e g f+63 d^2 g^2\right ) g+c^{3/2} \left (64 e^3 f^3-216 d e^2 g f^2+252 d^2 e g^2 f-105 d^3 g^3\right )\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right ),\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right ) g-\sqrt{c} \left (i \sqrt{c} f-\sqrt{a} g\right ) \left (21 a^2 e^3 g^4-3 a c e \left (10 e^2 f^2-39 d e g f+63 d^2 g^2\right ) g^2+c^2 f \left (-64 e^3 f^3+216 d e^2 g f^2-252 d^2 e g^2 f+105 d^3 g^3\right )\right ) \sqrt{\frac{g \left (x+\frac{i \sqrt{a}}{\sqrt{c}}\right )}{f+g x}} \sqrt{-\frac{\frac{i \sqrt{a} g}{\sqrt{c}}-g x}{f+g x}} (f+g x)^{3/2} E\left (i \sinh ^{-1}\left (\frac{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}}}{\sqrt{f+g x}}\right )|\frac{\sqrt{c} f-i \sqrt{a} g}{\sqrt{c} f+i \sqrt{a} g}\right )\right )}{\sqrt{-f-\frac{i \sqrt{a} g}{\sqrt{c}}} (f+g x)}\right )}{315 c^2 g^6 \sqrt{c x^2+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[f + g*x]*(-(c*g^2*(a + c*x^2)*(-2*a*e^2*g^2*(-11*e*f + 45*d*g + 7*e*g*x) + c*(-105*d^3*g^3 + 63*d^2*e*
g^2*(4*f - 3*g*x) - 27*d*e^2*g*(8*f^2 - 6*f*g*x + 5*g^2*x^2) + e^3*(64*f^3 - 48*f^2*g*x + 40*f*g^2*x^2 - 35*g^
3*x^3)))) - (2*(g^2*Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g +
63*d^2*g^2) + c^2*f*(-64*e^3*f^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3))*(a + c*x^2) - Sqrt[c]*(I*
Sqrt[c]*f - Sqrt[a]*g)*(21*a^2*e^3*g^4 - 3*a*c*e*g^2*(10*e^2*f^2 - 39*d*e*f*g + 63*d^2*g^2) + c^2*f*(-64*e^3*f
^3 + 216*d*e^2*f^2*g - 252*d^2*e*f*g^2 + 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a])/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((
I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-f - (I*Sqrt[a]*g)/Sqrt[c]]/S
qrt[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^3*g^3 - 9*a*Sqrt[c]*e^2*g^2*(2*e*f - 5*d*g) - (3*I)*Sqrt[a]*c*e*g*(16*e^2*f^2 - 54*d*e*f
*g + 63*d^2*g^2) + c^(3/2)*(64*e^3*f^3 - 216*d*e^2*f^2*g + 252*d^2*e*f*g^2 - 105*d^3*g^3))*Sqrt[(g*((I*Sqrt[a]
)/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*Sqrt[a]*g)/Sqrt[c] - g*x)/(f + g*x))]*(f + g*x)^(3/2)*EllipticF[I*ArcSin
h[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)]))/(Sqr
t[-f - (I*Sqrt[a]*g)/Sqrt[c]]*(f + g*x))))/(315*c^2*g^6*Sqrt[a + c*x^2])

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Maple [B]  time = 0.287, size = 5079, 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*(c*x^2+a)^(1/2)/(g*x+f)^(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 \frac{\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*(c*x^2+a)^(1/2)/(g*x+f)^(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 (\frac{{\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*(c*x^2+a)^(1/2)/(g*x+f)^(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 \frac{\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*(c*x**2+a)**(1/2)/(g*x+f)**(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 \frac{\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*(c*x^2+a)^(1/2)/(g*x+f)^(1/2),x, algorithm="giac")

[Out]

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