3.73 \(\int \frac{(a+b x)^{3/2} (A+B x+C x^2)}{\sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=838 \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{5/2}}{7 b d f}-\frac{2 (2 a C d f-b (7 B d f-6 C (d e+c f))) \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{3/2}}{35 b d^2 f^2}-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f)))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{105 b d^3 f^3}-\frac{2 \sqrt{a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 b c e+a d e+a c f) (2 a C d f-b (7 B d f-6 C (d e+c f))))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 a d f-4 b (d e+c f)) (2 a C d f-b (7 B d f-6 C (d e+c f))))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (b e-a f) \left (-\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}} \]

[Out]

(-2*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) + (3*a*d*f - 4*b*(d*e + c*f))*(2*a*C*d*f - b*(7*B*d*f
- 6*C*(d*e + c*f))))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b*d^3*f^3) - (2*(2*a*C*d*f - b*(7*B*d*f
- 6*C*(d*e + c*f)))*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(5/2)*Sqrt[c
+ d*x]*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A
*b*d*f) - (3*b*c*e + a*d*e + a*c*f)*(2*a*C*d*f - b*(7*B*d*f - 6*C*(d*e + c*f)))) + 2*((a*d*f)/2 - b*(d*e + c*f
))*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) + (3*a*d*f - 4*b*(d*e + c*f))*(2*a*C*d*f - b*(7*B*d*f
- 6*C*(d*e + c*f)))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b
*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(3*B*d*e + 2*B
*c*f - 5*A*d*f) - C*(16*d^2*e^2 + 8*c*d*e*f + 11*c^2*f^2)) - b^2*(C*(48*d^3*e^3 + 16*c*d^2*e^2*f + 17*c^2*d*e*
f^2 + 24*c^3*f^3) + 7*d*f*(5*A*d*f*(2*d*e + c*f) - B*(8*d^2*e^2 + 3*c*d*e*f + 4*c^2*f^2))))*Sqrt[(b*(c + d*x))
/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((
b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 2.16678, antiderivative size = 831, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 7, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.184, Rules used = {1615, 154, 158, 114, 113, 121, 120} \[ \frac{2 C \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{5/2}}{7 b d f}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) \sqrt{c+d x} \sqrt{e+f x} (a+b x)^{3/2}}{35 b d^2 f^2}-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{c+d x} \sqrt{e+f x} \sqrt{a+b x}}{105 b d^3 f^3}-\frac{2 \sqrt{a d-b c} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{a d-b c} (b e-a f) \left (-\left (C \left (48 d^3 e^3+16 c d^2 f e^2+17 c^2 d f^2 e+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d f e+4 c^2 f^2\right )\right )\right ) b^2-3 a d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d f e+11 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*a*d*f - 4*b*(d*e + c*f))*(7*b*B*d*f - 2*a*C*d*f
- 6*b*C*(d*e + c*f)))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(105*b*d^3*f^3) + (2*(7*b*B*d*f - 2*a*C*d*f -
6*b*C*(d*e + c*f))*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])/(35*b*d^2*f^2) + (2*C*(a + b*x)^(5/2)*Sqrt[c
+ d*x]*Sqrt[e + f*x])/(7*b*d*f) - (2*Sqrt[-(b*c) + a*d]*(3*b*d*f*(5*a*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A
*b*d*f) + (3*b*c*e + a*d*e + a*c*f)*(7*b*B*d*f - 2*a*C*d*f - 6*b*C*(d*e + c*f))) + 2*((a*d*f)/2 - b*(d*e + c*f
))*(5*b*d*f*(5*b*c*C*e + a*C*d*e + a*c*C*f - 7*A*b*d*f) - (3*a*d*f - 4*b*(d*e + c*f))*(7*b*B*d*f - 2*a*C*d*f -
6*b*C*(d*e + c*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/S
qrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b
*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(3*B*d*e + 2*B
*c*f - 5*A*d*f) - C*(16*d^2*e^2 + 8*c*d*e*f + 11*c^2*f^2)) - b^2*(C*(48*d^3*e^3 + 16*c*d^2*e^2*f + 17*c^2*d*e*
f^2 + 24*c^3*f^3) + 7*d*f*(5*A*d*f*(2*d*e + c*f) - B*(8*d^2*e^2 + 3*c*d*e*f + 4*c^2*f^2))))*Sqrt[(b*(c + d*x))
/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((
b*c - a*d)*f)/(d*(b*e - a*f))])/(105*b^2*d^(7/2)*f^4*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 1615

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[
{q = Expon[Px, x], k = Coeff[Px, x, Expon[Px, x]]}, Simp[(k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*(e + f*x)^
(p + 1))/(d*f*b^(q - 1)*(m + n + p + q + 1)), x] + Dist[1/(d*f*b^q*(m + n + p + q + 1)), Int[(a + b*x)^m*(c +
d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a +
b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*
(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; F
reeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x] && IntegersQ[2*m, 2*n, 2*p]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 158

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-((b*c - a*d)/d), 0]

Rule 113

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

Rule 121

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[(b*(c
+ d*x))/(b*c - a*d)]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-(b/d), 2]*Sqrt[(b*c - a*d)/b])], (f*(b*c - a*d))/(d*(b*e - a*f))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &
& SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x] && (PosQ[-((b*c - a*d)/d)] || NegQ[-((b*e - a*f)/f)
])

Rubi steps

\begin{align*} \int \frac{(a+b x)^{3/2} \left (A+B x+C x^2\right )}{\sqrt{c+d x} \sqrt{e+f x}} \, dx &=\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}+\frac{2 \int \frac{(a+b x)^{3/2} \left (-\frac{1}{2} b (5 b c C e+a C d e+a c C f-7 A b d f)+\frac{1}{2} b (7 b B d f-2 a C d f-6 b C (d e+c f)) x\right )}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{7 b^2 d f}\\ &=\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}+\frac{4 \int \frac{\sqrt{a+b x} \left (-\frac{1}{4} b (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))-\frac{1}{4} b (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) x\right )}{\sqrt{c+d x} \sqrt{e+f x}} \, dx}{35 b^2 d^2 f^2}\\ &=-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b d^3 f^3}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}+\frac{8 \int \frac{-\frac{1}{8} b (3 a d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))-(b c e+a d e+a c f) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))))-\frac{1}{8} b \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b^2 d^3 f^3}\\ &=-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b d^3 f^3}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}-\frac{\left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{105 b d^3 f^4}-\frac{\left ((b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{105 b d^3 f^4}\\ &=-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b d^3 f^3}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}-\frac{\left ((b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}}\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{e+f x}} \, dx}{105 b d^3 f^4 \sqrt{c+d x}}-\frac{\left (\left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x}\right ) \int \frac{\sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx}{105 b d^3 f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b d^3 f^3}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}-\frac{2 \sqrt{-b c+a d} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left ((b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}}\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}} \sqrt{\frac{b e}{b e-a f}+\frac{b f x}{b e-a f}}} \, dx}{105 b d^3 f^4 \sqrt{c+d x} \sqrt{e+f x}}\\ &=-\frac{2 (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f))) \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}}{105 b d^3 f^3}+\frac{2 (7 b B d f-2 a C d f-6 b C (d e+c f)) (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}}{35 b d^2 f^2}+\frac{2 C (a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}}{7 b d f}-\frac{2 \sqrt{-b c+a d} \left (3 b d f (5 a d f (5 b c C e+a C d e+a c C f-7 A b d f)+(3 b c e+a d e+a c f) (7 b B d f-2 a C d f-6 b C (d e+c f)))+2 \left (\frac{a d f}{2}-b (d e+c f)\right ) (5 b d f (5 b c C e+a C d e+a c C f-7 A b d f)-(3 a d f-4 b (d e+c f)) (7 b B d f-2 a C d f-6 b C (d e+c f)))\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{-b c+a d} (b e-a f) \left (3 a^2 C d^2 f^2 (d e-c f)-3 a b d f \left (7 d f (3 B d e+2 B c f-5 A d f)-C \left (16 d^2 e^2+8 c d e f+11 c^2 f^2\right )\right )-b^2 \left (C \left (48 d^3 e^3+16 c d^2 e^2 f+17 c^2 d e f^2+24 c^3 f^3\right )+7 d f \left (5 A d f (2 d e+c f)-B \left (8 d^2 e^2+3 c d e f+4 c^2 f^2\right )\right )\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{105 b^2 d^{7/2} f^4 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 13.8407, size = 1000, normalized size = 1.19 \[ \frac{2 \left (-\sqrt{\frac{b c}{d}-a} \left (\left (8 C \left (6 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e+6 c^3 f^3\right )+7 d f \left (10 A d f (d e+c f)-B \left (8 d^2 e^2+7 c d f e+8 c^2 f^2\right )\right )\right ) b^3-a d f \left (C \left (72 d^2 e^2+62 c d f e+72 c^2 f^2\right )+7 d f (20 A d f-13 B (d e+c f))\right ) b^2+3 a^2 d^2 f^2 (4 C (d e+c f)-7 B d f) b+6 a^3 C d^3 f^3\right ) (c+d x) (e+f x) b^2+\sqrt{\frac{b c}{d}-a} d f (a+b x) (c+d x) (e+f x) \left (\left (7 d f (5 A d f+B (-4 d e-4 c f+3 d f x))+C \left (3 \left (8 e^2-6 f x e+5 f^2 x^2\right ) d^2+c f (23 e-18 f x) d+24 c^2 f^2\right )\right ) b^2+3 a d f (14 B d f+C (-11 d e-11 c f+8 d f x)) b+3 a^2 C d^2 f^2\right ) b^2+i (b c-a d) f \left (\left (C \left (24 d^3 e^3+17 c d^2 f e^2+16 c^2 d f^2 e+48 c^3 f^3\right )+7 d f \left (5 A d f (d e+2 c f)-B \left (4 d^2 e^2+3 c d f e+8 c^2 f^2\right )\right )\right ) b^2-3 a d f \left (7 d f (-2 B d e-3 B c f+5 A d f)+C \left (11 d^2 e^2+8 c d f e+16 c^2 f^2\right )\right ) b+3 a^2 C d^2 f^2 (d e-c f)\right ) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right ) b-i (b c-a d) f \left (\left (8 C \left (6 d^3 e^3+5 c d^2 f e^2+5 c^2 d f^2 e+6 c^3 f^3\right )+7 d f \left (10 A d f (d e+c f)-B \left (8 d^2 e^2+7 c d f e+8 c^2 f^2\right )\right )\right ) b^3-a d f \left (C \left (72 d^2 e^2+62 c d f e+72 c^2 f^2\right )+7 d f (20 A d f-13 B (d e+c f))\right ) b^2+3 a^2 d^2 f^2 (4 C (d e+c f)-7 B d f) b+6 a^3 C d^3 f^3\right ) (a+b x)^{3/2} \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )}{105 b^3 \sqrt{\frac{b c}{d}-a} d^4 f^4 \sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^(3/2)*(A + B*x + C*x^2))/(Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(2*(-(b^2*Sqrt[-a + (b*c)/d]*(6*a^3*C*d^3*f^3 + 3*a^2*b*d^2*f^2*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(7
2*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 7*d*f*(20*A*d*f - 13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^
2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e + c*f) - B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(c +
d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(a + b*x)*(c + d*x)*(e + f*x)*(3*a^2*C*d^2*f^2 + 3*a*b*d*f*(14*B
*d*f + C*(-11*d*e - 11*c*f + 8*d*f*x)) + b^2*(7*d*f*(5*A*d*f + B*(-4*d*e - 4*c*f + 3*d*f*x)) + C*(24*c^2*f^2 +
c*d*f*(23*e - 18*f*x) + 3*d^2*(8*e^2 - 6*e*f*x + 5*f^2*x^2)))) - I*(b*c - a*d)*f*(6*a^3*C*d^3*f^3 + 3*a^2*b*d
^2*f^2*(-7*B*d*f + 4*C*(d*e + c*f)) - a*b^2*d*f*(C*(72*d^2*e^2 + 62*c*d*e*f + 72*c^2*f^2) + 7*d*f*(20*A*d*f -
13*B*(d*e + c*f))) + b^3*(8*C*(6*d^3*e^3 + 5*c*d^2*e^2*f + 5*c^2*d*e*f^2 + 6*c^3*f^3) + 7*d*f*(10*A*d*f*(d*e +
c*f) - B*(8*d^2*e^2 + 7*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e
+ f*x))/(f*(a + b*x))]*EllipticE[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)]
+ I*b*(b*c - a*d)*f*(3*a^2*C*d^2*f^2*(d*e - c*f) - 3*a*b*d*f*(7*d*f*(-2*B*d*e - 3*B*c*f + 5*A*d*f) + C*(11*d^
2*e^2 + 8*c*d*e*f + 16*c^2*f^2)) + b^2*(C*(24*d^3*e^3 + 17*c*d^2*e^2*f + 16*c^2*d*e*f^2 + 48*c^3*f^3) + 7*d*f*
(5*A*d*f*(d*e + 2*c*f) - B*(4*d^2*e^2 + 3*c*d*e*f + 8*c^2*f^2))))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b
*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)
/(b*c*f - a*d*f)]))/(105*b^3*Sqrt[-a + (b*c)/d]*d^4*f^4*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])

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Maple [B]  time = 0.051, size = 10546, normalized size = 12.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(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{{\left (C x^{2} + B x + A\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{\sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C b x^{3} +{\left (C a + B b\right )} x^{2} + A a +{\left (B a + A b\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{d f x^{2} + c e +{\left (d e + c f\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2)*(C*x^2+B*x+A)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

integral((C*b*x^3 + (C*a + B*b)*x^2 + A*a + (B*a + A*b)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(d*f*x^2
+ c*e + (d*e + c*f)*x), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2)*(C*x**2+B*x+A)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )}{\left (b x + a\right )}^{\frac{3}{2}}}{\sqrt{d x + c} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*x^2 + B*x + A)*(b*x + a)^(3/2)/(sqrt(d*x + c)*sqrt(f*x + e)), x)