\(\int \frac {g+h x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}} \, dx\) [173]

Optimal result

Integrand size = 29, antiderivative size = 287 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}} \, dx=-\frac {f (b (d e g-3 c f g+2 c e h)+a (2 d f g-3 d e h+c f h))}{(b c-a d) (b e-a f)^2 (d e-c f) \sqrt {e+f x}}-\frac {b g-a h}{(b c-a d) (b e-a f) (a+b x) \sqrt {e+f x}}+\frac {\sqrt {b} \left (3 a^2 d f h-a b f (5 d g+c h)+b^2 (2 d e g+3 c f g-2 c e h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^2 (b e-a f)^{5/2}}-\frac {2 d^{3/2} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^2 (d e-c f)^{3/2}} \] Output:

-f*(b*(2*c*e*h-3*c*f*g+d*e*g)+a*(c*f*h-3*d*e*h+2*d*f*g))/(-a*d+b*c)/(-a*f+ 
b*e)^2/(-c*f+d*e)/(f*x+e)^(1/2)-(-a*h+b*g)/(-a*d+b*c)/(-a*f+b*e)/(b*x+a)/( 
f*x+e)^(1/2)+b^(1/2)*(3*a^2*d*f*h-a*b*f*(c*h+5*d*g)+b^2*(-2*c*e*h+3*c*f*g+ 
2*d*e*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)^(5/2)-2*d^(3/2)*(-c*h+d*g)*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e 
)^(1/2))/(-a*d+b*c)^2/(-c*f+d*e)^(3/2)
 

Mathematica [A] (verified)

Time = 1.75 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.01 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}} \, dx=\frac {\frac {(b c-a d) \left (2 a^2 d f (-f g+e h)-b^2 d e g (e+f x)+a b c f (2 f g-3 e h-f h x)+a b d \left (e^2 h-2 f^2 g x+3 e f h x\right )+b^2 c f (3 f g x+e (g-2 h x))\right )}{(b e-a f)^2 (d e-c f) (a+b x) \sqrt {e+f x}}+\frac {\sqrt {b} \left (-3 a^2 d f h+a b f (5 d g+c h)+b^2 (-2 d e g-3 c f g+2 c e h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b e+a f)^{5/2}}-\frac {2 d^{3/2} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{(-d e+c f)^{3/2}}}{(b c-a d)^2} \] Input:

Integrate[(g + h*x)/((a + b*x)^2*(c + d*x)*(e + f*x)^(3/2)),x]
 

Output:

(((b*c - a*d)*(2*a^2*d*f*(-(f*g) + e*h) - b^2*d*e*g*(e + f*x) + a*b*c*f*(2 
*f*g - 3*e*h - f*h*x) + a*b*d*(e^2*h - 2*f^2*g*x + 3*e*f*h*x) + b^2*c*f*(3 
*f*g*x + e*(g - 2*h*x))))/((b*e - a*f)^2*(d*e - c*f)*(a + b*x)*Sqrt[e + f* 
x]) + (Sqrt[b]*(-3*a^2*d*f*h + a*b*f*(5*d*g + c*h) + b^2*(-2*d*e*g - 3*c*f 
*g + 2*c*e*h))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(-(b*e) 
 + a*f)^(5/2) - (2*d^(3/2)*(d*g - c*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt 
[-(d*e) + c*f]])/(-(d*e) + c*f)^(3/2))/(b*c - a*d)^2
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {168, 27, 169, 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.

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 169

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

Int[(g + h*x)/((a + b*x)^2*(c + d*x)*(e + f*x)^(3/2)),x]
 

Output:

-((b*g - a*h)/((b*c - a*d)*(b*e - a*f)*(a + b*x)*Sqrt[e + f*x])) + ((-2*f* 
(b*(d*e*g - 3*c*f*g + 2*c*e*h) + a*(2*d*f*g - 3*d*e*h + c*f*h)))/((b*e - a 
*f)*(d*e - c*f)*Sqrt[e + f*x]) - ((-2*Sqrt[b]*(d*e - c*f)*(3*a^2*d*f*h - a 
*b*f*(5*d*g + c*h) + b^2*(2*d*e*g + 3*c*f*g - 2*c*e*h))*ArcTanh[(Sqrt[b]*S 
qrt[e + f*x])/Sqrt[b*e - a*f]])/((b*c - a*d)*Sqrt[b*e - a*f]) + (4*d^(3/2) 
*(b*e - a*f)^2*(d*g - c*h)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f] 
])/((b*c - a*d)*Sqrt[d*e - c*f]))/((b*e - a*f)*(d*e - c*f)))/(2*(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 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 169

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

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 = 0.72 (sec) , antiderivative size = 277, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(2 f \left (\frac {\left (c h -d g \right ) d^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{2} f \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2} d f h -\frac {1}{2} a b c f h -\frac {1}{2} a b d f g +\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (3 a^{2} d f h -a b c f h -5 a b d f g -2 b^{2} c e h +3 b^{2} c f g +2 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{2} \left (a f -b e \right )^{2} f}-\frac {-e h +f g}{\left (c f -d e \right ) \left (a f -b e \right )^{2} \sqrt {f x +e}}\right )\)	\(277\)
default	\(2 f \left (\frac {\left (c h -d g \right ) d^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{2} f \left (c f -d e \right ) \sqrt {\left (c f -d e \right ) d}}-\frac {b \left (\frac {\left (\frac {1}{2} a^{2} d f h -\frac {1}{2} a b c f h -\frac {1}{2} a b d f g +\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (3 a^{2} d f h -a b c f h -5 a b d f g -2 b^{2} c e h +3 b^{2} c f g +2 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{2} \left (a f -b e \right )^{2} f}-\frac {-e h +f g}{\left (c f -d e \right ) \left (a f -b e \right )^{2} \sqrt {f x +e}}\right )\)	\(277\)
pseudoelliptic	\(\frac {-3 \left (\left (a \left (a h -\frac {5 b g}{3}\right ) f +\frac {2 b^{2} e g}{3}\right ) d -\frac {c b \left (\left (a h -3 b g \right ) f +2 e h b \right )}{3}\right ) \sqrt {\left (c f -d e \right ) d}\, \left (b x +a \right ) \left (c f -d e \right ) \sqrt {f x +e}\, b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+2 \left (\sqrt {f x +e}\, d^{2} \left (a f -b e \right )^{2} \left (b x +a \right ) \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (\left (-a g \left (b x +a \right ) f^{2}+e \left (\frac {3}{2} b h a x -\frac {1}{2} b^{2} g x +a^{2} h \right ) f +\frac {b \,e^{2} \left (a h -b g \right )}{2}\right ) d -\frac {3 c \left (\left (-x g b -\frac {2 a \left (-\frac {h x}{2}+g \right )}{3}\right ) f +\left (\frac {\left (2 h x -g \right ) b}{3}+a h \right ) e \right ) b f}{2}\right ) \left (a d -b c \right )\right ) \sqrt {\left (a f -b e \right ) b}}{\left (b x +a \right ) \left (a d -b c \right )^{2} \left (a f -b e \right )^{2} \sqrt {\left (a f -b e \right ) b}\, \left (c f -d e \right ) \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}}\)	\(354\)

Input:

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

Output:

2*f*((c*h-d*g)*d^2/(a*d-b*c)^2/f/(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))-b/(a*d-b*c)^2/(a*f-b*e)^2/f*((1/2*a^2*d*f 
*h-1/2*a*b*c*f*h-1/2*a*b*d*f*g+1/2*b^2*c*f*g)*(f*x+e)^(1/2)/((f*x+e)*b+a*f 
-b*e)+1/2*(3*a^2*d*f*h-a*b*c*f*h-5*a*b*d*f*g-2*b^2*c*e*h+3*b^2*c*f*g+2*b^2 
*d*e*g)/((a*f-b*e)*b)^(1/2)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2)))-( 
-e*h+f*g)/(c*f-d*e)/(a*f-b*e)^2/(f*x+e)^(1/2))
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((h*x+g)/(b*x+a)^2/(d*x+c)/(f*x+e)^(3/2),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(c*f-d*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. 623 vs. \(2 (263) = 526\).

Time = 0.16 (sec) , antiderivative size = 623, normalized size of antiderivative = 2.17 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}} \, dx=-\frac {{\left (2 \, b^{3} d e g + 3 \, b^{3} c f g - 5 \, a b^{2} d f g - 2 \, b^{3} c e h - a b^{2} c f h + 3 \, a^{2} b d f h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{4} c^{2} e^{2} - 2 \, a b^{3} c d e^{2} + a^{2} b^{2} d^{2} e^{2} - 2 \, a b^{3} c^{2} e f + 4 \, a^{2} b^{2} c d e f - 2 \, a^{3} b d^{2} e f + a^{2} b^{2} c^{2} f^{2} - 2 \, a^{3} b c d f^{2} + a^{4} d^{2} f^{2}\right )} \sqrt {-b^{2} e + a b f}} + \frac {2 \, {\left (d^{3} g - c d^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{2} c^{2} d e - 2 \, a b c d^{2} e + a^{2} d^{3} e - b^{2} c^{3} f + 2 \, a b c^{2} d f - a^{2} c d^{2} f\right )} \sqrt {-d^{2} e + c d f}} - \frac {{\left (f x + e\right )} b^{2} d e f g - 3 \, {\left (f x + e\right )} b^{2} c f^{2} g + 2 \, {\left (f x + e\right )} a b d f^{2} g + 2 \, b^{2} c e f^{2} g - 2 \, a b d e f^{2} g - 2 \, a b c f^{3} g + 2 \, a^{2} d f^{3} g + 2 \, {\left (f x + e\right )} b^{2} c e f h - 3 \, {\left (f x + e\right )} a b d e f h - 2 \, b^{2} c e^{2} f h + 2 \, a b d e^{2} f h + {\left (f x + e\right )} a b c f^{2} h + 2 \, a b c e f^{2} h - 2 \, a^{2} d e f^{2} h}{{\left (b^{3} c d e^{3} - a b^{2} d^{2} e^{3} - b^{3} c^{2} e^{2} f - a b^{2} c d e^{2} f + 2 \, a^{2} b d^{2} e^{2} f + 2 \, a b^{2} c^{2} e f^{2} - a^{2} b c d e f^{2} - a^{3} d^{2} e f^{2} - a^{2} b c^{2} f^{3} + a^{3} c d f^{3}\right )} {\left ({\left (f x + e\right )}^{\frac {3}{2}} b - \sqrt {f x + e} b e + \sqrt {f x + e} a f\right )}} \] Input:

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

Output:

-(2*b^3*d*e*g + 3*b^3*c*f*g - 5*a*b^2*d*f*g - 2*b^3*c*e*h - a*b^2*c*f*h + 
3*a^2*b*d*f*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^4*c^2*e^2 
- 2*a*b^3*c*d*e^2 + a^2*b^2*d^2*e^2 - 2*a*b^3*c^2*e*f + 4*a^2*b^2*c*d*e*f 
- 2*a^3*b*d^2*e*f + a^2*b^2*c^2*f^2 - 2*a^3*b*c*d*f^2 + a^4*d^2*f^2)*sqrt( 
-b^2*e + a*b*f)) + 2*(d^3*g - c*d^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-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)) - ((f*x + e)*b^2*d*e*f*g - 3*(f 
*x + e)*b^2*c*f^2*g + 2*(f*x + e)*a*b*d*f^2*g + 2*b^2*c*e*f^2*g - 2*a*b*d* 
e*f^2*g - 2*a*b*c*f^3*g + 2*a^2*d*f^3*g + 2*(f*x + e)*b^2*c*e*f*h - 3*(f*x 
 + e)*a*b*d*e*f*h - 2*b^2*c*e^2*f*h + 2*a*b*d*e^2*f*h + (f*x + e)*a*b*c*f^ 
2*h + 2*a*b*c*e*f^2*h - 2*a^2*d*e*f^2*h)/((b^3*c*d*e^3 - a*b^2*d^2*e^3 - b 
^3*c^2*e^2*f - a*b^2*c*d*e^2*f + 2*a^2*b*d^2*e^2*f + 2*a*b^2*c^2*e*f^2 - a 
^2*b*c*d*e*f^2 - a^3*d^2*e*f^2 - a^2*b*c^2*f^3 + a^3*c*d*f^3)*((f*x + e)^( 
3/2)*b - sqrt(f*x + e)*b*e + sqrt(f*x + e)*a*f))
 

Mupad [B] (verification not implemented)

Time = 27.36 (sec) , antiderivative size = 404752, normalized size of antiderivative = 1410.29 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan((((e + f*x)^(1/2)*(18*a^6*b^10*c^10*d^3*f^15*g^2 - 114*a^7*b^9*c^9* 
d^4*f^15*g^2 + 284*a^8*b^8*c^8*d^5*f^15*g^2 - 348*a^9*b^7*c^7*d^6*f^15*g^2 
 + 218*a^10*b^6*c^6*d^7*f^15*g^2 - 74*a^11*b^5*c^5*d^8*f^15*g^2 + 24*a^12* 
b^4*c^4*d^9*f^15*g^2 - 8*a^13*b^3*c^3*d^10*f^15*g^2 + 2*a^8*b^8*c^10*d^3*f 
^15*h^2 - 18*a^9*b^7*c^9*d^4*f^15*h^2 + 68*a^10*b^6*c^8*d^5*f^15*h^2 - 116 
*a^11*b^5*c^7*d^6*f^15*h^2 + 90*a^12*b^4*c^6*d^7*f^15*h^2 - 26*a^13*b^3*c^ 
5*d^8*f^15*h^2 + 16*a^3*b^13*d^13*e^13*f^2*g^2 - 168*a^4*b^12*d^13*e^12*f^ 
3*g^2 + 770*a^5*b^11*d^13*e^11*f^4*g^2 - 2020*a^6*b^10*d^13*e^10*f^5*g^2 + 
 3350*a^7*b^9*d^13*e^9*f^6*g^2 - 3664*a^8*b^8*d^13*e^8*f^7*g^2 + 2678*a^9* 
b^7*d^13*e^7*f^8*g^2 - 1300*a^10*b^6*d^13*e^6*f^9*g^2 + 410*a^11*b^5*d^13* 
e^5*f^10*g^2 - 80*a^12*b^4*d^13*e^4*f^11*g^2 + 8*a^13*b^3*d^13*e^3*f^12*g^ 
2 + 18*a^7*b^9*d^13*e^11*f^4*h^2 - 108*a^8*b^8*d^13*e^10*f^5*h^2 + 270*a^9 
*b^7*d^13*e^9*f^6*h^2 - 360*a^10*b^6*d^13*e^8*f^7*h^2 + 270*a^11*b^5*d^13* 
e^7*f^8*h^2 - 108*a^12*b^4*d^13*e^6*f^9*h^2 + 18*a^13*b^3*d^13*e^5*f^10*h^ 
2 - 16*b^16*c^3*d^10*e^13*f^2*g^2 + 40*b^16*c^4*d^9*e^12*f^3*g^2 - 2*b^16* 
c^5*d^8*e^11*f^4*g^2 - 62*b^16*c^6*d^7*e^10*f^5*g^2 + 20*b^16*c^7*d^6*e^9* 
f^6*g^2 + 68*b^16*c^8*d^5*e^8*f^7*g^2 - 66*b^16*c^9*d^4*e^7*f^8*g^2 + 18*b 
^16*c^10*d^3*e^6*f^9*g^2 - 16*b^16*c^5*d^8*e^13*f^2*h^2 + 64*b^16*c^6*d^7* 
e^12*f^3*h^2 - 104*b^16*c^7*d^6*e^11*f^4*h^2 + 88*b^16*c^8*d^5*e^10*f^5*h^ 
2 - 40*b^16*c^9*d^4*e^9*f^6*h^2 + 8*b^16*c^10*d^3*e^8*f^7*h^2 + 48*a*b^...
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 4163, normalized size of antiderivative = 14.51 \[ \int \frac {g+h x}{(a+b x)^2 (c+d x) (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b 
)*sqrt(a*f - b*e)))*a**3*c**2*d*f**3*h + 6*sqrt(b)*sqrt(e + f*x)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*c*d**2*e*f** 
2*h - 3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**3*d**3*e**2*f*h + sqrt(b)*sqrt(e + f*x)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c**3*f**3* 
h - 2*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b 
)*sqrt(a*f - b*e)))*a**2*b*c**2*d*e*f**2*h + 5*sqrt(b)*sqrt(e + f*x)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c**2*d 
*f**3*g - 3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c**2*d*f**3*h*x + sqrt(b)*sqrt(e + f*x)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c* 
d**2*e**2*f*h - 10*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f* 
x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c*d**2*e*f**2*g + 6*sqrt(b)*sqrt(e 
 + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a**2*b*c*d**2*e*f**2*h*x + 5*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*d**3*e**2*f*g - 3*sqrt(b 
)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a**2*b*d**3*e**2*f*h*x + 2*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**2*c**3*e*f**2*h -...