3.77 \(\int \frac{A+B x+C x^2}{(a+b x)^{5/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx\)

Optimal. Leaf size=642 \[ -\frac{2 \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right ),\frac{f (b c-a d)}{d (b e-a f)}\right )}{3 b^2 \sqrt{d} \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}+\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (a^2 b (B d f-4 C (c f+d e))+2 a^3 C d f+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{3 b \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (a^2 b (B d f-4 C (c f+d e))+2 a^3 C d f+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]

[Out]

(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + (2*(2
*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*
(d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[a + b*x]) - (2*Sqrt[d]*(2*a^3
*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*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])/Sqrt[-(b*c) +
a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e +
f*x))/(b*e - a*f)]) - (2*(a^2*C*d*(d*e - c*f) - b^2*(3*c^2*C*e - 3*B*c*d*e + 2*A*d^2*e + A*c*d*f) + a*b*(3*(c
^2*C + A*d^2)*f - B*d*(d*e + 2*c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*Elliptic
F[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*Sqrt[d]*(-(b*c)
+ a*d)^(3/2)*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[e + f*x])

________________________________________________________________________________________

Rubi [A]  time = 1.51666, antiderivative size = 642, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{\frac{b (e+f x)}{b e-a f}} \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 \sqrt{d} \sqrt{c+d x} \sqrt{e+f x} (a d-b c)^{3/2} (b e-a f)}+\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (a^2 b (B d f-4 C (c f+d e))+2 a^3 C d f+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{3 b \sqrt{a+b x} (b c-a d)^2 (b e-a f)^2}-\frac{2 \sqrt{d} \sqrt{e+f x} \sqrt{\frac{b (c+d x)}{b c-a d}} \left (a^2 b (B d f-4 C (c f+d e))+2 a^3 C d f+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|\frac{(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 \sqrt{c+d x} (a d-b c)^{3/2} (b e-a f)^2 \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \sqrt{c+d x} \sqrt{e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/((a + b*x)^(5/2)*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])/(3*b*(b*c - a*d)*(b*e - a*f)*(a + b*x)^(3/2)) + (2*(2
*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*
(d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*(b*c - a*d)^2*(b*e - a*f)^2*Sqrt[a + b*x]) - (2*Sqrt[d]*(2*a^3
*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*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])/Sqrt[-(b*c) +
a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*(-(b*c) + a*d)^(3/2)*(b*e - a*f)^2*Sqrt[c + d*x]*Sqrt[(b*(e +
f*x))/(b*e - a*f)]) - (2*(a^2*C*d*(d*e - c*f) - b^2*(3*c^2*C*e - 3*B*c*d*e + 2*A*d^2*e + A*c*d*f) + a*b*(3*(c
^2*C + A*d^2)*f - B*d*(d*e + 2*c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*Elliptic
F[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(3*b^2*Sqrt[d]*(-(b*c)
+ a*d)^(3/2)*(b*e - a*f)*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)^{5/2} \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}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}-\frac{2 \int \frac{-\frac{a^2 C (d e+c f)-a b (3 c C e+B d e+B c f-3 A d f)+b^2 (3 B c e-2 A (d e+c f))}{2 b}+\frac{1}{2} \left (-3 b c C e+3 a C d e+3 a c C f+A b d f-a B d f-\frac{2 a^2 C d f}{b}\right ) x}{(a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 (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}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt{a+b x}}+\frac{4 \int \frac{-\frac{a^3 C d f (d e+c f)-b^3 c e (3 c C e-A d f)+a b^2 \left (6 c^2 C e f+A d^2 e f+c d \left (6 C e^2-4 B e f+A f^2\right )\right )-a^2 b \left (C \left (3 d^2 e^2+5 c d e f+3 c^2 f^2\right )+d f (3 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 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) x}{4 b}}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 (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}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{\left (d \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right )\right ) \int \frac{\sqrt{e+f x}}{\sqrt{a+b x} \sqrt{c+d x}} \, dx}{3 b (b c-a d)^2 (b e-a f)^2}-\frac{\left (a^2 C d (d e-c f)-b^2 \left (3 c^2 C e-3 B c d e+2 A d^2 e+A c d f\right )+a b \left (3 \left (c^2 C+A d^2\right ) f-B d (d e+2 c f)\right )\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+f x}} \, dx}{3 b (b c-a d)^2 (b e-a f)}\\ &=-\frac{2 \left (A b^2-a (b B-a C)\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{\left (\left (a^2 C d (d e-c f)-b^2 \left (3 c^2 C e-3 B c d e+2 A d^2 e+A c d f\right )+a b \left (3 \left (c^2 C+A d^2\right ) f-B d (d e+2 c f)\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}{3 b (b c-a d)^2 (b e-a f) \sqrt{c+d x}}-\frac{\left (d \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 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}{3 b (b c-a d)^2 (b e-a f)^2 \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}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \sqrt{d} \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 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 )}{3 b^2 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{\left (\left (a^2 C d (d e-c f)-b^2 \left (3 c^2 C e-3 B c d e+2 A d^2 e+A c d f\right )+a b \left (3 \left (c^2 C+A d^2\right ) f-B d (d e+2 c f)\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}{3 b (b c-a d)^2 (b e-a f) \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}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac{2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt{c+d x} \sqrt{e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt{a+b x}}-\frac{2 \sqrt{d} \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 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 )}{3 b^2 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt{c+d x} \sqrt{\frac{b (e+f x)}{b e-a f}}}-\frac{2 \left (a^2 C d (d e-c f)-b^2 \left (3 c^2 C e-3 B c d e+2 A d^2 e+A c d f\right )+a b \left (3 \left (c^2 C+A d^2\right ) f-B d (d e+2 c f)\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 )}{3 b^2 \sqrt{d} (-b c+a d)^{3/2} (b e-a f) \sqrt{c+d x} \sqrt{e+f x}}\\ \end{align*}

Mathematica [C]  time = 10.9311, size = 699, normalized size = 1.09 \[ -\frac{2 \left (b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left ((a+b x) \left (a^2 b (4 C (c f+d e)-B d f)-2 a^3 C d f-a b^2 (-4 A d f+B c f+B d e+6 c C e)+b^3 (3 B c e-2 A (c f+d e))\right )+(b c-a d) (b e-a f) \left (a (a C-b B)+A b^2\right )\right )+(a+b x) \left (-i b (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (a^2 C f (d e-c f)+a b \left (f (-3 A d f+B c f+2 B d e)-3 C d e^2\right )+b^2 \left (c f (2 A f-3 B e)+A d e f+3 c C e^2\right )\right ) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right ),\frac{b d e-a d f}{b c f-a d f}\right )+b^2 (c+d x) (e+f x) \sqrt{\frac{b c}{d}-a} \left (a^2 b (B d f-4 C (c f+d e))+2 a^3 C d f+a b^2 (-4 A d f+B c f+B d e+6 c C e)+b^3 (2 A (c f+d e)-3 B c e)\right )+i f (a+b x)^{3/2} (b c-a d) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \sqrt{\frac{b (e+f x)}{f (a+b x)}} \left (a^2 b (B d f-4 C (c f+d e))+2 a^3 C d f+a b^2 (-4 A d f+B c f+B d e+6 c C e)+b^3 (2 A (c f+d e)-3 B c e)\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{b c}{d}-a}}{\sqrt{a+b x}}\right )|\frac{b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^3 (a+b x)^{3/2} \sqrt{c+d x} \sqrt{e+f x} \sqrt{\frac{b c}{d}-a} (b c-a d)^2 (b e-a f)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

(-2*(b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)*(b*e - a*f) + (-2*a^3*
C*d*f - a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(-(B*d*f) + 4*C*(d
*e + c*f)))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A
*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d
)*f*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) + a^2*b*(B*d*f
- 4*C*(d*e + c*f)))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ellip
ticE[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)*(a^2*C*f*
(d*e - c*f) + b^2*(3*c*C*e^2 + A*d*e*f + c*f*(-3*B*e + 2*A*f)) + a*b*(-3*C*d*e^2 + f*(2*B*d*e + B*c*f - 3*A*d*
f)))*(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[S
qrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b^3*Sqrt[-a + (b*c)/d]*(b*c - a*d)^2*
(b*e - a*f)^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])

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Maple [B]  time = 0.122, size = 12981, normalized size = 20.2 \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)^(5/2)/(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{C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac{5}{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)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate((C*x^2 + B*x + A)/((b*x + a)^(5/2)*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 x^{2} + B x + A\right )} \sqrt{b x + a} \sqrt{d x + c} \sqrt{f x + e}}{b^{3} d f x^{5} + a^{3} c e +{\left (b^{3} d e +{\left (b^{3} c + 3 \, a b^{2} d\right )} f\right )} x^{4} +{\left ({\left (b^{3} c + 3 \, a b^{2} d\right )} e + 3 \,{\left (a b^{2} c + a^{2} b d\right )} f\right )} x^{3} +{\left (3 \,{\left (a b^{2} c + a^{2} b d\right )} e +{\left (3 \, a^{2} b c + a^{3} d\right )} f\right )} x^{2} +{\left (a^{3} c f +{\left (3 \, a^{2} b c + a^{3} 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)^(5/2)/(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^3*d*f*x^5 + a^3*c*e + (b^3*d*e + (b^3*
c + 3*a*b^2*d)*f)*x^4 + ((b^3*c + 3*a*b^2*d)*e + 3*(a*b^2*c + a^2*b*d)*f)*x^3 + (3*(a*b^2*c + a^2*b*d)*e + (3*
a^2*b*c + a^3*d)*f)*x^2 + (a^3*c*f + (3*a^2*b*c + a^3*d)*e)*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((C*x**2+B*x+A)/(b*x+a)**(5/2)/(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{C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac{5}{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)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

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