\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx\) [39]

Optimal result

Integrand size = 29, antiderivative size = 355 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx=-\frac {(b g-a h) \sqrt {e+f x}}{2 b (b c-a d) (a+b x)^2}-\frac {\left (a^2 d f h+a b f (3 d g-5 c h)-b^2 (4 d e g-c f g-4 c e h)\right ) \sqrt {e+f x}}{4 b (b c-a d)^2 (b e-a f) (a+b x)}-\frac {\left (a^3 d^2 f^2 h+3 a^2 b d f^2 (d g-2 c h)-3 a b^2 f \left (4 d^2 e g+c^2 f h-2 c d (f g+2 e h)\right )+b^3 \left (8 d^2 e^2 g-c^2 f (f g-4 e h)-4 c d e (f g+2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 b^{3/2} (b c-a d)^3 (b e-a f)^{3/2}}+\frac {2 \sqrt {d} \sqrt {d e-c f} (d g-c h) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{(b c-a d)^3} \] Output:

-1/2*(-a*h+b*g)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(b*x+a)^2-1/4*(a^2*d*f*h+a*b*f* 
(-5*c*h+3*d*g)-b^2*(-4*c*e*h-c*f*g+4*d*e*g))*(f*x+e)^(1/2)/b/(-a*d+b*c)^2/ 
(-a*f+b*e)/(b*x+a)-1/4*(a^3*d^2*f^2*h+3*a^2*b*d*f^2*(-2*c*h+d*g)-3*a*b^2*f 
*(4*d^2*e*g+c^2*f*h-2*c*d*(2*e*h+f*g))+b^3*(8*d^2*e^2*g-c^2*f*(-4*e*h+f*g) 
-4*c*d*e*(2*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^( 
3/2)/(-a*d+b*c)^3/(-a*f+b*e)^(3/2)+2*d^(1/2)*(-c*f+d*e)^(1/2)*(-c*h+d*g)*a 
rctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 2.14 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.99 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx=-\frac {\frac {(b c-a d) \sqrt {e+f x} \left (-a^3 d f h-a b^2 (6 d e g+c f g-2 c e h-3 d f g x+5 c f h x)+b^3 (-4 d e g x+c f g x+2 c e (g+2 h x))+a^2 b (-3 c f h+d (5 f g+2 e h+f h x))\right )}{b (b e-a f) (a+b x)^2}+\frac {\left (a^3 d^2 f^2 h+3 a^2 b d f^2 (d g-2 c h)-3 a b^2 f \left (4 d^2 e g+c^2 f h-2 c d (f g+2 e h)\right )+b^3 \left (8 d^2 e^2 g-4 c d e (f g+2 e h)+c^2 f (-f g+4 e h)\right )\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{3/2} (-b e+a f)^{3/2}}-8 \sqrt {d} \sqrt {-d e+c f} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 (b c-a d)^3} \] Input:

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

Output:

-1/4*(((b*c - a*d)*Sqrt[e + f*x]*(-(a^3*d*f*h) - a*b^2*(6*d*e*g + c*f*g - 
2*c*e*h - 3*d*f*g*x + 5*c*f*h*x) + b^3*(-4*d*e*g*x + c*f*g*x + 2*c*e*(g + 
2*h*x)) + a^2*b*(-3*c*f*h + d*(5*f*g + 2*e*h + f*h*x))))/(b*(b*e - a*f)*(a 
 + b*x)^2) + ((a^3*d^2*f^2*h + 3*a^2*b*d*f^2*(d*g - 2*c*h) - 3*a*b^2*f*(4* 
d^2*e*g + c^2*f*h - 2*c*d*(f*g + 2*e*h)) + b^3*(8*d^2*e^2*g - 4*c*d*e*(f*g 
 + 2*e*h) + c^2*f*(-(f*g) + 4*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[- 
(b*e) + a*f]])/(b^(3/2)*(-(b*e) + a*f)^(3/2)) - 8*Sqrt[d]*Sqrt[-(d*e) + c* 
f]*(d*g - c*h)*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(b*c - 
a*d)^3
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*((b*g - a*h)*Sqrt[e + f*x])/(b*(b*c - a*d)*(a + b*x)^2) - (((a^2*d*f* 
h + a*b*f*(3*d*g - 5*c*h) - b^2*(4*d*e*g - c*f*g - 4*c*e*h))*Sqrt[e + f*x] 
)/((b*c - a*d)*(b*e - a*f)*(a + b*x)) + ((2*(a^3*d^2*f^2*h + 3*a^2*b*d*f^2 
*(d*g - 2*c*h) - 3*a*b^2*f*(4*d^2*e*g + c^2*f*h - 2*c*d*(f*g + 2*e*h)) + b 
^3*(8*d^2*e^2*g - c^2*f*(f*g - 4*e*h) - 4*c*d*e*(f*g + 2*e*h)))*ArcTanh[(S 
qrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)*Sqrt[b*e - a* 
f]) - (16*b*Sqrt[d]*(b*e - a*f)*Sqrt[d*e - c*f]*(d*g - c*h)*ArcTanh[(Sqrt[ 
d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(b*c - a*d))/(2*(b*c - a*d)*(b*e - a*f 
)))/(4*b*(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 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 = 0.82 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.16

method	result	size
pseudoelliptic	\(-\frac {-\sqrt {\left (c f -d e \right ) d}\, \left (b x +a \right )^{2} \left (\left (f^{2} a^{3} h +3 a^{2} b \,f^{2} g -12 a \,b^{2} e f g +8 b^{3} e^{2} g \right ) d^{2}-6 c \left (\frac {2 \left (2 e^{2} h +e f g \right ) b^{2}}{3}+a \left (-2 e f h -f^{2} g \right ) b +a^{2} f^{2} h \right ) b d -3 c^{2} b^{2} f \left (\frac {\left (-4 e h +f g \right ) b}{3}+a f h \right )\right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (-8 b d \left (b x +a \right )^{2} \left (a f -b e \right ) \left (c f -d e \right ) \left (c h -d g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (a d -b c \right ) \sqrt {f x +e}\, \left (\left (4 e g x \,b^{3}+6 a \left (-\frac {f x}{2}+e \right ) g \,b^{2}-2 a^{2} \left (\frac {\left (h x +5 g \right ) f}{2}+e h \right ) b +a^{3} f h \right ) d +3 c \left (\frac {\left (-f g x -2 e \left (2 h x +g \right )\right ) b^{2}}{3}-\frac {2 a \left (\frac {\left (-5 h x -g \right ) f}{2}+e h \right ) b}{3}+a^{2} f h \right ) b \right )\right )}{4 \sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (a f -b e \right ) \left (a d -b c \right )^{3} \left (b x +a \right )^{2} b}\)	\(413\)
derivativedivides	\(2 f^{2} \left (\frac {\left (c f -d e \right ) \left (c h -d g \right ) d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (c f -d e \right ) d}}-\frac {\frac {-\frac {f \left (a^{3} d^{2} f h -6 a^{2} b c d f h +3 a^{2} b \,d^{2} f g +5 a \,b^{2} c^{2} f h +4 a \,b^{2} c d e h -2 a \,b^{2} c d f g -4 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h -b^{3} c^{2} f g +4 b^{3} c d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (a f -b e \right )}+\frac {f \left (a^{3} d^{2} f h +2 a^{2} b c d f h -5 a^{2} b \,d^{2} f g -3 a \,b^{2} c^{2} f h -4 a \,b^{2} c d e h +6 a \,b^{2} c d f g +4 a \,b^{2} d^{2} e g +4 b^{3} c^{2} e h -b^{3} c^{2} f g -4 b^{3} c d e g \right ) \sqrt {f x +e}}{8 b}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}-\frac {\left (a^{3} d^{2} f^{2} h -6 a^{2} b c d \,f^{2} h +3 a^{2} b \,d^{2} f^{2} g -3 a \,b^{2} c^{2} f^{2} h +12 a \,b^{2} c d e f h +6 a \,b^{2} c d \,f^{2} g -12 a \,b^{2} d^{2} e f g +4 b^{3} c^{2} e f h -b^{3} c^{2} f^{2} g -8 b^{3} c d \,e^{2} h -4 b^{3} c d e f g +8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) b \sqrt {\left (a f -b e \right ) b}}}{\left (a d -b c \right )^{3} f^{2}}\right )\)	\(534\)
default	\(2 f^{2} \left (\frac {\left (c f -d e \right ) \left (c h -d g \right ) d \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (c f -d e \right ) d}}-\frac {\frac {-\frac {f \left (a^{3} d^{2} f h -6 a^{2} b c d f h +3 a^{2} b \,d^{2} f g +5 a \,b^{2} c^{2} f h +4 a \,b^{2} c d e h -2 a \,b^{2} c d f g -4 a \,b^{2} d^{2} e g -4 b^{3} c^{2} e h -b^{3} c^{2} f g +4 b^{3} c d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 \left (a f -b e \right )}+\frac {f \left (a^{3} d^{2} f h +2 a^{2} b c d f h -5 a^{2} b \,d^{2} f g -3 a \,b^{2} c^{2} f h -4 a \,b^{2} c d e h +6 a \,b^{2} c d f g +4 a \,b^{2} d^{2} e g +4 b^{3} c^{2} e h -b^{3} c^{2} f g -4 b^{3} c d e g \right ) \sqrt {f x +e}}{8 b}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}-\frac {\left (a^{3} d^{2} f^{2} h -6 a^{2} b c d \,f^{2} h +3 a^{2} b \,d^{2} f^{2} g -3 a \,b^{2} c^{2} f^{2} h +12 a \,b^{2} c d e f h +6 a \,b^{2} c d \,f^{2} g -12 a \,b^{2} d^{2} e f g +4 b^{3} c^{2} e f h -b^{3} c^{2} f^{2} g -8 b^{3} c d \,e^{2} h -4 b^{3} c d e f g +8 b^{3} d^{2} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8 \left (a f -b e \right ) b \sqrt {\left (a f -b e \right ) b}}}{\left (a d -b c \right )^{3} f^{2}}\right )\)	\(534\)

Input:

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

Output:

-1/4/((a*f-b*e)*b)^(1/2)/((c*f-d*e)*d)^(1/2)*(-((c*f-d*e)*d)^(1/2)*(b*x+a) 
^2*((a^3*f^2*h+3*a^2*b*f^2*g-12*a*b^2*e*f*g+8*b^3*e^2*g)*d^2-6*c*(2/3*(2*e 
^2*h+e*f*g)*b^2+a*(-2*e*f*h-f^2*g)*b+a^2*f^2*h)*b*d-3*c^2*b^2*f*(1/3*(-4*e 
*h+f*g)*b+a*f*h))*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+((a*f-b*e)*b 
)^(1/2)*(-8*b*d*(b*x+a)^2*(a*f-b*e)*(c*f-d*e)*(c*h-d*g)*arctan(d*(f*x+e)^( 
1/2)/((c*f-d*e)*d)^(1/2))+((c*f-d*e)*d)^(1/2)*(a*d-b*c)*(f*x+e)^(1/2)*((4* 
e*g*x*b^3+6*a*(-1/2*f*x+e)*g*b^2-2*a^2*(1/2*(h*x+5*g)*f+e*h)*b+a^3*f*h)*d+ 
3*c*(1/3*(-f*g*x-2*e*(2*h*x+g))*b^2-2/3*a*(1/2*(-5*h*x-g)*f+e*h)*b+a^2*f*h 
)*b)))/(a*f-b*e)/(a*d-b*c)^3/(b*x+a)^2/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1610 vs. \(2 (325) = 650\).

Time = 26.66 (sec) , antiderivative size = 6490, normalized size of antiderivative = 18.28 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)^3/(d*x+c),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. 743 vs. \(2 (325) = 650\).

Time = 0.16 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.09 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx=\frac {{\left (8 \, b^{3} d^{2} e^{2} g - 4 \, b^{3} c d e f g - 12 \, a b^{2} d^{2} e f g - b^{3} c^{2} f^{2} g + 6 \, a b^{2} c d f^{2} g + 3 \, a^{2} b d^{2} f^{2} g - 8 \, b^{3} c d e^{2} h + 4 \, b^{3} c^{2} e f h + 12 \, a b^{2} c d e f h - 3 \, a b^{2} c^{2} f^{2} h - 6 \, a^{2} b c d f^{2} h + a^{3} d^{2} f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{4 \, {\left (b^{5} c^{3} e - 3 \, a b^{4} c^{2} d e + 3 \, a^{2} b^{3} c d^{2} e - a^{3} b^{2} d^{3} e - a b^{4} c^{3} f + 3 \, a^{2} b^{3} c^{2} d f - 3 \, a^{3} b^{2} c d^{2} f + a^{4} b d^{3} f\right )} \sqrt {-b^{2} e + a b f}} - \frac {2 \, {\left (d^{3} e g - c d^{2} f g - c d^{2} e h + c^{2} d f h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-d^{2} e + c d f}} + \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d e f g - 4 \, \sqrt {f x + e} b^{3} d e^{2} f g - {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c f^{2} g - 3 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d f^{2} g - \sqrt {f x + e} b^{3} c e f^{2} g + 9 \, \sqrt {f x + e} a b^{2} d e f^{2} g + \sqrt {f x + e} a b^{2} c f^{3} g - 5 \, \sqrt {f x + e} a^{2} b d f^{3} g - 4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c e f h + 4 \, \sqrt {f x + e} b^{3} c e^{2} f h + 5 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c f^{2} h - {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d f^{2} h - 7 \, \sqrt {f x + e} a b^{2} c e f^{2} h - \sqrt {f x + e} a^{2} b d e f^{2} h + 3 \, \sqrt {f x + e} a^{2} b c f^{3} h + \sqrt {f x + e} a^{3} d f^{3} h}{4 \, {\left (b^{4} c^{2} e - 2 \, a b^{3} c d e + a^{2} b^{2} d^{2} e - a b^{3} c^{2} f + 2 \, a^{2} b^{2} c d f - a^{3} b d^{2} f\right )} {\left ({\left (f x + e\right )} b - b e + a f\right )}^{2}} \] Input:

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

Output:

1/4*(8*b^3*d^2*e^2*g - 4*b^3*c*d*e*f*g - 12*a*b^2*d^2*e*f*g - b^3*c^2*f^2* 
g + 6*a*b^2*c*d*f^2*g + 3*a^2*b*d^2*f^2*g - 8*b^3*c*d*e^2*h + 4*b^3*c^2*e* 
f*h + 12*a*b^2*c*d*e*f*h - 3*a*b^2*c^2*f^2*h - 6*a^2*b*c*d*f^2*h + a^3*d^2 
*f^2*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*c^3*e - 3*a*b^4 
*c^2*d*e + 3*a^2*b^3*c*d^2*e - a^3*b^2*d^3*e - a*b^4*c^3*f + 3*a^2*b^3*c^2 
*d*f - 3*a^3*b^2*c*d^2*f + a^4*b*d^3*f)*sqrt(-b^2*e + a*b*f)) - 2*(d^3*e*g 
 - c*d^2*f*g - c*d^2*e*h + c^2*d*f*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + 
 c*d*f))/((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-d^2*e 
+ c*d*f)) + 1/4*(4*(f*x + e)^(3/2)*b^3*d*e*f*g - 4*sqrt(f*x + e)*b^3*d*e^2 
*f*g - (f*x + e)^(3/2)*b^3*c*f^2*g - 3*(f*x + e)^(3/2)*a*b^2*d*f^2*g - sqr 
t(f*x + e)*b^3*c*e*f^2*g + 9*sqrt(f*x + e)*a*b^2*d*e*f^2*g + sqrt(f*x + e) 
*a*b^2*c*f^3*g - 5*sqrt(f*x + e)*a^2*b*d*f^3*g - 4*(f*x + e)^(3/2)*b^3*c*e 
*f*h + 4*sqrt(f*x + e)*b^3*c*e^2*f*h + 5*(f*x + e)^(3/2)*a*b^2*c*f^2*h - ( 
f*x + e)^(3/2)*a^2*b*d*f^2*h - 7*sqrt(f*x + e)*a*b^2*c*e*f^2*h - sqrt(f*x 
+ e)*a^2*b*d*e*f^2*h + 3*sqrt(f*x + e)*a^2*b*c*f^3*h + sqrt(f*x + e)*a^3*d 
*f^3*h)/((b^4*c^2*e - 2*a*b^3*c*d*e + a^2*b^2*d^2*e - a*b^3*c^2*f + 2*a^2* 
b^2*c*d*f - a^3*b*d^2*f)*((f*x + e)*b - b*e + a*f)^2)
 

Mupad [B] (verification not implemented)

Time = 15.06 (sec) , antiderivative size = 278260, normalized size of antiderivative = 783.83 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((8*a*b^10*c^8*d^2*f^6*g - 40*a^8*b^3*c*d^9*f^6*g + 8*a^9*b^2*c*d^9 
*f^6*h + 40*a^8*b^3*d^10*e*f^5*g - 8*a^9*b^2*d^10*e*f^5*h - 8*b^11*c^8*d^2 
*e*f^5*g - 88*a^2*b^9*c^7*d^3*f^6*g + 360*a^3*b^8*c^6*d^4*f^6*g - 760*a^4* 
b^7*c^5*d^5*f^6*g + 920*a^5*b^6*c^4*d^6*f^6*g - 648*a^6*b^5*c^3*d^7*f^6*g 
+ 248*a^7*b^4*c^2*d^8*f^6*g + 24*a^2*b^9*c^8*d^2*f^6*h - 136*a^3*b^8*c^7*d 
^3*f^6*h + 312*a^4*b^7*c^6*d^4*f^6*h - 360*a^5*b^6*c^5*d^5*f^6*h + 200*a^6 
*b^5*c^4*d^6*f^6*h - 24*a^7*b^4*c^3*d^7*f^6*h - 24*a^8*b^3*c^2*d^8*f^6*h + 
 32*a^6*b^5*d^10*e^3*f^3*g - 72*a^7*b^4*d^10*e^2*f^4*g + 8*a^8*b^3*d^10*e^ 
2*f^4*h + 32*b^11*c^6*d^4*e^3*f^3*g - 24*b^11*c^7*d^3*e^2*f^4*g - 32*b^11* 
c^7*d^3*e^3*f^3*h + 32*b^11*c^8*d^2*e^2*f^4*h + 480*a^2*b^9*c^4*d^6*e^3*f^ 
3*g + 72*a^2*b^9*c^5*d^5*e^2*f^4*g - 640*a^3*b^8*c^3*d^7*e^3*f^3*g - 600*a 
^3*b^8*c^4*d^6*e^2*f^4*g + 480*a^4*b^7*c^2*d^8*e^3*f^3*g + 1080*a^4*b^7*c^ 
3*d^7*e^2*f^4*g - 936*a^5*b^6*c^2*d^8*e^2*f^4*g - 480*a^2*b^9*c^5*d^5*e^3* 
f^3*h + 152*a^2*b^9*c^6*d^4*e^2*f^4*h + 640*a^3*b^8*c^4*d^6*e^3*f^3*h + 15 
2*a^3*b^8*c^5*d^5*e^2*f^4*h - 480*a^4*b^7*c^3*d^7*e^3*f^3*h - 520*a^4*b^7* 
c^4*d^6*e^2*f^4*h + 192*a^5*b^6*c^2*d^8*e^3*f^3*h + 488*a^5*b^6*c^3*d^7*e^ 
2*f^4*h - 184*a^6*b^5*c^2*d^8*e^2*f^4*h + 112*a*b^10*c^7*d^3*e*f^5*g - 176 
*a^7*b^4*c*d^9*e*f^5*g - 56*a*b^10*c^8*d^2*e*f^5*h + 16*a^8*b^3*c*d^9*e*f^ 
5*h - 192*a*b^10*c^5*d^5*e^3*f^3*g + 72*a*b^10*c^6*d^4*e^2*f^4*g - 464*a^2 
*b^9*c^6*d^4*e*f^5*g + 880*a^3*b^8*c^5*d^5*e*f^5*g - 800*a^4*b^7*c^4*d^...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 3843, normalized size of antiderivative = 10.83 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x)^3 (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*a**5*d**2*f**2*h - 6*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**4*b*c*d*f**2*h + 3*sqrt(b)*sqrt(a*f - b*e)*atan(( 
sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d**2*f**2*g + 2*sqrt(b) 
*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b* 
d**2*f**2*h*x - 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)* 
sqrt(a*f - b*e)))*a**3*b**2*c**2*f**2*h + 12*sqrt(b)*sqrt(a*f - b*e)*atan( 
(sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*d*e*f*h + 6*sqrt( 
b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3* 
b**2*c*d*f**2*g - 12*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt( 
b)*sqrt(a*f - b*e)))*a**3*b**2*c*d*f**2*h*x - 12*sqrt(b)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2*e*f*g + 6* 
sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))* 
a**3*b**2*d**2*f**2*g*x + sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d**2*f**2*h*x**2 + 4*sqrt(b)*sqrt(a*f 
- b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**3*c**2*e* 
f*h - sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b 
*e)))*a**2*b**3*c**2*f**2*g - 6*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**3*c**2*f**2*h*x - 8*sqrt(b)*sqrt(a 
*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**3*c...