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

Optimal. Leaf size=1116 \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \sqrt{d} \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 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 (10 B e-7 A f) c+8 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^2 (a d-b c)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{d} \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\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^2 (a d-b c)^{5/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 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 (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}} \]

[Out]

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(5*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(5/2)) + (2*(2
*a^3*C*d*f + a*b^2*(10*c*C*e + B*d*e + B*c*f - 8*A*d*f) - b^3*(5*B*c*e - 4*A*(d*e + c*f)) + 3*a^2*b*(B*d*f - 2
*C*(d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)) + (2*(2*a^4*C
*d^2*f^2 + a^3*b*d*f*(3*B*d*f - 7*C*(d*e + c*f)) - b^4*(8*A*d^2*e^2 - c*d*e*(10*B*e - 7*A*f) + c^2*(15*C*e^2 -
10*B*e*f + 8*A*f^2)) - a*b^3*(d^2*e*(2*B*e - 23*A*f) - 2*c^2*f*(5*C*e - B*f) - c*d*(10*C*e^2 - 33*B*e*f + 23*
A*f^2)) - a^2*b^2*(C*(3*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + d*f*(23*A*d*f - 7*B*(d*e + c*f))))*Sqrt[c + d*x]*S
qrt[e + f*x])/(15*b*(b*c - a*d)^3*(b*e - a*f)^3*Sqrt[a + b*x]) + (2*Sqrt[d]*(2*a^4*C*d^2*f^2 + a^3*b*d*f*(3*B*
d*f - 7*C*(d*e + c*f)) - b^4*(8*A*d^2*e^2 - c*d*e*(10*B*e - 7*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a*
b^3*(d^2*e*(2*B*e - 23*A*f) - 2*c^2*f*(5*C*e - B*f) - c*d*(10*C*e^2 - 33*B*e*f + 23*A*f^2)) - a^2*b^2*(C*(3*d^
2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + d*f*(23*A*d*f - 7*B*(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])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^2*(
-(b*c) + a*d)^(5/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(a^3*C*d*f*(d*e
- c*f) + b^3*(8*A*d^2*e^2 - c*d*e*(10*B*e - 3*A*f) + c^2*(15*C*e^2 - 5*B*e*f + 4*A*f^2)) + a*b^2*(d^2*e*(2*B*e
- 19*A*f) - c^2*f*(20*C*e - B*f) - c*d*(10*C*e^2 - 27*B*e*f + 11*A*f^2)) - 3*a^2*b*(d*f*(2*B*d*e + 3*B*c*f -
5*A*d*f) - C*(d^2*e^2 + c*d*e*f + 3*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))])/(15*b^2*(-(b*
c) + a*d)^(5/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 3.34186, antiderivative size = 1116, normalized size of antiderivative = 1., 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 = {1614, 152, 158, 114, 113, 121, 120} \[ -\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \sqrt{d} \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+f x} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^2 (a d-b c)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{d} \left (C d f (d e-c f) a^3-3 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d f e+3 c^2 f^2\right )\right ) a^2+b^2 \left (-f (20 C e-B f) c^2-d \left (10 C e^2-27 B f e+11 A f^2\right ) c+d^2 e (2 B e-19 A f)\right ) a+b^3 \left (\left (15 C e^2-5 B f e+4 A f^2\right ) c^2-d e (10 B e-3 A f) c+8 A d^2 e^2\right )\right ) \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{15 b^2 (a d-b c)^{5/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}+\frac{2 \left (2 C d^2 f^2 a^4+b d f (3 B d f-7 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-13 c d f e+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right ) a^2-b^3 \left (-2 f (5 C e-B f) c^2-d \left (10 C e^2-33 B f e+23 A f^2\right ) c+d^2 e (2 B e-23 A f)\right ) a-b^4 \left (\left (15 C e^2-10 B f e+8 A f^2\right ) c^2-d e (10 B e-7 A f) c+8 A d^2 e^2\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \left (2 C d f a^3+3 b (B d f-2 C (d e+c f)) a^2+b^2 (10 c C e+B d e+B c f-8 A d f) a-b^3 (5 B c e-4 A (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(5*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(5/2)) + (2*(2
*a^3*C*d*f + a*b^2*(10*c*C*e + B*d*e + B*c*f - 8*A*d*f) - b^3*(5*B*c*e - 4*A*(d*e + c*f)) + 3*a^2*b*(B*d*f - 2
*C*(d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^(3/2)) + (2*(2*a^4*C
*d^2*f^2 + a^3*b*d*f*(3*B*d*f - 7*C*(d*e + c*f)) - b^4*(8*A*d^2*e^2 - c*d*e*(10*B*e - 7*A*f) + c^2*(15*C*e^2 -
10*B*e*f + 8*A*f^2)) - a*b^3*(d^2*e*(2*B*e - 23*A*f) - 2*c^2*f*(5*C*e - B*f) - c*d*(10*C*e^2 - 33*B*e*f + 23*
A*f^2)) - a^2*b^2*(C*(3*d^2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + d*f*(23*A*d*f - 7*B*(d*e + c*f))))*Sqrt[c + d*x]*S
qrt[e + f*x])/(15*b*(b*c - a*d)^3*(b*e - a*f)^3*Sqrt[a + b*x]) + (2*Sqrt[d]*(2*a^4*C*d^2*f^2 + a^3*b*d*f*(3*B*
d*f - 7*C*(d*e + c*f)) - b^4*(8*A*d^2*e^2 - c*d*e*(10*B*e - 7*A*f) + c^2*(15*C*e^2 - 10*B*e*f + 8*A*f^2)) - a*
b^3*(d^2*e*(2*B*e - 23*A*f) - 2*c^2*f*(5*C*e - B*f) - c*d*(10*C*e^2 - 33*B*e*f + 23*A*f^2)) - a^2*b^2*(C*(3*d^
2*e^2 - 13*c*d*e*f + 3*c^2*f^2) + d*f*(23*A*d*f - 7*B*(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])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^2*(
-(b*c) + a*d)^(5/2)*(b*e - a*f)^3*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) + (2*Sqrt[d]*(a^3*C*d*f*(d*e
- c*f) + b^3*(8*A*d^2*e^2 - c*d*e*(10*B*e - 3*A*f) + c^2*(15*C*e^2 - 5*B*e*f + 4*A*f^2)) + a*b^2*(d^2*e*(2*B*e
- 19*A*f) - c^2*f*(20*C*e - B*f) - c*d*(10*C*e^2 - 27*B*e*f + 11*A*f^2)) - 3*a^2*b*(d*f*(2*B*d*e + 3*B*c*f -
5*A*d*f) - C*(d^2*e^2 + c*d*e*f + 3*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))])/(15*b^2*(-(b*
c) + a*d)^(5/2)*(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 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{A+B x+C x^2}{(a+b x)^{7/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac{2 \int \frac{-\frac{a^2 C (d e+c f)-a b (5 c C e+B d e+B c f-5 A d f)+b^2 (5 B c e-4 A (d e+c f))}{2 b}+\frac{1}{2} \left (-5 b c C e+5 a C d e+5 a c C f+3 A b d f-3 a B d f-\frac{2 a^2 C d f}{b}\right ) x}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{5 (b c-a d) (b e-a f)}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{4 \int \frac{\frac{a^3 C d f (d e+c f)+b^3 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )+a b^2 \left (d^2 e (2 B e-19 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-28 B e f+19 A f^2\right )\right )+3 a^2 b \left (C \left (d^2 e^2-c d e f+c^2 f^2\right )+d f (5 A d f-2 B (d e+c f))\right )}{4 b}+\frac{d f \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) x}{4 b}}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 (b c-a d)^2 (b e-a f)^2}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}-\frac{8 \int \frac{\frac{d f \left (a^4 C d f (d e+c f)+b^4 c e (5 B c e-4 A (d e+c f))-a b^3 \left (4 A d^2 e^2-c d e (4 B e+9 A f)+c^2 \left (25 C e^2-4 B e f+4 A f^2\right )\right )-a^2 b^2 \left (d^2 e (B e-11 A f)-c^2 f (26 C e-B f)-c d \left (26 C e^2-29 B e f+11 A f^2\right )\right )-a^3 b \left (C \left (9 d^2 e^2+11 c d e f+9 c^2 f^2\right )+3 d f (5 A d f-3 B (d e+c f))\right )\right )}{8 b}+\frac{d f \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) x}{8 b}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}-\frac{\left (d \left (a^3 C d f (d e-c f)+b^3 \left (8 A d^2 e^2-c d e (10 B e-3 A f)+c^2 \left (15 C e^2-5 B e f+4 A f^2\right )\right )+a b^2 \left (d^2 e (2 B e-19 A f)-c^2 f (20 C e-B f)-c d \left (10 C e^2-27 B e f+11 A f^2\right )\right )-3 a^2 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d e f+3 c^2 f^2\right )\right )\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{15 b (b c-a d)^3 (b e-a f)^2}-\frac{\left (d \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right )\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{15 b (b c-a d)^3 (b e-a f)^3}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}-\frac{\left (d \left (a^3 C d f (d e-c f)+b^3 \left (8 A d^2 e^2-c d e (10 B e-3 A f)+c^2 \left (15 C e^2-5 B e f+4 A f^2\right )\right )+a b^2 \left (d^2 e (2 B e-19 A f)-c^2 f (20 C e-B f)-c d \left (10 C e^2-27 B e f+11 A f^2\right )\right )-3 a^2 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d e f+3 c^2 f^2\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}{15 b (b c-a d)^3 (b e-a f)^2 \sqrt{c+d x}}-\frac{\left (d \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\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 (b c-a d)^3 (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \sqrt{d} \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\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^2 (-b c+a d)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left (d \left (a^3 C d f (d e-c f)+b^3 \left (8 A d^2 e^2-c d e (10 B e-3 A f)+c^2 \left (15 C e^2-5 B e f+4 A f^2\right )\right )+a b^2 \left (d^2 e (2 B e-19 A f)-c^2 f (20 C e-B f)-c d \left (10 C e^2-27 B e f+11 A f^2\right )\right )-3 a^2 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d e f+3 c^2 f^2\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}{15 b (b c-a d)^3 (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (10 c C e+B d e+B c f-8 A d f)-b^3 (5 B c e-4 A (d e+c f))+3 a^2 b (B d f-2 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2}}+\frac{2 \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\right )\right ) \sqrt{c+d x} \sqrt{e+f x}}{15 b (b c-a d)^3 (b e-a f)^3 \sqrt{a+b x}}+\frac{2 \sqrt{d} \left (2 a^4 C d^2 f^2+a^3 b d f (3 B d f-7 C (d e+c f))-b^4 \left (8 A d^2 e^2-c d e (10 B e-7 A f)+c^2 \left (15 C e^2-10 B e f+8 A f^2\right )\right )-a b^3 \left (d^2 e (2 B e-23 A f)-2 c^2 f (5 C e-B f)-c d \left (10 C e^2-33 B e f+23 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-13 c d e f+3 c^2 f^2\right )+d f (23 A d f-7 B (d e+c f))\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^2 (-b c+a d)^{5/2} (b e-a f)^3 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}+\frac{2 \sqrt{d} \left (a^3 C d f (d e-c f)+b^3 \left (8 A d^2 e^2-c d e (10 B e-3 A f)+c^2 \left (15 C e^2-5 B e f+4 A f^2\right )\right )+a b^2 \left (d^2 e (2 B e-19 A f)-c^2 f (20 C e-B f)-c d \left (10 C e^2-27 B e f+11 A f^2\right )\right )-3 a^2 b \left (d f (2 B d e+3 B c f-5 A d f)-C \left (d^2 e^2+c d e f+3 c^2 f^2\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 )}{15 b^2 (-b c+a d)^{5/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 16.5791, size = 8844, normalized size = 7.92 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

________________________________________________________________________________________

Maple [B]  time = 0.297, size = 34102, normalized size = 30.6 \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)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(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{C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac{7}{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((C*x^2+B*x+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)/((b*x + a)^(7/2)*sqrt(d*x + c)*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} d f x^{6} + a^{4} c e +{\left (b^{4} d e +{\left (b^{4} c + 4 \, a b^{3} d\right )} f\right )} x^{5} +{\left ({\left (b^{4} c + 4 \, a b^{3} d\right )} e + 2 \,{\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} f\right )} x^{4} + 2 \,{\left ({\left (2 \, a b^{3} c + 3 \, a^{2} b^{2} d\right )} e +{\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} f\right )} x^{3} +{\left (2 \,{\left (3 \, a^{2} b^{2} c + 2 \, a^{3} b d\right )} e +{\left (4 \, a^{3} b c + a^{4} d\right )} f\right )} x^{2} +{\left (a^{4} c f +{\left (4 \, a^{3} b c + a^{4} d\right )} e\right )} x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(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*d*f*x^6 + a^4*c*e + (b^4*d*e + (b^4*
c + 4*a*b^3*d)*f)*x^5 + ((b^4*c + 4*a*b^3*d)*e + 2*(2*a*b^3*c + 3*a^2*b^2*d)*f)*x^4 + 2*((2*a*b^3*c + 3*a^2*b^
2*d)*e + (3*a^2*b^2*c + 2*a^3*b*d)*f)*x^3 + (2*(3*a^2*b^2*c + 2*a^3*b*d)*e + (4*a^3*b*c + a^4*d)*f)*x^2 + (a^4
*c*f + (4*a^3*b*c + a^4*d)*e)*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)/(b*x+a)**(7/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac{7}{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((C*x^2+B*x+A)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

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