\(\int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx\) [55]

Optimal result

Integrand size = 29, antiderivative size = 360 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\frac {(d g-c h) \sqrt {e+f x}}{2 d (b c-a d) (c+d x)^2}-\frac {\left (a d (d f g+4 d e h-5 c f h)-b \left (4 d^2 e g-3 c d f g-c^2 f h\right )\right ) \sqrt {e+f x}}{4 d (b c-a d)^2 (d e-c f) (c+d x)}-\frac {2 \sqrt {b} \sqrt {b e-a f} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{(b c-a d)^3}-\frac {\left (a^2 d^2 f (d f g-4 d e h+3 c f h)-b^2 \left (8 d^3 e^2 g-12 c d^2 e f g+3 c^2 d f^2 g+c^3 f^2 h\right )+2 a b d \left (3 c^2 f^2 h+2 d^2 e (f g+2 e h)-3 c d f (f g+2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{3/2} (b c-a d)^3 (d e-c f)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\frac {1}{4} \left (\frac {\sqrt {e+f x} \left (b \left (c^3 f h+4 d^3 e g x+3 c d^2 g (2 e-f x)-c^2 d (5 f g+2 e h+f h x)\right )+a d \left (3 c^2 f h-d^2 (2 e g+f g x+4 e h x)+c d (f g-2 e h+5 f h x)\right )\right )}{d (b c-a d)^2 (d e-c f) (c+d x)^2}-\frac {8 \sqrt {b} \sqrt {-b e+a f} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(b c-a d)^3}-\frac {\left (-a^2 d^2 f (d f g-4 d e h+3 c f h)+b^2 \left (8 d^3 e^2 g-12 c d^2 e f g+3 c^2 d f^2 g+c^3 f^2 h\right )-2 a b d \left (3 c^2 f^2 h+2 d^2 e (f g+2 e h)-3 c d f (f g+2 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (-b c+a d)^3 (-d e+c f)^{3/2}}\right ) \] Input:

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

Output:

((Sqrt[e + f*x]*(b*(c^3*f*h + 4*d^3*e*g*x + 3*c*d^2*g*(2*e - f*x) - c^2*d* 
(5*f*g + 2*e*h + f*h*x)) + a*d*(3*c^2*f*h - d^2*(2*e*g + f*g*x + 4*e*h*x) 
+ c*d*(f*g - 2*e*h + 5*f*h*x))))/(d*(b*c - a*d)^2*(d*e - c*f)*(c + d*x)^2) 
 - (8*Sqrt[b]*Sqrt[-(b*e) + a*f]*(b*g - a*h)*ArcTan[(Sqrt[b]*Sqrt[e + f*x] 
)/Sqrt[-(b*e) + a*f]])/(b*c - a*d)^3 - ((-(a^2*d^2*f*(d*f*g - 4*d*e*h + 3* 
c*f*h)) + b^2*(8*d^3*e^2*g - 12*c*d^2*e*f*g + 3*c^2*d*f^2*g + c^3*f^2*h) - 
 2*a*b*d*(3*c^2*f^2*h + 2*d^2*e*(f*g + 2*e*h) - 3*c*d*f*(f*g + 2*e*h)))*Ar 
cTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(3/2)*(-(b*c) + a*d)^ 
3*(-(d*e) + c*f)^(3/2)))/4
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 405, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {\left (a^2 d^2 f (3 c f h-4 d e h+d f g)+2 a b d \left (3 c^2 f^2 h-3 c d f (2 e h+f g)+2 d^2 e (2 e h+f g)\right )-\left (b^2 \left (c^3 f^2 h+3 c^2 d f^2 g-12 c d^2 e f g+8 d^3 e^2 g\right )\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}+\frac {8 b d (b e-a f) (b g-a h) (d e-c f) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}}{2 (b c-a d) (d e-c f)}-\frac {\sqrt {e+f x} \left (a d (-5 c f h+4 d e h+d f g)-b \left (c^2 (-f) h-3 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {2 \left (a^2 d^2 f (3 c f h-4 d e h+d f g)+2 a b d \left (3 c^2 f^2 h-3 c d f (2 e h+f g)+2 d^2 e (2 e h+f g)\right )-\left (b^2 \left (c^3 f^2 h+3 c^2 d f^2 g-12 c d^2 e f g+8 d^3 e^2 g\right )\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}+\frac {16 b d (b e-a f) (b g-a h) (d e-c f) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{2 (b c-a d) (d e-c f)}-\frac {\sqrt {e+f x} \left (a d (-5 c f h+4 d e h+d f g)-b \left (c^2 (-f) h-3 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a^2 d^2 f (3 c f h-4 d e h+d f g)+2 a b d \left (3 c^2 f^2 h-3 c d f (2 e h+f g)+2 d^2 e (2 e h+f g)\right )-\left (b^2 \left (c^3 f^2 h+3 c^2 d f^2 g-12 c d^2 e f g+8 d^3 e^2 g\right )\right )\right )}{\sqrt {d} (b c-a d) \sqrt {d e-c f}}-\frac {16 \sqrt {b} d \sqrt {b e-a f} (b g-a h) (d e-c f) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b c-a d}}{2 (b c-a d) (d e-c f)}-\frac {\sqrt {e+f x} \left (a d (-5 c f h+4 d e h+d f g)-b \left (c^2 (-f) h-3 c d f g+4 d^2 e g\right )\right )}{(c+d x) (b c-a d) (d e-c f)}}{4 d (b c-a d)}+\frac {\sqrt {e+f x} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

Input:

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

Output:

((d*g - c*h)*Sqrt[e + f*x])/(2*d*(b*c - a*d)*(c + d*x)^2) + (-(((a*d*(d*f* 
g + 4*d*e*h - 5*c*f*h) - b*(4*d^2*e*g - 3*c*d*f*g - c^2*f*h))*Sqrt[e + f*x 
])/((b*c - a*d)*(d*e - c*f)*(c + d*x))) + ((-16*Sqrt[b]*d*Sqrt[b*e - a*f]* 
(d*e - c*f)*(b*g - a*h)*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/ 
(b*c - a*d) - (2*(a^2*d^2*f*(d*f*g - 4*d*e*h + 3*c*f*h) - b^2*(8*d^3*e^2*g 
 - 12*c*d^2*e*f*g + 3*c^2*d*f^2*g + c^3*f^2*h) + 2*a*b*d*(3*c^2*f^2*h + 2* 
d^2*e*(f*g + 2*e*h) - 3*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*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 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 = 0.80 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(-\frac {3 \left (-\sqrt {\left (a f -b e \right ) b}\, \left (x d +c \right )^{2} \left (\left (-\frac {1}{3} c^{3} f^{2} h -c^{2} d \,f^{2} g +4 c \,d^{2} e f g -\frac {8}{3} d^{3} e^{2} g \right ) b^{2}+2 a d \left (\frac {2 \left (2 e^{2} h +e f g \right ) d^{2}}{3}+c \left (-2 e f h -f^{2} g \right ) d +c^{2} f^{2} h \right ) b +a^{2} d^{2} \left (\frac {\left (-4 e h +f g \right ) d}{3}+c f h \right ) f \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \left (\frac {8 b d \left (x d +c \right )^{2} \left (c f -d e \right ) \left (a f -b e \right ) \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{3}+\sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right ) \sqrt {f x +e}\, \left (\left (\frac {4 d^{3} e g x}{3}+2 c \left (-\frac {f x}{2}+e \right ) g \,d^{2}-\frac {2 c^{2} \left (\frac {\left (h x +5 g \right ) f}{2}+e h \right ) d}{3}+\frac {c^{3} f h}{3}\right ) b +a \left (\frac {\left (-f g x -2 e \left (2 h x +g \right )\right ) d^{2}}{3}-\frac {2 c \left (\frac {\left (-5 h x -g \right ) f}{2}+e h \right ) d}{3}+c^{2} f h \right ) d \right )\right )\right )}{4 \sqrt {\left (c f -d e \right ) d}\, \sqrt {\left (a f -b e \right ) b}\, \left (x d +c \right )^{2} \left (a d -b c \right )^{3} \left (c f -d e \right ) d}\)	\(413\)
derivativedivides	\(2 f^{2} \left (-\frac {\left (a f -b e \right ) \left (a h -b g \right ) b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}}-\frac {\frac {\frac {f \left (5 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -a^{2} d^{3} f g -6 a b \,c^{2} d f h +4 a b c \,d^{2} e h -2 a b c \,d^{2} f g +4 a b \,d^{3} e g +b^{2} c^{3} f h +3 b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 c f -8 d e}+\frac {f \left (3 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +a^{2} d^{3} f g -2 a b \,c^{2} d f h +4 a b c \,d^{2} e h -6 a b c \,d^{2} f g +4 a b \,d^{3} e g -b^{2} c^{3} f h +5 b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \sqrt {f x +e}}{8 d}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (3 a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +a^{2} d^{3} f^{2} g +6 a b \,c^{2} d \,f^{2} h -12 a b c \,d^{2} e f h -6 a b c \,d^{2} f^{2} g +8 e^{2} h b a \,d^{3}+4 a b \,d^{3} e f g -b^{2} c^{3} f^{2} h -3 b^{2} c^{2} d \,f^{2} g +12 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c f -d e \right ) d \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{3} f^{2}}\right )\)	\(535\)
default	\(2 f^{2} \left (-\frac {\left (a f -b e \right ) \left (a h -b g \right ) b \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}}-\frac {\frac {\frac {f \left (5 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -a^{2} d^{3} f g -6 a b \,c^{2} d f h +4 a b c \,d^{2} e h -2 a b c \,d^{2} f g +4 a b \,d^{3} e g +b^{2} c^{3} f h +3 b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 c f -8 d e}+\frac {f \left (3 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h +a^{2} d^{3} f g -2 a b \,c^{2} d f h +4 a b c \,d^{2} e h -6 a b c \,d^{2} f g +4 a b \,d^{3} e g -b^{2} c^{3} f h +5 b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \sqrt {f x +e}}{8 d}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}-\frac {\left (3 a^{2} c \,d^{2} f^{2} h -4 a^{2} d^{3} e f h +a^{2} d^{3} f^{2} g +6 a b \,c^{2} d \,f^{2} h -12 a b c \,d^{2} e f h -6 a b c \,d^{2} f^{2} g +8 e^{2} h b a \,d^{3}+4 a b \,d^{3} e f g -b^{2} c^{3} f^{2} h -3 b^{2} c^{2} d \,f^{2} g +12 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \left (c f -d e \right ) d \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{3} f^{2}}\right )\)	\(535\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1611 vs. \(2 (330) = 660\).

Time = 27.20 (sec) , antiderivative size = 6495, normalized size of antiderivative = 18.04 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(1/2)*(h*x+g)/(b*x+a)/(d*x+c)^3,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. 743 vs. \(2 (330) = 660\).

Time = 0.17 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\frac {2 \, {\left (b^{3} e g - a b^{2} f g - a b^{2} e h + a^{2} b f h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} e + a b f}} - \frac {{\left (8 \, b^{2} d^{3} e^{2} g - 12 \, b^{2} c d^{2} e f g - 4 \, a b d^{3} e f g + 3 \, b^{2} c^{2} d f^{2} g + 6 \, a b c d^{2} f^{2} g - a^{2} d^{3} f^{2} g - 8 \, a b d^{3} e^{2} h + 12 \, a b c d^{2} e f h + 4 \, a^{2} d^{3} e f h + b^{2} c^{3} f^{2} h - 6 \, a b c^{2} d f^{2} h - 3 \, a^{2} c d^{2} f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{4 \, {\left (b^{3} c^{3} d^{2} e - 3 \, a b^{2} c^{2} d^{3} e + 3 \, a^{2} b c d^{4} e - a^{3} d^{5} e - b^{3} c^{4} d f + 3 \, a b^{2} c^{3} d^{2} f - 3 \, a^{2} b c^{2} d^{3} f + a^{3} c d^{4} f\right )} \sqrt {-d^{2} e + c d f}} + \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b d^{3} e f g - 4 \, \sqrt {f x + e} b d^{3} e^{2} f g - 3 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d^{2} f^{2} g - {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} f^{2} g + 9 \, \sqrt {f x + e} b c d^{2} e f^{2} g - \sqrt {f x + e} a d^{3} e f^{2} g - 5 \, \sqrt {f x + e} b c^{2} d f^{3} g + \sqrt {f x + e} a c d^{2} f^{3} g - 4 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} e f h + 4 \, \sqrt {f x + e} a d^{3} e^{2} f h - {\left (f x + e\right )}^{\frac {3}{2}} b c^{2} d f^{2} h + 5 \, {\left (f x + e\right )}^{\frac {3}{2}} a c d^{2} f^{2} h - \sqrt {f x + e} b c^{2} d e f^{2} h - 7 \, \sqrt {f x + e} a c d^{2} e f^{2} h + \sqrt {f x + e} b c^{3} f^{3} h + 3 \, \sqrt {f x + e} a c^{2} d f^{3} h}{4 \, {\left (b^{2} c^{2} d^{2} e - 2 \, a b c d^{3} e + a^{2} d^{4} e - b^{2} c^{3} d f + 2 \, a b c^{2} d^{2} f - a^{2} c d^{3} f\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 17.27 (sec) , antiderivative size = 278260, normalized size of antiderivative = 772.94 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((8*a^8*b^2*c*d^10*f^6*g - 40*a*b^9*c^8*d^3*f^6*g + 8*a*b^9*c^9*d^2 
*f^6*h - 8*a^8*b^2*d^11*e*f^5*g + 40*b^10*c^8*d^3*e*f^5*g - 8*b^10*c^9*d^2 
*e*f^5*h + 248*a^2*b^8*c^7*d^4*f^6*g - 648*a^3*b^7*c^6*d^5*f^6*g + 920*a^4 
*b^6*c^5*d^6*f^6*g - 760*a^5*b^5*c^4*d^7*f^6*g + 360*a^6*b^4*c^3*d^8*f^6*g 
 - 88*a^7*b^3*c^2*d^9*f^6*g - 24*a^2*b^8*c^8*d^3*f^6*h - 24*a^3*b^7*c^7*d^ 
4*f^6*h + 200*a^4*b^6*c^6*d^5*f^6*h - 360*a^5*b^5*c^5*d^6*f^6*h + 312*a^6* 
b^4*c^4*d^7*f^6*h - 136*a^7*b^3*c^3*d^8*f^6*h + 24*a^8*b^2*c^2*d^9*f^6*h + 
 32*a^6*b^4*d^11*e^3*f^3*g - 24*a^7*b^3*d^11*e^2*f^4*g - 32*a^7*b^3*d^11*e 
^3*f^3*h + 32*a^8*b^2*d^11*e^2*f^4*h + 32*b^10*c^6*d^5*e^3*f^3*g - 72*b^10 
*c^7*d^4*e^2*f^4*g + 8*b^10*c^8*d^3*e^2*f^4*h + 480*a^2*b^8*c^4*d^7*e^3*f^ 
3*g - 936*a^2*b^8*c^5*d^6*e^2*f^4*g - 640*a^3*b^7*c^3*d^8*e^3*f^3*g + 1080 
*a^3*b^7*c^4*d^7*e^2*f^4*g + 480*a^4*b^6*c^2*d^9*e^3*f^3*g - 600*a^4*b^6*c 
^3*d^8*e^2*f^4*g + 72*a^5*b^5*c^2*d^9*e^2*f^4*g + 192*a^2*b^8*c^5*d^6*e^3* 
f^3*h - 184*a^2*b^8*c^6*d^5*e^2*f^4*h - 480*a^3*b^7*c^4*d^7*e^3*f^3*h + 48 
8*a^3*b^7*c^5*d^6*e^2*f^4*h + 640*a^4*b^6*c^3*d^8*e^3*f^3*h - 520*a^4*b^6* 
c^4*d^7*e^2*f^4*h - 480*a^5*b^5*c^2*d^9*e^3*f^3*h + 152*a^5*b^5*c^3*d^8*e^ 
2*f^4*h + 152*a^6*b^4*c^2*d^9*e^2*f^4*h - 176*a*b^9*c^7*d^4*e*f^5*g + 112* 
a^7*b^3*c*d^10*e*f^5*g + 16*a*b^9*c^8*d^3*e*f^5*h - 56*a^8*b^2*c*d^10*e*f^ 
5*h - 192*a*b^9*c^5*d^6*e^3*f^3*g + 408*a*b^9*c^6*d^5*e^2*f^4*g + 208*a^2* 
b^8*c^6*d^5*e*f^5*g + 208*a^3*b^7*c^5*d^6*e*f^5*g - 800*a^4*b^6*c^4*d^7...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3843, normalized size of antiderivative = 10.68 \[ \int \frac {\sqrt {e+f x} (g+h x)}{(a+b x) (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b 
*e)))*a*c**4*d**2*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*c**3*d**3*e*f*h - 16*sqrt(b)*sqrt(a*f - b* 
e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*c**3*d**3*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*c**2*d**4*e**2*h + 32*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b) 
/(sqrt(b)*sqrt(a*f - b*e)))*a*c**2*d**4*e*f*h*x - 8*sqrt(b)*sqrt(a*f - b*e 
)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*c**2*d**4*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*c*d**5*e**2*h*x + 16*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)* 
b)/(sqrt(b)*sqrt(a*f - b*e)))*a*c*d**5*e*f*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*d**6*e**2*h*x**2 + 
 8*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*b*c**4*d**2*f**2*g - 16*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/ 
(sqrt(b)*sqrt(a*f - b*e)))*b*c**3*d**3*e*f*g + 16*sqrt(b)*sqrt(a*f - b*e)* 
atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b*c**3*d**3*f**2*g*x + 8 
*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e))) 
*b*c**2*d**4*e**2*g - 32*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(s 
qrt(b)*sqrt(a*f - b*e)))*b*c**2*d**4*e*f*g*x + 8*sqrt(b)*sqrt(a*f - b*e)*a 
tan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b*c**2*d**4*f**2*g*x**...