\(\int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx\) [136]

Optimal result

Integrand size = 29, antiderivative size = 207 \[ \int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx=\frac {(d g-c h) \sqrt {e+f x}}{(b c-a d) (d e-c f) (c+d x)}-\frac {2 \sqrt {b} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^2 \sqrt {b e-a f}}+\frac {\left (a d (d f g-2 d e h+c f h)+b \left (2 d^2 e g-3 c d f g+c^2 f h\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{\sqrt {d} (b c-a d)^2 (d e-c f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.95 \[ \int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx=\frac {\frac {(b c-a d) (-d g+c h) \sqrt {e+f x}}{(-d e+c f) (c+d x)}+\frac {2 \sqrt {b} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{\sqrt {-b e+a f}}+\frac {\left (a d (d f g-2 d e h+c f h)+b \left (2 d^2 e g-3 c d f g+c^2 f h\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d} (-d e+c f)^{3/2}}}{(b c-a d)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {168, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((d*g - c*h)*Sqrt[e + f*x])/((b*c - a*d)*(d*e - c*f)*(c + d*x)) + ((-4*Sqr 
t[b]*(d*e - c*f)*(b*g - a*h)*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a* 
f]])/((b*c - a*d)*Sqrt[b*e - a*f]) + (2*(a*d*(d*f*g - 2*d*e*h + c*f*h) + b 
*(2*d^2*e*g - 3*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*(b*c - a*d)*(d*e - c 
*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 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 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 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 = 0.62 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(2 f \left (-\frac {\left (a h -b g \right ) b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}-\frac {\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 \left (c f -d e \right ) \left (\left (f x +e \right ) d +c f -d e \right )}-\frac {\left (a c d f h -2 a \,d^{2} e h +a \,d^{2} f g +b \,c^{2} f h -3 b c d f g +2 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{2} f}\right )\)	\(231\)
default	\(2 f \left (-\frac {\left (a h -b g \right ) b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}-\frac {\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 \left (c f -d e \right ) \left (\left (f x +e \right ) d +c f -d e \right )}-\frac {\left (a c d f h -2 a \,d^{2} e h +a \,d^{2} f g +b \,c^{2} f h -3 b c d f g +2 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{2} f}\right )\)	\(231\)
pseudoelliptic	\(-\frac {-\left (x d +c \right ) \sqrt {\left (a f -b e \right ) b}\, \left (b \left (c^{2} f h -3 c d f g +2 d^{2} e g \right )+a \left (\left (-2 e h +f g \right ) d +c f h \right ) d \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (2 b \left (x d +c \right ) \left (c f -d e \right ) \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}\, \left (c h -d g \right )\right ) \sqrt {\left (c f -d e \right ) d}}{\sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (a d -b c \right )^{2} \left (c f -d e \right ) \left (x d +c \right )}\)	\(233\)

Input:

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

Output:

2*f*(-1/f*(a*h-b*g)*b/(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)/(c*f-d*e)*(f*x+e)^(1/2)/((f*x+e)*d+c*f-d*e)-1/2*(a*c*d*f*h-2*a*d^2* 
e*h+a*d^2*f*g+b*c^2*f*h-3*b*c*d*f*g+2*b*d^2*e*g)/(c*f-d*e)/((c*f-d*e)*d)^( 
1/2)*arctan(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. 628 vs. \(2 (185) = 370\).

Time = 9.44 (sec) , antiderivative size = 2596, normalized size of antiderivative = 12.54 \[ \int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.36 \[ \int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx=\frac {2 \, {\left (b^{2} g - a b h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt {-b^{2} e + a b f}} - \frac {{\left (2 \, b d^{2} e g - 3 \, b c d f g + a d^{2} f g - 2 \, a d^{2} e h + b c^{2} f h + a c d f h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{2} c^{2} d e - 2 \, a b c d^{2} e + a^{2} d^{3} e - b^{2} c^{3} f + 2 \, a b c^{2} d f - a^{2} c d^{2} f\right )} \sqrt {-d^{2} e + c d f}} + \frac {\sqrt {f x + e} d f g - \sqrt {f x + e} c f h}{{\left (b c d e - a d^{2} e - b c^{2} f + a c d f\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 6.91 (sec) , antiderivative size = 24052, normalized size of antiderivative = 116.19 \[ \int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

atan((((b^3*g^2 + a^2*b*h^2 - 2*a*b^2*g*h)/(b^5*c^4*e - a^5*d^4*f - a*b^4* 
c^4*f + a^4*b*d^4*e - 4*a*b^4*c^3*d*e + 4*a^4*b*c*d^3*f - 4*a^3*b^2*c*d^3* 
e + 4*a^2*b^3*c^3*d*f + 6*a^2*b^3*c^2*d^2*e - 6*a^3*b^2*c^2*d^2*f))^(1/2)* 
(((2*(4*b^7*c^6*d^2*f^5*g - 18*a*b^6*c^5*d^3*f^5*g - 2*a^5*b^2*c*d^7*f^5*g 
 - 2*a*b^6*c^6*d^2*f^5*h + 2*a^5*b^2*d^8*e*f^4*g - 6*b^7*c^5*d^3*e*f^4*g - 
 2*b^7*c^6*d^2*e*f^4*h + 32*a^2*b^5*c^4*d^4*f^5*g - 28*a^3*b^4*c^3*d^5*f^5 
*g + 12*a^4*b^3*c^2*d^6*f^5*g + 8*a^2*b^5*c^5*d^3*f^5*h - 12*a^3*b^4*c^4*d 
^4*f^5*h + 8*a^4*b^3*c^3*d^5*f^5*h - 2*a^5*b^2*c^2*d^6*f^5*h + 2*a^4*b^3*d 
^8*e^2*f^3*g - 4*a^5*b^2*d^8*e^2*f^3*h + 2*b^7*c^4*d^4*e^2*f^3*g + 2*b^7*c 
^5*d^3*e^2*f^3*h + 12*a^2*b^5*c^2*d^6*e^2*f^3*g + 28*a^2*b^5*c^3*d^5*e^2*f 
^3*h - 32*a^3*b^4*c^2*d^6*e^2*f^3*h + 26*a*b^6*c^4*d^4*e*f^4*g - 14*a^4*b^ 
3*c*d^7*e*f^4*g + 14*a*b^6*c^5*d^3*e*f^4*h + 6*a^5*b^2*c*d^7*e*f^4*h - 8*a 
*b^6*c^3*d^5*e^2*f^3*g - 44*a^2*b^5*c^3*d^5*e*f^4*g - 8*a^3*b^4*c*d^7*e^2* 
f^3*g + 36*a^3*b^4*c^2*d^6*e*f^4*g - 12*a*b^6*c^4*d^4*e^2*f^3*h - 36*a^2*b 
^5*c^4*d^4*e*f^4*h + 44*a^3*b^4*c^3*d^5*e*f^4*h + 18*a^4*b^3*c*d^7*e^2*f^3 
*h - 26*a^4*b^3*c^2*d^6*e*f^4*h))/(a^3*d^5*e^2 - b^3*c^5*f^2 + a^3*c^2*d^3 
*f^2 - b^3*c^3*d^2*e^2 - 2*a^3*c*d^4*e*f + 2*b^3*c^4*d*e*f - 3*a^2*b*c*d^4 
*e^2 + 3*a*b^2*c^4*d*f^2 + 3*a*b^2*c^2*d^3*e^2 - 3*a^2*b*c^3*d^2*f^2 - 6*a 
*b^2*c^3*d^2*e*f + 6*a^2*b*c^2*d^3*e*f) - (2*(e + f*x)^(1/2)*((b^3*g^2 + a 
^2*b*h^2 - 2*a*b^2*g*h)/(b^5*c^4*e - a^5*d^4*f - a*b^4*c^4*f + a^4*b*d^...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 2158, normalized size of antiderivative = 10.43 \[ \int \frac {g+h x}{(a+b x) (c+d x)^2 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((h*x+g)/(b*x+a)/(d*x+c)^2/(f*x+e)^(1/2),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**3*d*f**2*h + 4*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*e*f*h - 2*sqrt(b)*sqrt(a*f - b*e)*at 
an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*c**2*d**2*f**2*h*x - 2*s 
qrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a 
*c*d**3*e**2*h + 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b) 
*sqrt(a*f - b*e)))*a*c*d**3*e*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**2*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**3*d*f**2* 
g - 4*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*e*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**2*d**2*f**2*g*x + 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**2*g - 4*s 
qrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b 
*c*d**3*e*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**2*g*x + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e 
 + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c**2*d*f**2*h - 2*sqrt(d)*sqrt( 
c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*c*d**2*e 
*f*h + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*a**2*c*d**2*f**2*g + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)...