\(\int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx\) [145]

Optimal result

Integrand size = 29, antiderivative size = 363 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx=\frac {(d g-c h) \sqrt {e+f x}}{2 (b c-a d) (d e-c f) (c+d x)^2}+\frac {\left (a d (3 d f g-4 d e h+c f h)+b \left (4 d^2 e g-7 c d f g+3 c^2 f h\right )\right ) \sqrt {e+f x}}{4 (b c-a d)^2 (d e-c f)^2 (c+d x)}-\frac {2 b^{3/2} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^3 \sqrt {b e-a f}}+\frac {\left (a^2 d^2 f (3 d f g-4 d e h+c f h)+b^2 \left (8 d^3 e^2 g-20 c d^2 e f g+15 c^2 d f^2 g-3 c^3 f^2 h\right )-2 a b d \left (3 c^2 f^2 h-2 d^2 e (f g-2 e h)+5 c d f (f g-2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 \sqrt {d} (b c-a d)^3 (d e-c f)^{5/2}} \] Output:

1/2*(-c*h+d*g)*(f*x+e)^(1/2)/(-a*d+b*c)/(-c*f+d*e)/(d*x+c)^2+1/4*(a*d*(c*f 
*h-4*d*e*h+3*d*f*g)+b*(3*c^2*f*h-7*c*d*f*g+4*d^2*e*g))*(f*x+e)^(1/2)/(-a*d 
+b*c)^2/(-c*f+d*e)^2/(d*x+c)-2*b^(3/2)*(-a*h+b*g)*arctanh(b^(1/2)*(f*x+e)^ 
(1/2)/(-a*f+b*e)^(1/2))/(-a*d+b*c)^3/(-a*f+b*e)^(1/2)+1/4*(a^2*d^2*f*(c*f* 
h-4*d*e*h+3*d*f*g)+b^2*(-3*c^3*f^2*h+15*c^2*d*f^2*g-20*c*d^2*e*f*g+8*d^3*e 
^2*g)-2*a*b*d*(3*c^2*f^2*h-2*d^2*e*(-2*e*h+f*g)+5*c*d*f*(-2*e*h+f*g)))*arc 
tanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(1/2)/(-a*d+b*c)^3/(-c*f+d* 
e)^(5/2)
 

Mathematica [A] (verified)

Time = 2.22 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.02 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx=\frac {1}{4} \left (\frac {\sqrt {e+f x} \left (b \left (5 c^3 f h+4 d^3 e g x+c d^2 g (6 e-7 f x)+c^2 d (-9 f g-2 e h+3 f h x)\right )+a d \left (-c^2 f h+c d (5 f g-2 e h+f h x)+d^2 (3 f g x-2 e (g+2 h x))\right )\right )}{(b c-a d)^2 (d e-c f)^2 (c+d x)^2}+\frac {8 b^{3/2} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(b c-a d)^3 \sqrt {-b e+a f}}+\frac {\left (a^2 d^2 f (3 d f g-4 d e h+c f h)+b^2 \left (8 d^3 e^2 g-20 c d^2 e f g+15 c^2 d f^2 g-3 c^3 f^2 h\right )-2 a b d \left (3 c^2 f^2 h+5 c d f (f g-2 e h)+2 d^2 e (-f g+2 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{\sqrt {d} (-b c+a d)^3 (-d e+c f)^{5/2}}\right ) \] Input:

Integrate[(g + h*x)/((a + b*x)*(c + d*x)^3*Sqrt[e + f*x]),x]
 

Output:

((Sqrt[e + f*x]*(b*(5*c^3*f*h + 4*d^3*e*g*x + c*d^2*g*(6*e - 7*f*x) + c^2* 
d*(-9*f*g - 2*e*h + 3*f*h*x)) + a*d*(-(c^2*f*h) + c*d*(5*f*g - 2*e*h + f*h 
*x) + d^2*(3*f*g*x - 2*e*(g + 2*h*x)))))/((b*c - a*d)^2*(d*e - c*f)^2*(c + 
 d*x)^2) + (8*b^(3/2)*(b*g - a*h)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b* 
e) + a*f]])/((b*c - a*d)^3*Sqrt[-(b*e) + a*f]) + ((a^2*d^2*f*(3*d*f*g - 4* 
d*e*h + c*f*h) + b^2*(8*d^3*e^2*g - 20*c*d^2*e*f*g + 15*c^2*d*f^2*g - 3*c^ 
3*f^2*h) - 2*a*b*d*(3*c^2*f^2*h + 5*c*d*f*(f*g - 2*e*h) + 2*d^2*e*(-(f*g) 
+ 2*e*h)))*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(Sqrt[d]*(- 
(b*c) + a*d)^3*(-(d*e) + c*f)^(5/2)))/4
 

Rubi [A] (verified)

Time = 0.70 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.15, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {168, 27, 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.

\(\displaystyle \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

\(\displaystyle \frac {\frac {\int \frac {d f (3 d f g-4 d e h+c f h) a^2-b \left (-4 e (f g-2 e h) d^2+c f (7 f g-16 e h) d+5 c^2 f^2 h\right ) a+8 b^2 (d e-c f)^2 g+b f \left (a d (3 d f g-4 d e h+c f h)+b \left (3 f h c^2-7 d f g c+4 d^2 e g\right )\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)+b \left (3 c^2 f h-7 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} (d g-c h)}{2 (c+d x)^2 (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {d f (3 d f g-4 d e h+c f h) a^2-b \left (-4 e (f g-2 e h) d^2+c f (7 f g-16 e h) d+5 c^2 f^2 h\right ) a+8 b^2 (d e-c f)^2 g+b f \left (a d (3 d f g-4 d e h+c f h)+b \left (3 f h c^2-7 d f g c+4 d^2 e g\right )\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{2 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)+b \left (3 c^2 f h-7 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} (d g-c h)}{2 (c+d x)^2 (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {8 b^2 (b g-a h) (d e-c f)^2 \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {\left (a^2 d^2 f (c f h-4 d e h+3 d f g)-2 a b d \left (3 c^2 f^2 h+5 c d f (f g-2 e h)-2 d^2 e (f g-2 e h)\right )+b^2 \left (-3 c^3 f^2 h+15 c^2 d f^2 g-20 c d^2 e f g+8 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{2 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)+b \left (3 c^2 f h-7 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} (d g-c h)}{2 (c+d x)^2 (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {16 b^2 (b g-a h) (d e-c f)^2 \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 \left (a^2 d^2 f (c f h-4 d e h+3 d f g)-2 a b d \left (3 c^2 f^2 h+5 c d f (f g-2 e h)-2 d^2 e (f g-2 e h)\right )+b^2 \left (-3 c^3 f^2 h+15 c^2 d f^2 g-20 c d^2 e f g+8 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{2 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)+b \left (3 c^2 f h-7 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} (d g-c h)}{2 (c+d x)^2 (b c-a d) (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a^2 d^2 f (c f h-4 d e h+3 d f g)-2 a b d \left (3 c^2 f^2 h+5 c d f (f g-2 e h)-2 d^2 e (f g-2 e h)\right )+b^2 \left (-3 c^3 f^2 h+15 c^2 d f^2 g-20 c d^2 e f g+8 d^3 e^2 g\right )\right )}{\sqrt {d} (b c-a d) \sqrt {d e-c f}}-\frac {16 b^{3/2} (b g-a h) (d e-c f)^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d) \sqrt {b e-a f}}}{2 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} \left (a d (c f h-4 d e h+3 d f g)+b \left (3 c^2 f h-7 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 (b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} (d g-c h)}{2 (c+d x)^2 (b c-a d) (d e-c f)}\)

Input:

Int[(g + h*x)/((a + b*x)*(c + d*x)^3*Sqrt[e + f*x]),x]
 

Output:

((d*g - c*h)*Sqrt[e + f*x])/(2*(b*c - a*d)*(d*e - c*f)*(c + d*x)^2) + (((a 
*d*(3*d*f*g - 4*d*e*h + c*f*h) + b*(4*d^2*e*g - 7*c*d*f*g + 3*c^2*f*h))*Sq 
rt[e + f*x])/((b*c - a*d)*(d*e - c*f)*(c + d*x)) + ((-16*b^(3/2)*(d*e - c* 
f)^2*(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^2*d^2*f*(3*d*f*g - 4*d*e*h + c*f*h) + b^2*( 
8*d^3*e^2*g - 20*c*d^2*e*f*g + 15*c^2*d*f^2*g - 3*c^3*f^2*h) - 2*a*b*d*(3* 
c^2*f^2*h - 2*d^2*e*(f*g - 2*e*h) + 5*c*d*f*(f*g - 2*e*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)))/(4*(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 = 0.88 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {\left (-\left (\left (\left (7 c g x \,d^{2}+9 c^{2} \left (-\frac {h x}{3}+g \right ) d -5 c^{3} h \right ) b +a d \left (-3 d^{2} g x -5 c \left (\frac {h x}{5}+g \right ) d +h \,c^{2}\right )\right ) f +2 d \left (\left (-2 d^{2} g x +h \,c^{2}-3 c d g \right ) b +a \left (\left (2 h x +g \right ) d +c h \right ) d \right ) e \right ) \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}+\arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) \left (x d +c \right )^{2} \left (\left (\left (-3 c^{3} h +15 c^{2} d g \right ) b^{2}-6 a c d \left (c h +\frac {5 d g}{3}\right ) b +a^{2} d^{2} \left (c h +3 d g \right )\right ) f^{2}-4 d^{2} e \left (a h -b g \right ) \left (a d -5 b c \right ) f -8 b \,d^{3} e^{2} \left (a h -b g \right )\right )\right ) \sqrt {\left (a f -b e \right ) b}+8 b^{2} \left (x d +c \right )^{2} \left (c f -d e \right )^{2} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right ) \sqrt {\left (c f -d e \right ) d}}{4 \sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (c f -d e \right )^{2} \left (x d +c \right )^{2} \left (a d -b c \right )^{3}}\)	\(384\)
derivativedivides	\(2 f^{2} \left (-\frac {\frac {-\frac {f d \left (a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +3 a^{2} d^{3} f g +2 a b \,c^{2} d f h +4 a b c \,d^{2} e h -10 a b c \,d^{2} f g +4 a b \,d^{3} e g -3 b^{2} c^{3} f h +7 b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a^{2} c \,d^{2} f h +4 a^{2} d^{3} e h -5 a^{2} d^{3} f g -6 a b \,c^{2} d f h -4 a b c \,d^{2} e h +14 a b c \,d^{2} f g -4 a b \,d^{3} e g +5 b^{2} c^{3} f h -9 b^{2} c^{2} d f g +4 b^{2} c \,d^{2} e g \right ) f \sqrt {f x +e}}{8 c f -8 d e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +3 a^{2} d^{3} f^{2} g -6 a b \,c^{2} d \,f^{2} h +20 a b c \,d^{2} e f h -10 a b c \,d^{2} f^{2} g -8 e^{2} h b a \,d^{3}+4 a b \,d^{3} e f g -3 b^{2} c^{3} f^{2} h +15 b^{2} c^{2} d \,f^{2} g -20 b^{2} c \,d^{2} e f g +8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{3} f^{2}}+\frac {b^{2} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}}\right )\)	\(559\)
default	\(2 f^{2} \left (-\frac {\frac {-\frac {f d \left (a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +3 a^{2} d^{3} f g +2 a b \,c^{2} d f h +4 a b c \,d^{2} e h -10 a b c \,d^{2} f g +4 a b \,d^{3} e g -3 b^{2} c^{3} f h +7 b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a^{2} c \,d^{2} f h +4 a^{2} d^{3} e h -5 a^{2} d^{3} f g -6 a b \,c^{2} d f h -4 a b c \,d^{2} e h +14 a b c \,d^{2} f g -4 a b \,d^{3} e g +5 b^{2} c^{3} f h -9 b^{2} c^{2} d f g +4 b^{2} c \,d^{2} e g \right ) f \sqrt {f x +e}}{8 c f -8 d e}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +3 a^{2} d^{3} f^{2} g -6 a b \,c^{2} d \,f^{2} h +20 a b c \,d^{2} e f h -10 a b c \,d^{2} f^{2} g -8 e^{2} h b a \,d^{3}+4 a b \,d^{3} e f g -3 b^{2} c^{3} f^{2} h +15 b^{2} c^{2} d \,f^{2} g -20 b^{2} c \,d^{2} e f g +8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{3} f^{2}}+\frac {b^{2} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}}\right )\)	\(559\)

Input:

int((h*x+g)/(b*x+a)/(d*x+c)^3/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/4/((a*f-b*e)*b)^(1/2)/((c*f-d*e)*d)^(1/2)*((-(((7*c*g*x*d^2+9*c^2*(-1/3* 
h*x+g)*d-5*c^3*h)*b+a*d*(-3*d^2*g*x-5*c*(1/5*h*x+g)*d+h*c^2))*f+2*d*((-2*d 
^2*g*x+c^2*h-3*c*d*g)*b+a*((2*h*x+g)*d+c*h)*d)*e)*(a*d-b*c)*((c*f-d*e)*d)^ 
(1/2)*(f*x+e)^(1/2)+arctan(d*(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))*(d*x+c)^2* 
(((-3*c^3*h+15*c^2*d*g)*b^2-6*a*c*d*(c*h+5/3*d*g)*b+a^2*d^2*(c*h+3*d*g))*f 
^2-4*d^2*e*(a*h-b*g)*(a*d-5*b*c)*f-8*b*d^3*e^2*(a*h-b*g)))*((a*f-b*e)*b)^( 
1/2)+8*b^2*(d*x+c)^2*(c*f-d*e)^2*(a*h-b*g)*arctan(b*(f*x+e)^(1/2)/((a*f-b* 
e)*b)^(1/2))*((c*f-d*e)*d)^(1/2))/(c*f-d*e)^2/(d*x+c)^2/(a*d-b*c)^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1905 vs. \(2 (333) = 666\).

Time = 172.56 (sec) , antiderivative size = 7703, normalized size of antiderivative = 21.22 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((h*x+g)/(b*x+a)/(d*x+c)**3/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)/(d*x+c)^3/(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 835 vs. \(2 (333) = 666\).

Time = 0.16 (sec) , antiderivative size = 835, normalized size of antiderivative = 2.30 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

2*(b^3*g - a*b^2*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^3*c^3 
 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b^2*e + a*b*f)) - 1/4*(8 
*b^2*d^3*e^2*g - 20*b^2*c*d^2*e*f*g + 4*a*b*d^3*e*f*g + 15*b^2*c^2*d*f^2*g 
 - 10*a*b*c*d^2*f^2*g + 3*a^2*d^3*f^2*g - 8*a*b*d^3*e^2*h + 20*a*b*c*d^2*e 
*f*h - 4*a^2*d^3*e*f*h - 3*b^2*c^3*f^2*h - 6*a*b*c^2*d*f^2*h + a^2*c*d^2*f 
^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^3*c^3*d^2*e^2 - 3*a 
*b^2*c^2*d^3*e^2 + 3*a^2*b*c*d^4*e^2 - a^3*d^5*e^2 - 2*b^3*c^4*d*e*f + 6*a 
*b^2*c^3*d^2*e*f - 6*a^2*b*c^2*d^3*e*f + 2*a^3*c*d^4*e*f + b^3*c^5*f^2 - 3 
*a*b^2*c^4*d*f^2 + 3*a^2*b*c^3*d^2*f^2 - a^3*c^2*d^3*f^2)*sqrt(-d^2*e + c* 
d*f)) + 1/4*(4*(f*x + e)^(3/2)*b*d^3*e*f*g - 4*sqrt(f*x + e)*b*d^3*e^2*f*g 
 - 7*(f*x + e)^(3/2)*b*c*d^2*f^2*g + 3*(f*x + e)^(3/2)*a*d^3*f^2*g + 13*sq 
rt(f*x + e)*b*c*d^2*e*f^2*g - 5*sqrt(f*x + e)*a*d^3*e*f^2*g - 9*sqrt(f*x + 
 e)*b*c^2*d*f^3*g + 5*sqrt(f*x + e)*a*c*d^2*f^3*g - 4*(f*x + e)^(3/2)*a*d^ 
3*e*f*h + 4*sqrt(f*x + e)*a*d^3*e^2*f*h + 3*(f*x + e)^(3/2)*b*c^2*d*f^2*h 
+ (f*x + e)^(3/2)*a*c*d^2*f^2*h - 5*sqrt(f*x + e)*b*c^2*d*e*f^2*h - 3*sqrt 
(f*x + e)*a*c*d^2*e*f^2*h + 5*sqrt(f*x + e)*b*c^3*f^3*h - sqrt(f*x + e)*a* 
c^2*d*f^3*h)/((b^2*c^2*d^2*e^2 - 2*a*b*c*d^3*e^2 + a^2*d^4*e^2 - 2*b^2*c^3 
*d*e*f + 4*a*b*c^2*d^2*e*f - 2*a^2*c*d^3*e*f + b^2*c^4*f^2 - 2*a*b*c^3*d*f 
^2 + a^2*c^2*d^2*f^2)*((f*x + e)*d - d*e + c*f)^2)
 

Mupad [B] (verification not implemented)

Time = 18.45 (sec) , antiderivative size = 351830, normalized size of antiderivative = 969.23 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

(((e + f*x)^(1/2)*(5*a*d^2*f^2*g + 5*b*c^2*f^2*h - a*c*d*f^2*h - 9*b*c*d*f 
^2*g - 4*a*d^2*e*f*h + 4*b*d^2*e*f*g))/(4*(c*f - d*e)*(a^2*d^2 + b^2*c^2 - 
 2*a*b*c*d)) + (d*(e + f*x)^(3/2)*(3*a*d^2*f^2*g + 3*b*c^2*f^2*h + a*c*d*f 
^2*h - 7*b*c*d*f^2*g - 4*a*d^2*e*f*h + 4*b*d^2*e*f*g))/(4*(c*f - d*e)^2*(a 
^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(d^2*(e + f*x)^2 - (e + f*x)*(2*d^2*e - 2* 
c*d*f) + c^2*f^2 + d^2*e^2 - 2*c*d*e*f) - atan(((((64*b^10*c^10*d^2*f^7*g 
- 440*a*b^9*c^9*d^3*f^7*g - 40*a*b^9*c^10*d^2*f^7*h - 200*b^10*c^9*d^3*e*f 
^6*g - 24*b^10*c^10*d^2*e*f^6*h + 1320*a^2*b^8*c^8*d^4*f^7*g - 2264*a^3*b^ 
7*c^7*d^5*f^7*g + 2440*a^4*b^6*c^6*d^6*f^7*g - 1704*a^5*b^5*c^5*d^7*f^7*g 
+ 760*a^6*b^4*c^4*d^8*f^7*g - 200*a^7*b^3*c^3*d^9*f^7*g + 24*a^8*b^2*c^2*d 
^10*f^7*g + 248*a^2*b^8*c^9*d^3*f^7*h - 648*a^3*b^7*c^8*d^4*f^7*h + 920*a^ 
4*b^6*c^7*d^5*f^7*h - 760*a^5*b^5*c^6*d^6*f^7*h + 360*a^6*b^4*c^5*d^7*f^7* 
h - 88*a^7*b^3*c^4*d^8*f^7*h + 8*a^8*b^2*c^3*d^9*f^7*h + 32*a^6*b^4*d^12*e 
^4*f^3*g + 8*a^7*b^3*d^12*e^3*f^4*g + 24*a^8*b^2*d^12*e^2*f^5*g - 32*a^7*b 
^3*d^12*e^4*f^3*h - 32*a^8*b^2*d^12*e^3*f^4*h + 32*b^10*c^6*d^6*e^4*f^3*g 
- 136*b^10*c^7*d^5*e^3*f^4*g + 240*b^10*c^8*d^4*e^2*f^5*g - 24*b^10*c^8*d^ 
4*e^3*f^4*h + 48*b^10*c^9*d^3*e^2*f^5*h + 480*a^2*b^8*c^4*d^8*e^4*f^3*g - 
2088*a^2*b^8*c^5*d^7*e^3*f^4*g + 4056*a^2*b^8*c^6*d^6*e^2*f^5*g - 640*a^3* 
b^7*c^3*d^9*e^4*f^3*g + 2840*a^3*b^7*c^4*d^8*e^3*f^4*g - 6024*a^3*b^7*c^5* 
d^7*e^2*f^5*g + 480*a^4*b^6*c^2*d^10*e^4*f^3*g - 2200*a^4*b^6*c^3*d^9*e...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 6278, normalized size of antiderivative = 17.29 \[ \int \frac {g+h x}{(a+b x) (c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(8*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a*b*c**5*d*f**3*h - 24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a*b*c**4*d**2*e*f**2*h + 16*sqrt(b)*sqrt(a*f - b 
*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c**4*d**2*f**3*h 
*x + 24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a*b*c**3*d**3*e**2*f*h - 48*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c**3*d**3*e*f**2*h*x + 8*sqrt(b)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c**3* 
d**3*f**3*h*x**2 - 8*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt( 
b)*sqrt(a*f - b*e)))*a*b*c**2*d**4*e**3*h + 48*sqrt(b)*sqrt(a*f - b*e)*ata 
n((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c**2*d**4*e**2*f*h*x - 
24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a*b*c**2*d**4*e*f**2*h*x**2 - 16*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
 f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c*d**5*e**3*h*x + 24*sqrt(b)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c*d**5*e* 
*2*f*h*x**2 - 8*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq 
rt(a*f - b*e)))*a*b*d**6*e**3*h*x**2 - 8*sqrt(b)*sqrt(a*f - b*e)*atan((sqr 
t(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2*c**5*d*f**3*g + 24*sqrt(b)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2*c**4 
*d**2*e*f**2*g - 16*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqr...