3.60 \(\int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx\)

Optimal. Leaf size=826 \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}+\frac {\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {\left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2} \]

[Out]

1/8*(b^3*(5*A*d^3*e^3-3*c*d^2*e^2*(-A*f+2*B*e)+c^2*d*e*(3*A*f^2-4*B*e*f+8*C*e^2)+c^3*f*(5*A*f^2-6*B*e*f+8*C*e^
2))+a*b^2*(d^3*e^2*(-18*A*f+B*e)-c^3*f^2*(-B*f+4*C*e)-c*d^2*e*(12*A*f^2-23*B*e*f+4*C*e^2)-c^2*d*f*(18*A*f^2-23
*B*e*f+40*C*e^2))-2*a^3*d*f*(C*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+4*d*f*(2*A*d*f-B*(c*f+d*e)))+a^2*b*(C*(c^3*f^3+
23*c^2*d*e*f^2+23*c*d^2*e^2*f+d^3*e^3)+4*d*f*(6*A*d*f*(c*f+d*e)-B*(c^2*f^2+10*c*d*e*f+d^2*e^2))))*arctanh((-a*
f+b*e)^(1/2)*(d*x+c)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/(-a*d+b*c)^(7/2)/(-a*f+b*e)^(7/2)-1/3*(A*b^2-a*(B*b
-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)^3+1/12*(2*a^3*C*d*f+a*b^2*(-10*A*d*f+B*c*f+
B*d*e+12*C*c*e)-b^3*(6*B*c*e-5*A*(c*f+d*e))+a^2*b*(4*B*d*f-7*C*(c*f+d*e)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d
+b*c)^2/(-a*f+b*e)^2/(b*x+a)^2+1/24*(4*a^4*C*d^2*f^2+8*a^3*b*d*f*(B*d*f-2*C*(c*f+d*e))-b^4*(15*A*d^2*e^2-2*c*d
*e*(-7*A*f+9*B*e)+3*c^2*(5*A*f^2-6*B*e*f+8*C*e^2))-a*b^3*(d^2*e*(-44*A*f+3*B*e)-3*c^2*f*(-B*f+4*C*e)-2*c*d*(22
*A*f^2-29*B*e*f+6*C*e^2))-a^2*b^2*(C*(3*c^2*f^2-34*c*d*e*f+3*d^2*e^2)+2*d*f*(22*A*d*f-5*B*(c*f+d*e))))*(d*x+c)
^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)^3/(-a*f+b*e)^3/(b*x+a)

________________________________________________________________________________________

Rubi [A]  time = 2.43, antiderivative size = 826, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {1613, 151, 12, 93, 208} \[ -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (b c-a d) (b e-a f) (a+b x)^3}+\frac {\left (-2 d f \left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^3+b \left (C \left (d^3 e^3+23 c d^2 f e^2+23 c^2 d f^2 e+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d f e+c^2 f^2\right )\right )\right ) a^2+b^2 \left (-f^2 (4 C e-B f) c^3-d f \left (40 C e^2-23 B f e+18 A f^2\right ) c^2-d^2 e \left (4 C e^2-23 B f e+12 A f^2\right ) c+d^3 e^2 (B e-18 A f)\right ) a+b^3 \left (f \left (8 C e^2-6 B f e+5 A f^2\right ) c^3+d e \left (8 C e^2-4 B f e+3 A f^2\right ) c^2-3 d^2 e^2 (2 B e-A f) c+5 A d^3 e^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}+\frac {\left (4 C d^2 f^2 a^4+8 b d f (B d f-2 C (d e+c f)) a^3-b^2 \left (C \left (3 d^2 e^2-34 c d f e+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right ) a^2-b^3 \left (-3 f (4 C e-B f) c^2-2 d \left (6 C e^2-29 B f e+22 A f^2\right ) c+d^2 e (3 B e-44 A f)\right ) a-b^4 \left (3 \left (8 C e^2-6 B f e+5 A f^2\right ) c^2-2 d e (9 B e-7 A f) c+15 A d^2 e^2\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {\left (2 C d f a^3+b (4 B d f-7 C (d e+c f)) a^2+b^2 (12 c C e+B d e+B c f-10 A d f) a-b^3 (6 B c e-5 A (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/((a + b*x)^4*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

-((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*a^3*C*d
*f + a*b^2*(12*c*C*e + B*d*e + B*c*f - 10*A*d*f) - b^3*(6*B*c*e - 5*A*(d*e + c*f)) + a^2*b*(4*B*d*f - 7*C*(d*e
+ c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/(12*b*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^2) + ((4*a^4*C*d^2*f^2 + 8*
a^3*b*d*f*(B*d*f - 2*C*(d*e + c*f)) - b^4*(15*A*d^2*e^2 - 2*c*d*e*(9*B*e - 7*A*f) + 3*c^2*(8*C*e^2 - 6*B*e*f +
5*A*f^2)) - a*b^3*(d^2*e*(3*B*e - 44*A*f) - 3*c^2*f*(4*C*e - B*f) - 2*c*d*(6*C*e^2 - 29*B*e*f + 22*A*f^2)) -
a^2*b^2*(C*(3*d^2*e^2 - 34*c*d*e*f + 3*c^2*f^2) + 2*d*f*(22*A*d*f - 5*B*(d*e + c*f))))*Sqrt[c + d*x]*Sqrt[e +
f*x])/(24*b*(b*c - a*d)^3*(b*e - a*f)^3*(a + b*x)) + ((b^3*(5*A*d^3*e^3 - 3*c*d^2*e^2*(2*B*e - A*f) + c^2*d*e*
(8*C*e^2 - 4*B*e*f + 3*A*f^2) + c^3*f*(8*C*e^2 - 6*B*e*f + 5*A*f^2)) + a*b^2*(d^3*e^2*(B*e - 18*A*f) - c^3*f^2
*(4*C*e - B*f) - c*d^2*e*(4*C*e^2 - 23*B*e*f + 12*A*f^2) - c^2*d*f*(40*C*e^2 - 23*B*e*f + 18*A*f^2)) - 2*a^3*d
*f*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f))) + a^2*b*(C*(d^3*e^3 + 23*c*d^2*e^
2*f + 23*c^2*d*e*f^2 + c^3*f^3) + 4*d*f*(6*A*d*f*(d*e + c*f) - B*(d^2*e^2 + 10*c*d*e*f + c^2*f^2))))*ArcTanh[(
Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(8*(b*c - a*d)^(7/2)*(b*e - a*f)^(7/2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 151

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] && IntegerQ[m]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 1613

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] && ILtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{(a+b x)^4 \sqrt {c+d x} \sqrt {e+f x}} \, dx &=-\frac {\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}-\frac {\int \frac {-\frac {a^2 C (d e+c f)-a b (6 c C e+B d e+B c f-6 A d f)+b^2 (6 B c e-5 A (d e+c f))}{2 b}+\left (-3 b c C e+3 a C d e+3 a c C f+2 A b d f-2 a B d f-\frac {a^2 C d f}{b}\right ) x}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx}{3 (b c-a d) (b e-a f)}\\ &=-\frac {\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}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\int \frac {\frac {2 a^3 C d f (d e+c f)+b^3 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^2 e (3 B e-34 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-23 B e f+17 A f^2\right )\right )+a^2 b \left (C \left (3 d^2 e^2-10 c d e f+3 c^2 f^2\right )+8 d f (3 A d f-B (d e+c f))\right )}{4 b}+\frac {d f \left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) x}{2 b}}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 (b c-a d)^2 (b e-a f)^2}\\ &=-\frac {\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}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\int \frac {3 \left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right )}{8 (a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{6 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac {\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}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}} \, dx}{16 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac {\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}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}-\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-b c+a d-(-b e+a f) x^2} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{8 (b c-a d)^3 (b e-a f)^3}\\ &=-\frac {\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}+\frac {\left (2 a^3 C d f+a b^2 (12 c C e+B d e+B c f-10 A d f)-b^3 (6 B c e-5 A (d e+c f))+a^2 b (4 B d f-7 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{12 b (b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (4 a^4 C d^2 f^2+8 a^3 b d f (B d f-2 C (d e+c f))-b^4 \left (15 A d^2 e^2-2 c d e (9 B e-7 A f)+3 c^2 \left (8 C e^2-6 B e f+5 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-44 A f)-3 c^2 f (4 C e-B f)-2 c d \left (6 C e^2-29 B e f+22 A f^2\right )\right )-a^2 b^2 \left (C \left (3 d^2 e^2-34 c d e f+3 c^2 f^2\right )+2 d f (22 A d f-5 B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{24 b (b c-a d)^3 (b e-a f)^3 (a+b x)}+\frac {\left (b^3 \left (5 A d^3 e^3-3 c d^2 e^2 (2 B e-A f)+c^2 d e \left (8 C e^2-4 B e f+3 A f^2\right )+c^3 f \left (8 C e^2-6 B e f+5 A f^2\right )\right )+a b^2 \left (d^3 e^2 (B e-18 A f)-c^3 f^2 (4 C e-B f)-c d^2 e \left (4 C e^2-23 B e f+12 A f^2\right )-c^2 d f \left (40 C e^2-23 B e f+18 A f^2\right )\right )-2 a^3 d f \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )+a^2 b \left (C \left (d^3 e^3+23 c d^2 e^2 f+23 c^2 d e f^2+c^3 f^3\right )+4 d f \left (6 A d f (d e+c f)-B \left (d^2 e^2+10 c d e f+c^2 f^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{8 (b c-a d)^{7/2} (b e-a f)^{7/2}}\\ \end {align*}

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Mathematica [A]  time = 6.11, size = 794, normalized size = 0.96 \[ -\frac {\frac {\left (a (a C-b B)+A b^2\right ) \left (\frac {b \sqrt {c+d x} \sqrt {e+f x} \left (44 a^2 d^2 f^2-44 a b d f (c f+d e)+b^2 \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )}{(a+b x) (b c-a d) (b e-a f)}+\frac {3 \left (8 a^2 d^2 f^2-8 a b d f (c f+d e)+b^2 \left (5 c^2 f^2-2 c d e f+5 d^2 e^2\right )\right ) (-2 a d f+b c f+b d e) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{(a d-b c)^{3/2} (a f-b e)^{3/2}}-\frac {10 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{(a+b x)^2}\right )}{2 (b c-a d)^2 (b e-a f)^2}-\frac {3 (b B-2 a C) \left (\frac {\left (8 a^2 d^2 f^2-8 a b d f (c f+d e)+b^2 \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{(a d-b c)^{3/2} (a f-b e)^{3/2}}+\frac {3 b \sqrt {c+d x} \sqrt {e+f x} (-2 a d f+b c f+b d e)}{(a+b x) (b c-a d) (b e-a f)}\right )}{(b c-a d) (b e-a f)}+\frac {4 b \sqrt {c+d x} \sqrt {e+f x} \left (a (a C-b B)+A b^2\right )}{(a+b x)^3 (b c-a d) (b e-a f)}+\frac {6 b \sqrt {c+d x} \sqrt {e+f x} (b B-2 a C)}{(a+b x)^2 (b c-a d) (b e-a f)}+\frac {12 b C \sqrt {c+d x} \sqrt {e+f x}}{(a+b x) (b c-a d) (b e-a f)}+\frac {12 C (-2 a d f+b c f+b d e) \tanh ^{-1}\left (\frac {\sqrt {c+d x} \sqrt {a f-b e}}{\sqrt {e+f x} \sqrt {a d-b c}}\right )}{(a d-b c)^{3/2} (a f-b e)^{3/2}}}{12 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/((a + b*x)^4*Sqrt[c + d*x]*Sqrt[e + f*x]),x]

[Out]

-1/12*((4*b*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^3) + (6
*b*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)^2) + (12*b*C*Sqrt[c + d*x]*Sq
rt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (12*C*(b*d*e + b*c*f - 2*a*d*f)*ArcTanh[(Sqrt[-(b*e) + a*f]
*Sqrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2)) - (3*(b*B - 2
*a*C)*((3*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((8*a
^2*d^2*f^2 - 8*a*b*d*f*(d*e + c*f) + b^2*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt
[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2))))/((b*c - a*d)*(b*
e - a*f)) + ((A*b^2 + a*(-(b*B) + a*C))*((-10*b*(b*d*e + b*c*f - 2*a*d*f)*Sqrt[c + d*x]*Sqrt[e + f*x])/(a + b*
x)^2 + (b*(44*a^2*d^2*f^2 - 44*a*b*d*f*(d*e + c*f) + b^2*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))*Sqrt[c + d*x]
*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + (3*(b*d*e + b*c*f - 2*a*d*f)*(8*a^2*d^2*f^2 - 8*a*b*d*f*
(d*e + c*f) + b^2*(5*d^2*e^2 - 2*c*d*e*f + 5*c^2*f^2))*ArcTanh[(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x])/(Sqrt[-(b*c)
+ a*d]*Sqrt[e + f*x])])/((-(b*c) + a*d)^(3/2)*(-(b*e) + a*f)^(3/2))))/(2*(b*c - a*d)^2*(b*e - a*f)^2))/b^2

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fricas [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)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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giac [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)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.31, size = 18802, normalized size = 22.76 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/(b*x+a)^4/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume((a*d-b*c)>0)', see `assume?` f
or more details)Is (a*d-b*c)    *(a*f-b*e)    positive, negative or zero?

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

\text{Hanged}

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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)/(b*x+a)**4/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Timed out

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