\(\int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx\) [81]

Optimal result

Integrand size = 36, antiderivative size = 403 \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{2 b (b c-a d) (b e-a f) (a+b x)^2}+\frac {\left (2 a^3 C d f+a b^2 (8 c C e+B d e+B c f-6 A d f)-b^3 (4 B c e-3 A (d e+c f))+a^2 b (2 B d f-5 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f)^2 (a+b x)}-\frac {\left (2 (2 b c e-a (d e+c f)) \left (a B d f+\frac {a^2 C d f}{b}-2 a C (d e+c f)+b (2 c C e-A d f)\right )-\frac {(b d e+b c f-2 a d f) \left (a^2 C (d e+c f)-a b (4 c C e+B d e+B c f-4 A d f)+b^2 (4 B c e-3 A (d e+c f))\right )}{b}\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{4 (b c-a d)^{5/2} (b e-a f)^{5/2}} \] Output:

-1/2*(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)^2+1/4*(2*a^3*C*d*f+a*b^2*(-6*A*d*f+B*c*f+B*d*e+8*C*c*e)-b^3*(4* 
B*c*e-3*A*(c*f+d*e))+a^2*b*(2*B*d*f-5*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)-1/4*(2*(2*b*c*e-a*(c*f+d*e))*(a* 
B*d*f+a^2*C*d*f/b-2*a*C*(c*f+d*e)+b*(-A*d*f+2*C*c*e))-(-2*a*d*f+b*c*f+b*d* 
e)*(a^2*C*(c*f+d*e)-a*b*(-4*A*d*f+B*c*f+B*d*e+4*C*c*e)+b^2*(4*B*c*e-3*A*(c 
*f+d*e)))/b)*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)^(5/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 2.44 (sec) , antiderivative size = 420, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {1}{4} \left (-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (4 b^3 B c e x-a b^2 (8 c C e x+B (-2 c e+d e x+c f x))+a^2 b (5 C d e x+B d (e-2 f x)+c (-6 C e+B f+5 C f x))+a^3 (-4 B d f+C (3 d e+3 c f-2 d f x))+A b \left (8 a^2 d f+b^2 (2 c e-3 d e x-3 c f x)+a b (-5 d e-5 c f+6 d f x)\right )\right )}{(b c-a d)^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (b^2 \left (3 A d^2 e^2+2 c d e (-2 B e+A f)+c^2 \left (8 C e^2-4 B e f+3 A f^2\right )\right )+a b \left (d^2 e (B e-8 A f)+c^2 f (-8 C e+B f)-2 c d \left (4 C e^2-7 B e f+4 A f^2\right )\right )+a^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )\right ) \arctan \left (\frac {\sqrt {b c-a d} \sqrt {e+f x}}{\sqrt {-b e+a f} \sqrt {c+d x}}\right )}{(b c-a d)^{5/2} (-b e+a f)^{5/2}}\right ) \] Input:

Integrate[(A + B*x + C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

(-((Sqrt[c + d*x]*Sqrt[e + f*x]*(4*b^3*B*c*e*x - a*b^2*(8*c*C*e*x + B*(-2* 
c*e + d*e*x + c*f*x)) + a^2*b*(5*C*d*e*x + B*d*(e - 2*f*x) + c*(-6*C*e + B 
*f + 5*C*f*x)) + a^3*(-4*B*d*f + C*(3*d*e + 3*c*f - 2*d*f*x)) + A*b*(8*a^2 
*d*f + b^2*(2*c*e - 3*d*e*x - 3*c*f*x) + a*b*(-5*d*e - 5*c*f + 6*d*f*x)))) 
/((b*c - a*d)^2*(b*e - a*f)^2*(a + b*x)^2)) + ((b^2*(3*A*d^2*e^2 + 2*c*d*e 
*(-2*B*e + A*f) + c^2*(8*C*e^2 - 4*B*e*f + 3*A*f^2)) + a*b*(d^2*e*(B*e - 8 
*A*f) + c^2*f*(-8*C*e + B*f) - 2*c*d*(4*C*e^2 - 7*B*e*f + 4*A*f^2)) + a^2* 
(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)))) 
*ArcTan[(Sqrt[b*c - a*d]*Sqrt[e + f*x])/(Sqrt[-(b*e) + a*f]*Sqrt[c + d*x]) 
])/((b*c - a*d)^(5/2)*(-(b*e) + a*f)^(5/2)))/4
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.10, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2116, 27, 168, 27, 104, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 2116

\(\displaystyle -\frac {\int -\frac {C (d e+c f) a^2-b (4 c C e+B d e+B c f-4 A d f) a+b^2 (4 B c e-3 A (d e+c f))+2 b \left (\frac {C d f a^2}{b}-2 C d e a-2 c C f a+B d f a+2 b c C e-A b d f\right ) x}{2 b (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {C (d e+c f) a^2-b (4 c C e+B d e+B c f-4 A d f) a+b^2 (4 B c e-3 A (d e+c f))+2 b \left (\frac {C d f a^2}{b}+B d f a-2 C (d e+c f) a+b (2 c C e-A d f)\right ) x}{(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}dx}{4 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{(a+b x) (b c-a d) (b e-a f)}-\frac {\int -\frac {b \left (\left (C \left (3 d^2 e^2+2 c d f e+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right ) a^2+b \left (-f (8 C e-B f) c^2-2 d \left (4 C e^2-7 B f e+4 A f^2\right ) c+d^2 e (B e-8 A f)\right ) a+b^2 \left (\left (8 C e^2-4 B f e+3 A f^2\right ) c^2-2 d e (2 B e-A f) c+3 A d^2 e^2\right )\right )}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 (b c-a d) (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{(a+b x) (b c-a d) (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {\frac {b \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{(b c-a d) (b e-a f)}+\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{(a+b x) (b c-a d) (b e-a f)}}{4 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (2 B d f-5 C (c f+d e))+a b^2 (-6 A d f+B c f+B d e+8 c C e)-b^3 (4 B c e-3 A (c f+d e))\right )}{(a+b x) (b c-a d) (b e-a f)}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (a^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+a b \left (-2 c d \left (4 A f^2-7 B e f+4 C e^2\right )+d^2 e (B e-8 A f)+c^2 (-f) (8 C e-B f)\right )+b^2 \left (c^2 \left (3 A f^2-4 B e f+8 C e^2\right )-2 c d e (2 B e-A f)+3 A d^2 e^2\right )\right )}{(b c-a d)^{3/2} (b e-a f)^{3/2}}}{4 b (b c-a d) (b e-a f)}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{2 b (a+b x)^2 (b c-a d) (b e-a f)}\)

Input:

Int[(A + B*x + C*x^2)/((a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
 

Output:

-1/2*((A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(b*(b*c - a*d)* 
(b*e - a*f)*(a + b*x)^2) + (((2*a^3*C*d*f + a*b^2*(8*c*C*e + B*d*e + B*c*f 
 - 6*A*d*f) - b^3*(4*B*c*e - 3*A*(d*e + c*f)) + a^2*b*(2*B*d*f - 5*C*(d*e 
+ c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*(a + b*x)) 
- (b*(b^2*(3*A*d^2*e^2 - 2*c*d*e*(2*B*e - A*f) + c^2*(8*C*e^2 - 4*B*e*f + 
3*A*f^2)) + a*b*(d^2*e*(B*e - 8*A*f) - c^2*f*(8*C*e - B*f) - 2*c*d*(4*C*e^ 
2 - 7*B*e*f + 4*A*f^2)) + a^2*(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))))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d*x])/(Sq 
rt[b*c - a*d]*Sqrt[e + f*x])])/((b*c - a*d)^(3/2)*(b*e - a*f)^(3/2)))/(4*b 
*(b*c - a*d)*(b*e - a*f))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] + S 
imp[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] && ILtQ[m, -1]
 

rule 221

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

rule 2116

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] + Si 
mp[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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(7118\) vs. \(2(377)=754\).

Time = 1.12 (sec) , antiderivative size = 7119, normalized size of antiderivative = 17.67

method	result	size
default	\(\text {Expression too large to display}\)	\(7119\)

Input:

int((C*x^2+B*x+A)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETURNVE 
RBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1998 vs. \(2 (376) = 752\).

Time = 157.37 (sec) , antiderivative size = 4058, normalized size of antiderivative = 10.07 \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm 
="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm 
="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume((a*d-b*c)>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7939 vs. \(2 (376) = 752\).

Time = 33.36 (sec) , antiderivative size = 7939, normalized size of antiderivative = 19.70 \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/(b*x+a)^3/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algorithm 
="giac")
 

Output:

-1/4*(8*sqrt(d*f)*C*b^2*c^2*d^2*e^2 - 8*sqrt(d*f)*C*a*b*c*d^3*e^2 - 4*sqrt 
(d*f)*B*b^2*c*d^3*e^2 + 3*sqrt(d*f)*C*a^2*d^4*e^2 + sqrt(d*f)*B*a*b*d^4*e^ 
2 + 3*sqrt(d*f)*A*b^2*d^4*e^2 - 8*sqrt(d*f)*C*a*b*c^2*d^2*e*f - 4*sqrt(d*f 
)*B*b^2*c^2*d^2*e*f + 2*sqrt(d*f)*C*a^2*c*d^3*e*f + 14*sqrt(d*f)*B*a*b*c*d 
^3*e*f + 2*sqrt(d*f)*A*b^2*c*d^3*e*f - 4*sqrt(d*f)*B*a^2*d^4*e*f - 8*sqrt( 
d*f)*A*a*b*d^4*e*f + 3*sqrt(d*f)*C*a^2*c^2*d^2*f^2 + sqrt(d*f)*B*a*b*c^2*d 
^2*f^2 + 3*sqrt(d*f)*A*b^2*c^2*d^2*f^2 - 4*sqrt(d*f)*B*a^2*c*d^3*f^2 - 8*s 
qrt(d*f)*A*a*b*c*d^3*f^2 + 8*sqrt(d*f)*A*a^2*d^4*f^2)*arctan(-1/2*(b*d^2*e 
 + b*c*d*f - 2*a*d^2*f - (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c) 
*d*f - c*d*f))^2*b)/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d 
^2*f^2)*d))/((b^4*c^2*e^2*abs(d) - 2*a*b^3*c*d*e^2*abs(d) + a^2*b^2*d^2*e^ 
2*abs(d) - 2*a*b^3*c^2*e*f*abs(d) + 4*a^2*b^2*c*d*e*f*abs(d) - 2*a^3*b*d^2 
*e*f*abs(d) + a^2*b^2*c^2*f^2*abs(d) - 2*a^3*b*c*d*f^2*abs(d) + a^4*d^2*f^ 
2*abs(d))*sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - a^2*d^2*f^2)*d) 
+ 1/2*(8*sqrt(d*f)*C*a*b^4*c*d^9*e^5 - 4*sqrt(d*f)*B*b^5*c*d^9*e^5 - 5*sqr 
t(d*f)*C*a^2*b^3*d^10*e^5 + sqrt(d*f)*B*a*b^4*d^10*e^5 + 3*sqrt(d*f)*A*b^5 
*d^10*e^5 - 32*sqrt(d*f)*C*a*b^4*c^2*d^8*e^4*f + 16*sqrt(d*f)*B*b^5*c^2*d^ 
8*e^4*f + 15*sqrt(d*f)*C*a^2*b^3*c*d^9*e^4*f - 3*sqrt(d*f)*B*a*b^4*c*d^9*e 
^4*f - 9*sqrt(d*f)*A*b^5*c*d^9*e^4*f + 2*sqrt(d*f)*C*a^3*b^2*d^10*e^4*f + 
2*sqrt(d*f)*B*a^2*b^3*d^10*e^4*f - 6*sqrt(d*f)*A*a*b^4*d^10*e^4*f + 48*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Hanged} \] Input:

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

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{(a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {C \,x^{2}+B x +A}{\left (b x +a \right )^{3} \sqrt {d x +c}\, \sqrt {f x +e}}d x \] Input:

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

Output:

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