\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx\) [46]

Optimal result

Integrand size = 29, antiderivative size = 204 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\frac {(d g-c h) \sqrt {e+f x}}{d (b c-a d) (c+d x)}-\frac {2 \sqrt {b e-a f} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d)^2}-\frac {\left (a d (d f g+2 d e h-3 c f h)-b \left (2 d^2 e g-c d f g-c^2 f h\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (b c-a d)^2 \sqrt {d e-c f}} \] Output:

(-c*h+d*g)*(f*x+e)^(1/2)/d/(-a*d+b*c)/(d*x+c)-2*(-a*f+b*e)^(1/2)*(-a*h+b*g 
)*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(1/2)/(-a*d+b*c)^2-(a* 
d*(-3*c*f*h+2*d*e*h+d*f*g)-b*(-c^2*f*h-c*d*f*g+2*d^2*e*g))*arctanh(d^(1/2) 
*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(3/2)/(-a*d+b*c)^2/(-c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\frac {(-d g+c h) \sqrt {e+f x}}{d (-b c+a d) (c+d x)}-\frac {2 \sqrt {-b e+a f} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{\sqrt {b} (b c-a d)^2}+\frac {\left (a d (d f g+2 d e h-3 c f h)+b \left (-2 d^2 e g+c d f g+c^2 f h\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (b c-a d)^2 \sqrt {-d e+c f}} \] Input:

Integrate[(Sqrt[e + f*x]*(g + h*x))/((a + b*x)*(c + d*x)^2),x]
 

Output:

((-(d*g) + c*h)*Sqrt[e + f*x])/(d*(-(b*c) + a*d)*(c + d*x)) - (2*Sqrt[-(b* 
e) + a*f]*(b*g - a*h)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/ 
(Sqrt[b]*(b*c - a*d)^2) + ((a*d*(d*f*g + 2*d*e*h - 3*c*f*h) + b*(-2*d^2*e* 
g + c*d*f*g + c^2*f*h))*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]] 
)/(d^(3/2)*(b*c - a*d)^2*Sqrt[-(d*e) + c*f])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.09, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {166, 27, 174, 73, 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.

\(\displaystyle \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {\sqrt {e+f x} (d g-c h)}{d (c+d x) (b c-a d)}-\frac {\int -\frac {2 b d e g-a (d f g+2 d e h-c f h)+f (b d g+b c h-2 a d h) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{d (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {2 b d e g+a c f h-a d (f g+2 e h)+f (b d g+b c h-2 a d h) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{2 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{d (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\left (a d (-3 c f h+2 d e h+d f g)-b \left (c^2 (-f) h-c d f g+2 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}+\frac {2 d (b e-a f) (b g-a h) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}}{2 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{d (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 \left (a d (-3 c f h+2 d e h+d f g)-b \left (c^2 (-f) h-c d f g+2 d^2 e g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}+\frac {4 d (b e-a f) (b g-a h) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{2 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{d (c+d x) (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a d (-3 c f h+2 d e h+d f g)-b \left (c^2 (-f) h-c d f g+2 d^2 e g\right )\right )}{\sqrt {d} (b c-a d) \sqrt {d e-c f}}-\frac {4 d \sqrt {b e-a f} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d)}}{2 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{d (c+d x) (b c-a d)}\)

Input:

Int[(Sqrt[e + f*x]*(g + h*x))/((a + b*x)*(c + d*x)^2),x]
 

Output:

((d*g - c*h)*Sqrt[e + f*x])/(d*(b*c - a*d)*(c + d*x)) + ((-4*d*Sqrt[b*e - 
a*f]*(b*g - a*h)*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b 
]*(b*c - a*d)) - (2*(a*d*(d*f*g + 2*d*e*h - 3*c*f*h) - b*(2*d^2*e*g - c*d* 
f*g - c^2*f*h))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(Sqrt[d] 
*(b*c - a*d)*Sqrt[d*e - c*f]))/(2*d*(b*c - a*d))
 

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 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 166

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[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] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

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]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.11

method	result	size
derivativedivides	\(2 f \left (\frac {\left (a f -b e \right ) \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}-\frac {-\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 d \left (\left (f x +e \right ) d +c f -d e \right )}+\frac {\left (3 a c d f h -2 a \,d^{2} e h -a \,d^{2} f g -b \,c^{2} f h -b c d f g +2 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 d \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{2} f}\right )\)	\(226\)
default	\(2 f \left (\frac {\left (a f -b e \right ) \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}-\frac {-\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 d \left (\left (f x +e \right ) d +c f -d e \right )}+\frac {\left (3 a c d f h -2 a \,d^{2} e h -a \,d^{2} f g -b \,c^{2} f h -b c d f g +2 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 d \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{2} f}\right )\)	\(226\)
pseudoelliptic	\(\frac {-3 \sqrt {\left (a f -b e \right ) b}\, \left (x d +c \right ) \left (\frac {2 \left (b e g -a \left (e h +\frac {f g}{2}\right )\right ) d^{2}}{3}+c f \left (a h -\frac {b g}{3}\right ) d -\frac {b \,c^{2} f h}{3}\right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (2 d \left (a h -b g \right ) \left (a f -b e \right ) \left (x d +c \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right ) \sqrt {f x +e}\, \left (c h -d g \right )\right )}{\sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (a d -b c \right )^{2} \left (x d +c \right ) d}\)	\(226\)

Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)/(d*x+c)^2,x,method=_RETURNVERBOSE)
 

Output:

2*f*((a*f-b*e)/f*(a*h-b*g)/(a*d-b*c)^2/((a*f-b*e)*b)^(1/2)*arctan(b*(f*x+e 
)^(1/2)/((a*f-b*e)*b)^(1/2))-1/(a*d-b*c)^2/f*(-1/2*f*(a*c*d*h-a*d^2*g-b*c^ 
2*h+b*c*d*g)/d*(f*x+e)^(1/2)/((f*x+e)*d+c*f-d*e)+1/2*(3*a*c*d*f*h-2*a*d^2* 
e*h-a*d^2*f*g-b*c^2*f*h-b*c*d*f*g+2*b*d^2*e*g)/d/((c*f-d*e)*d)^(1/2)*arcta 
n(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (182) = 364\).

Time = 1.57 (sec) , antiderivative size = 2088, normalized size of antiderivative = 10.24 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/2*(2*((b*c*d^3*e - b*c^2*d^2*f)*g - (a*c*d^3*e - a*c^2*d^2*f)*h + ((b* 
d^4*e - b*c*d^3*f)*g - (a*d^4*e - a*c*d^3*f)*h)*x)*sqrt((b*e - a*f)/b)*log 
((b*f*x + 2*b*e - a*f + 2*sqrt(f*x + e)*b*sqrt((b*e - a*f)/b))/(b*x + a)) 
- sqrt(d^2*e - c*d*f)*((2*b*c*d^2*e - (b*c^2*d + a*c*d^2)*f)*g - (2*a*c*d^ 
2*e + (b*c^3 - 3*a*c^2*d)*f)*h + ((2*b*d^3*e - (b*c*d^2 + a*d^3)*f)*g - (2 
*a*d^3*e + (b*c^2*d - 3*a*c*d^2)*f)*h)*x)*log((d*f*x + 2*d*e - c*f + 2*sqr 
t(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) - 2*(((b*c*d^3 - a*d^4)*e - (b* 
c^2*d^2 - a*c*d^3)*f)*g - ((b*c^2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2) 
*f)*h)*sqrt(f*x + e))/((b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e - (b^2* 
c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*f + ((b^2*c^2*d^4 - 2*a*b*c*d^5 + a 
^2*d^6)*e - (b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*f)*x), -1/2*(4*((b*c 
*d^3*e - b*c^2*d^2*f)*g - (a*c*d^3*e - a*c^2*d^2*f)*h + ((b*d^4*e - b*c*d^ 
3*f)*g - (a*d^4*e - a*c*d^3*f)*h)*x)*sqrt(-(b*e - a*f)/b)*arctan(-sqrt(f*x 
 + e)*b*sqrt(-(b*e - a*f)/b)/(b*e - a*f)) - sqrt(d^2*e - c*d*f)*((2*b*c*d^ 
2*e - (b*c^2*d + a*c*d^2)*f)*g - (2*a*c*d^2*e + (b*c^3 - 3*a*c^2*d)*f)*h + 
 ((2*b*d^3*e - (b*c*d^2 + a*d^3)*f)*g - (2*a*d^3*e + (b*c^2*d - 3*a*c*d^2) 
*f)*h)*x)*log((d*f*x + 2*d*e - c*f + 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/ 
(d*x + c)) - 2*(((b*c*d^3 - a*d^4)*e - (b*c^2*d^2 - a*c*d^3)*f)*g - ((b*c^ 
2*d^2 - a*c*d^3)*e - (b*c^3*d - a*c^2*d^2)*f)*h)*sqrt(f*x + e))/((b^2*c^3* 
d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e - (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2...
 

Sympy [F(-1)]

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

integrate((f*x+e)**(1/2)*(h*x+g)/(b*x+a)/(d*x+c)**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)/(d*x+c)^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*f-b*e>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\frac {2 \, {\left (b^{2} e g - a b f g - a b e h + a^{2} f h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} e + a b f}} - \frac {{\left (2 \, b d^{2} e g - b c d f g - a d^{2} f g - 2 \, a d^{2} e h - b c^{2} f h + 3 \, a c d f h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {-d^{2} e + c d f}} + \frac {\sqrt {f x + e} d f g - \sqrt {f x + e} c f h}{{\left (b c d - a d^{2}\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}} \] Input:

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

Output:

2*(b^2*e*g - a*b*f*g - a*b*e*h + a^2*f*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2 
*e + a*b*f))/((b^2*c^2 - 2*a*b*c*d + a^2*d^2)*sqrt(-b^2*e + a*b*f)) - (2*b 
*d^2*e*g - b*c*d*f*g - a*d^2*f*g - 2*a*d^2*e*h - b*c^2*f*h + 3*a*c*d*f*h)* 
arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^2*c^2*d - 2*a*b*c*d^2 + a 
^2*d^3)*sqrt(-d^2*e + c*d*f)) + (sqrt(f*x + e)*d*f*g - sqrt(f*x + e)*c*f*h 
)/((b*c*d - a*d^2)*((f*x + e)*d - d*e + c*f))
 

Mupad [B] (verification not implemented)

Time = 5.27 (sec) , antiderivative size = 11557, normalized size of antiderivative = 56.65 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)^(1/2)*(g + h*x))/((a + b*x)*(c + d*x)^2),x)
 

Output:

(atan(((((2*(e + f*x)^(1/2)*(b^5*c^4*f^4*h^2 + 5*a^2*b^3*d^4*f^4*g^2 + b^5 
*c^2*d^2*f^4*g^2 + 8*b^5*d^4*e^2*f^2*g^2 + 4*a^4*b*d^4*f^4*h^2 + 2*a*b^4*c 
*d^3*f^4*g^2 - 6*a*b^4*c^3*d*f^4*h^2 - 12*a*b^4*d^4*e*f^3*g^2 - 4*b^5*c*d^ 
3*e*f^3*g^2 - 8*a^3*b^2*d^4*f^4*g*h - 8*a^3*b^2*d^4*e*f^3*h^2 + 2*b^5*c^3* 
d*f^4*g*h + 9*a^2*b^3*c^2*d^2*f^4*h^2 + 8*a^2*b^3*d^4*e^2*f^2*h^2 - 4*a*b^ 
4*c^2*d^2*f^4*g*h - 6*a^2*b^3*c*d^3*f^4*g*h - 16*a*b^4*d^4*e^2*f^2*g*h + 2 
0*a^2*b^3*d^4*e*f^3*g*h - 4*b^5*c^2*d^2*e*f^3*g*h + 4*a*b^4*c^2*d^2*e*f^3* 
h^2 - 12*a^2*b^3*c*d^3*e*f^3*h^2 + 16*a*b^4*c*d^3*e*f^3*g*h))/(a^2*d^3 + b 
^2*c^2*d - 2*a*b*c*d^2) + ((-d^3*(c*f - d*e))^(1/2)*((2*(2*a^5*b^2*d^7*f^4 
*g + 2*a*b^6*c^4*d^3*f^4*g - 8*a^4*b^3*c*d^6*f^4*g - 2*a*b^6*c^5*d^2*f^4*h 
 - 2*a^5*b^2*c*d^6*f^4*h - 2*a^4*b^3*d^7*e*f^3*g - 2*b^7*c^4*d^3*e*f^3*g + 
 2*b^7*c^5*d^2*e*f^3*h - 8*a^2*b^5*c^3*d^4*f^4*g + 12*a^3*b^4*c^2*d^5*f^4* 
g + 8*a^2*b^5*c^4*d^3*f^4*h - 12*a^3*b^4*c^3*d^4*f^4*h + 8*a^4*b^3*c^2*d^5 
*f^4*h + 8*a*b^6*c^3*d^4*e*f^3*g + 8*a^3*b^4*c*d^6*e*f^3*g - 8*a*b^6*c^4*d 
^3*e*f^3*h + 2*a^4*b^3*c*d^6*e*f^3*h - 12*a^2*b^5*c^2*d^5*e*f^3*g + 12*a^2 
*b^5*c^3*d^4*e*f^3*h - 8*a^3*b^4*c^2*d^5*e*f^3*h))/(a^3*d^4 - b^3*c^3*d + 
3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3) + ((e + f*x)^(1/2)*(-d^3*(c*f - d*e))^(1/ 
2)*(2*a*d^2*e*h + a*d^2*f*g - 2*b*d^2*e*g + b*c^2*f*h - 3*a*c*d*f*h + b*c* 
d*f*g)*(4*a^5*b^2*d^8*f^3 + 4*b^7*c^5*d^3*f^3 + 8*a^2*b^5*c^3*d^5*f^3 + 8* 
a^3*b^4*c^2*d^6*f^3 - 12*a*b^6*c^4*d^4*f^3 - 12*a^4*b^3*c*d^7*f^3 - 8*a...
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1168, normalized size of antiderivative = 5.73 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^2} \, dx =\text {Too large to display} \] Input:

int((f*x+e)^(1/2)*(h*x+g)/(b*x+a)/(d*x+c)^2,x)
 

Output:

(2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a*c**2*d**2*f*h - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqr 
t(b)*sqrt(a*f - b*e)))*a*c*d**3*e*h + 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt 
(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*c*d**3*f*h*x - 2*sqrt(b)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*d**4*e*h*x - 
2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)) 
)*b*c**2*d**2*f*g + 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*b*c*d**3*e*g - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt( 
e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b*c*d**3*f*g*x + 2*sqrt(b)*sqrt(a*f 
 - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b*d**4*e*g*x - 3 
*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e))) 
*a*b*c**2*d*f*h + 2*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d 
)*sqrt(c*f - d*e)))*a*b*c*d**2*e*h + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e 
+ f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c*d**2*f*g - 3*sqrt(d)*sqrt(c*f - 
 d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b*c*d**2*f*h*x + 
 2*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e) 
))*a*b*d**3*e*h*x + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d 
)*sqrt(c*f - d*e)))*a*b*d**3*f*g*x + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e 
+ f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**2*c**3*f*h + sqrt(d)*sqrt(c*f - d* 
e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b**2*c**2*d*f*g + ...