3.72 \(\int \frac{\sqrt{c+d x} (A+B x+C x^2)}{(a+b x)^{7/2} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=1034 \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{e+f x} (c+d x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^3 \sqrt{\frac{b (e+f x)}{b e-a f}} \sqrt{c+d x}}+\frac{2 \sqrt{d} (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^2 \sqrt{e+f x} \sqrt{c+d x}} \]

[Out]

(2*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(8*
C*d*e + 6*c*C*f - B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(3/2)) - (2
*(8*a^4*C*d^2*f^2 - a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2
*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f +
3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[
c + d*x]*Sqrt[e + f*x])/(15*b^2*(b*c - a*d)^2*(b*e - a*f)^3*Sqrt[a + b*x]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d
*x)^(3/2)*Sqrt[e + f*x])/(5*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(5/2)) + (2*Sqrt[d]*(8*a^4*C*d^2*f^2 - a^3*b*d
*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A
*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f + 3*c^2*f^2)) - a*b^3*(d^2*e*
(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*
Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/
(15*b^3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e -
c*f)*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(
8*C*d*e + 6*c*C*f - B*d*f))*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))])/(15*b^3*(-(b*c) + a*d)^(3/2)*(b*
e - a*f)^2*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 3.15979, antiderivative size = 1034, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {1614, 150, 152, 158, 114, 113, 121, 120} \[ -\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{e+f x} (c+d x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{e+f x} \sqrt{c+d x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \sqrt{d} \left (8 C d^2 f^2 a^4-b d f (23 C d e+13 c C f-2 B d f) a^3-b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d f e+3 c^2 f^2\right )\right ) a^2-b^3 \left (2 f (5 C e-B f) c^2+d \left (40 C e^2-13 f (B e-A f)\right ) c+d^2 e (3 B e-7 A f)\right ) a-b^4 \left (-\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (5 B e-3 A f) c+2 A d^2 e^2\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^3 \sqrt{\frac{b (e+f x)}{b e-a f}} \sqrt{c+d x}}+\frac{2 \sqrt{d} (d e-c f) \left (4 C d f a^3-b (8 C d e+6 c C f-B d f) a^2+b^2 (10 c C e+3 B d e+B c f-6 A d f) a-b^3 (5 B c e-2 A d e-4 A c f)\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (a d-b c)^{3/2} (b e-a f)^2 \sqrt{e+f x} \sqrt{c+d x}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(7/2)*Sqrt[e + f*x]),x]

[Out]

(2*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(8*
C*d*e + 6*c*C*f - B*d*f))*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b^2*(b*c - a*d)*(b*e - a*f)^2*(a + b*x)^(3/2)) - (2
*(8*a^4*C*d^2*f^2 - a^3*b*d*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2
*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f +
3*c^2*f^2)) - a*b^3*(d^2*e*(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[
c + d*x]*Sqrt[e + f*x])/(15*b^2*(b*c - a*d)^2*(b*e - a*f)^3*Sqrt[a + b*x]) - (2*(A*b^2 - a*(b*B - a*C))*(c + d
*x)^(3/2)*Sqrt[e + f*x])/(5*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(5/2)) + (2*Sqrt[d]*(8*a^4*C*d^2*f^2 - a^3*b*d
*f*(23*C*d*e + 13*c*C*f - 2*B*d*f) - b^4*(2*A*d^2*e^2 - c*d*e*(5*B*e - 3*A*f) - c^2*(15*C*e^2 - 10*B*e*f + 8*A
*f^2)) - a^2*b^2*(d*f*(7*B*d*e + 2*B*c*f - 3*A*d*f) - C*(23*d^2*e^2 + 37*c*d*e*f + 3*c^2*f^2)) - a*b^3*(d^2*e*
(3*B*e - 7*A*f) + 2*c^2*f*(5*C*e - B*f) + c*d*(40*C*e^2 - 13*f*(B*e - A*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*
Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/
(15*b^3*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(d*e -
c*f)*(4*a^3*C*d*f - b^3*(5*B*c*e - 2*A*d*e - 4*A*c*f) + a*b^2*(10*c*C*e + 3*B*d*e + B*c*f - 6*A*d*f) - a^2*b*(
8*C*d*e + 6*c*C*f - B*d*f))*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))])/(15*b^3*(-(b*c) + a*d)^(3/2)*(b*
e - a*f)^2*Sqrt[c + d*x]*Sqrt[e + f*x])

Rule 1614

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> With[{
Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[(b*R*(a + b*x)^(m + 1)
*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e
- a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f
*R*(m + 1) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x], x], x]] /; FreeQ[{a, b,
c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 150

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

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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{\sqrt{c+d x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2} \sqrt{e+f x}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \int \frac{\sqrt{c+d x} \left (-\frac{a^2 C (3 d e+c f)+b^2 (5 B c e-2 A d e-4 A c f)-a b (5 c C e+3 B d e+B c f-5 A d f)}{2 b}+\frac{1}{2} \left (-5 b c C e+5 a C d e+5 a c C f+A b d f-a B d f-\frac{4 a^2 C d f}{b}\right ) x\right )}{(a+b x)^{5/2} \sqrt{e+f x}} \, dx}{5 (b c-a d) (b e-a f)}\\ &=\frac{2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{4 \int \frac{\frac{4 a^3 C d f (d e+c f)+b^3 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )+a b^2 \left (3 d^2 e (B e-2 A f)+2 c^2 f (5 C e-B f)+c d \left (25 C e^2-8 B e f+9 A f^2\right )\right )+a^2 b \left (B d f (d e+c f)-C \left (8 d^2 e^2+17 c d e f+3 c^2 f^2\right )\right )}{4 b}+\frac{d \left (8 a^3 C d f^2-a^2 b f (19 C d e+9 c C f-2 B d f)-b^3 \left (15 c C e^2-A d e f-c f (5 B e-4 A f)\right )+a b^2 (5 C e (3 d e+4 c f)-f (6 B d e+B c f-3 A d f))\right ) x}{4 b}}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 b (b c-a d) (b e-a f)^2}\\ &=\frac{2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac{2 \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{8 \int \frac{\frac{d \left (4 a^4 C d f^2 (d e+c f)+b^4 c e \left (15 c C e^2-A d e f-c f (5 B e-4 A f)\right )+a^3 b f \left (B d f (d e+c f)-C \left (11 d^2 e^2+19 c d e f+6 c^2 f^2\right )\right )-a b^3 \left (A d^2 e^2 f+c d e \left (30 C e^2-f (16 B e-9 A f)\right )+4 c^2 f \left (5 C e^2+f (B e-A f)\right )\right )+a^2 b^2 \left (C e \left (15 d^2 e^2+29 c d e f+19 c^2 f^2\right )+f \left (3 A d f (3 d e-2 c f)-B \left (9 d^2 e^2+c d e f-c^2 f^2\right )\right )\right )\right )}{8 b}+\frac{d f \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) x}{8 b}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 b (b c-a d)^2 (b e-a f)^3}\\ &=\frac{2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac{2 \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{\left (d (d e-c f) \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 b^2 (b c-a d)^2 (b e-a f)^2}+\frac{\left (d \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right )\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{15 b^2 (b c-a d)^2 (b e-a f)^3}\\ &=\frac{2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac{2 \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{\left (d (d e-c f) \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\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}{15 b^2 (b c-a d)^2 (b e-a f)^2 \sqrt{c+d x}}+\frac{\left (d \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\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}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=\frac{2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac{2 \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \sqrt{d} \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\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 )}{15 b^3 (-b c+a d)^{3/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{\left (d (d e-c f) \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\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}{15 b^2 (b c-a d)^2 (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}\\ &=\frac{2 \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d f)\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac{2 \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b^2 (b c-a d)^2 (b e-a f)^3 \sqrt{a+b x}}-\frac{2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \sqrt{d} \left (8 a^4 C d^2 f^2-a^3 b d f (23 C d e+13 c C f-2 B d f)-b^4 \left (2 A d^2 e^2-c d e (5 B e-3 A f)-c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a^2 b^2 \left (d f (7 B d e+2 B c f-3 A d f)-C \left (23 d^2 e^2+37 c d e f+3 c^2 f^2\right )\right )-a b^3 \left (d^2 e (3 B e-7 A f)+2 c^2 f (5 C e-B f)+c d \left (40 C e^2-13 f (B e-A f)\right )\right )\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 )}{15 b^3 (-b c+a d)^{3/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{d} (d e-c f) \left (4 a^3 C d f-b^3 (5 B c e-2 A d e-4 A c f)+a b^2 (10 c C e+3 B d e+B c f-6 A d f)-a^2 b (8 C d e+6 c C f-B d 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{-b c+a d}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 16.5889, size = 9186, normalized size = 8.88 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[c + d*x]*(A + B*x + C*x^2))/((a + b*x)^(7/2)*Sqrt[e + f*x]),x]

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.213, size = 33007, normalized size = 31.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x)

[Out]

result too large to display

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{7}{2}} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(7/2)*sqrt(f*x + e)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C x^{2} + B x + A\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b^{4} f x^{5} + a^{4} e +{\left (b^{4} e + 4 \, a b^{3} f\right )} x^{4} + 2 \,{\left (2 \, a b^{3} e + 3 \, a^{2} b^{2} f\right )} x^{3} + 2 \,{\left (3 \, a^{2} b^{2} e + 2 \, a^{3} b f\right )} x^{2} +{\left (4 \, a^{3} b e + a^{4} f\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

integral((C*x^2 + B*x + A)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)/(b^4*f*x^5 + a^4*e + (b^4*e + 4*a*b^3*f)*
x^4 + 2*(2*a*b^3*e + 3*a^2*b^2*f)*x^3 + 2*(3*a^2*b^2*e + 2*a^3*b*f)*x^2 + (4*a^3*b*e + a^4*f)*x), x)

________________________________________________________________________________________

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((C*x**2+B*x+A)*(d*x+c)**(1/2)/(b*x+a)**(7/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C x^{2} + B x + A\right )} \sqrt{d x + c}}{{\left (b x + a\right )}^{\frac{7}{2}} \sqrt{f x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)*(d*x+c)^(1/2)/(b*x+a)^(7/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate((C*x^2 + B*x + A)*sqrt(d*x + c)/((b*x + a)^(7/2)*sqrt(f*x + e)), x)