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

Optimal. Leaf size=964 \[ -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \sqrt {d} \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right ) \sqrt {e+f x}}{15 b^4 (a d-b c)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 \sqrt {d} (a d-b c)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \]

[Out]

-2/5*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^(5/2)+2/15*(6*a^3*C*d*f+a
*b^2*(-4*A*d*f+3*B*c*f+3*B*d*e+10*C*c*e)-b^3*(5*B*c*e-2*A*(c*f+d*e))-a^2*b*(B*d*f+8*C*(c*f+d*e)))*(f*x+e)^(3/2
)*(d*x+c)^(1/2)/b^2/(-a*d+b*c)/(-a*f+b*e)^2/(b*x+a)^(3/2)+2/15*(24*a^3*C*d^2*f-a^2*b*d*(4*B*d*f+41*C*c*f+23*C*
d*e)-b^3*(15*c^2*C*e-2*A*d^2*e+c*d*(A*f+5*B*e))+a*b^2*(15*c^2*C*f+d^2*(-A*f+3*B*e)+c*(6*B*d*f+40*C*d*e)))*(d*x
+c)^(1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^2/(-a*f+b*e)/(b*x+a)^(1/2)+2/15*(48*a^4*C*d^2*f^2-8*a^3*b*d*f*(B*d*f+11
*C*(c*f+d*e))-b^4*(2*A*d^2*e^2-c*d*e*(2*A*f+5*B*e)-c^2*(-2*A*f^2+5*B*e*f+30*C*e^2))-a*b^3*(d^2*e*(-2*A*f+3*B*e
)+c^2*f*(3*B*f+70*C*e)+2*c*d*(-A*f^2+11*B*e*f+35*C*e^2))+a^2*b^2*(2*C*(19*c^2*f^2+81*c*d*e*f+19*d^2*e^2)-d*f*(
2*A*d*f-13*B*(c*f+d*e))))*EllipticE(d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*d
^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^4/(a*d-b*c)^(3/2)/(-a*f+b*e)^2/(d*x+c)^(1/2)/(b*(f*x+e)/(-
a*f+b*e))^(1/2)+2/15*(-c*f+d*e)*(24*a^3*C*d^2*f-a^2*b*d*(4*B*d*f+41*C*c*f+23*C*d*e)-b^3*(15*c^2*C*e-2*A*d^2*e+
c*d*(A*f+5*B*e))+a*b^2*(15*c^2*C*f+d^2*(-A*f+3*B*e)+c*(6*B*d*f+40*C*d*e)))*EllipticF(d^(1/2)*(b*x+a)^(1/2)/(a*
d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^4/
(a*d-b*c)^(3/2)/(-a*f+b*e)/d^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 3.12, antiderivative size = 964, normalized size of antiderivative = 1.00, 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, 150, 158, 114, 113, 121, 120} \[ -\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}+\frac {2 \sqrt {d} \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (30 C e^2+5 B f e-2 A f^2\right ) c^2-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right ) \sqrt {e+f x}}{15 b^4 (a d-b c)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 \sqrt {d} (a d-b c)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(24*a^3*C*d^2*f - a^2*b*d*(23*C*d*e + 41*c*C*f + 4*B*d*f) - b^3*(15*c^2*C*e - 2*A*d^2*e + c*d*(5*B*e + A*f)
) + a*b^2*(15*c^2*C*f + d^2*(3*B*e - A*f) + c*(40*C*d*e + 6*B*d*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(15*b^3*(b*c
- a*d)^2*(b*e - a*f)*Sqrt[a + b*x]) + (2*(6*a^3*C*d*f + a*b^2*(10*c*C*e + 3*B*d*e + 3*B*c*f - 4*A*d*f) - b^3*
(5*B*c*e - 2*A*(d*e + c*f)) - a^2*b*(B*d*f + 8*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(15*b^2*(b*c - a
*d)*(b*e - a*f)^2*(a + b*x)^(3/2)) - (2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(5*b*(b*c - a
*d)*(b*e - a*f)*(a + b*x)^(5/2)) + (2*Sqrt[d]*(48*a^4*C*d^2*f^2 - 8*a^3*b*d*f*(B*d*f + 11*C*(d*e + c*f)) - b^4
*(2*A*d^2*e^2 - c*d*e*(5*B*e + 2*A*f) - c^2*(30*C*e^2 + 5*B*e*f - 2*A*f^2)) - a*b^3*(d^2*e*(3*B*e - 2*A*f) + c
^2*f*(70*C*e + 3*B*f) + 2*c*d*(35*C*e^2 + 11*B*e*f - A*f^2)) + a^2*b^2*(2*C*(19*d^2*e^2 + 81*c*d*e*f + 19*c^2*
f^2) - d*f*(2*A*d*f - 13*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^4*(-(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*(d*e - c*f)*(24*a^3*C*d^2*f - a^2*b*d*(23*C*d*e + 4
1*c*C*f + 4*B*d*f) - b^3*(15*c^2*C*e - 2*A*d^2*e + c*d*(5*B*e + A*f)) + a*b^2*(15*c^2*C*f + d^2*(3*B*e - A*f)
+ c*(40*C*d*e + 6*B*d*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[ArcSin[(S
qrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(15*b^4*Sqrt[d]*(-(b*c) + a*d)^(3
/2)*(b*e - a*f)*Sqrt[c + d*x]*Sqrt[e + f*x])

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 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 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)
])

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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx &=-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac {2 \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3 a^2 C (d e+c f)-a b (5 c C e+3 B d e+3 B c f-5 A d f)+b^2 (5 B c e-2 A (d e+c f))}{2 b}+\frac {1}{2} \left (a B d f-\frac {6 a^2 C d f}{b}+5 a C (d e+c f)-b (5 c C e+A d f)\right ) x\right )}{(a+b x)^{5/2}} \, dx}{5 (b c-a d) (b e-a f)}\\ &=\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{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} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac {4 \int \frac {\sqrt {e+f x} \left (\frac {6 a^3 C d f (d e+3 c f)-b^3 e \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (30 c^2 C e f+d^2 e (3 B e-4 A f)+c d \left (25 C e^2+6 B e f+3 A f^2\right )\right )-a^2 b \left (B d f (d e+3 c f)+C \left (8 d^2 e^2+41 c d e f+15 c^2 f^2\right )\right )}{4 b}+\frac {d \left (24 a^3 C d f^2-a^2 b f (41 C d e+23 c C f+4 B d f)-b^3 \left (15 c C e^2+A d e f+c f (5 B e-2 A f)\right )+a b^2 (5 C e (3 d e+8 c f)+f (6 B d e+3 B c f-A d f))\right ) x}{4 b}\right )}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{15 b (b c-a d) (b e-a f)^2}\\ &=\frac {2 \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{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} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}-\frac {8 \int \frac {-\frac {24 a^4 C d^2 f^2 (d e+c f)+b^4 c e \left (15 c^2 C e f-A d^2 e f+c d \left (15 C e^2+10 B e f-A f^2\right )\right )-a b^3 \left (A d^3 e^2 f+30 c^3 C e f^2+2 c d^2 e \left (15 C e^2+7 B e f-3 A f^2\right )+c^2 d f \left (80 C e^2+14 B e f+A f^2\right )\right )-a^3 b d f \left (4 B d f (d e+c f)+C \left (41 d^2 e^2+94 c d e f+41 c^2 f^2\right )\right )+a^2 b^2 \left (C \left (15 d^3 e^3+104 c d^2 e^2 f+104 c^2 d e f^2+15 c^3 f^3\right )-d f \left (A d f (d e+c f)-2 B \left (3 d^2 e^2+7 c d e f+3 c^2 f^2\right )\right )\right )}{8 b}-\frac {d f \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (2 A d^2 e^2-c d e (5 B e+2 A f)-c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )-d f (2 A d f-13 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^2 (b c-a d)^2 (b e-a f)^2}\\ &=\frac {2 \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{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} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {\left ((d e-c f) \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 b^3 (b c-a d)^2 (b e-a f)}+\frac {\left (d \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (2 A d^2 e^2-c d e (5 B e+2 A f)-c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )-d f (2 A d f-13 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^3 (b c-a d)^2 (b e-a f)^2}\\ &=\frac {2 \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{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} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {\left ((d e-c f) \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d 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}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {c+d x}}+\frac {\left (d \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (2 A d^2 e^2-c d e (5 B e+2 A f)-c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )-d f (2 A d f-13 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^3 (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 (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{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} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \sqrt {d} \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (2 A d^2 e^2-c d e (5 B e+2 A f)-c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )-d f (2 A d f-13 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^4 (-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 ((d e-c f) \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d 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}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x}}\\ &=\frac {2 \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{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} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \sqrt {d} \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (2 A d^2 e^2-c d e (5 B e+2 A f)-c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )-d f (2 A d f-13 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^4 (-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 (d e-c f) \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d 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 )}{15 b^4 \sqrt {d} (-b c+a d)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}

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Mathematica [C]  time = 16.42, size = 9529, normalized size = 9.88 \[ \text {Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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fricas [F]  time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b^{4} x^{4} + 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.24, size = 34389, normalized size = 35.67 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{7/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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