\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx\) [92]

Optimal result

Integrand size = 29, antiderivative size = 315 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {(d e-c f) (2 b d g-b c h-a d h) \sqrt {e+f x}}{b d (b c-a d)^2 (c+d x)}-\frac {(b g-a h) (e+f x)^{3/2}}{b (b c-a d) (a+b x) (c+d x)}-\frac {\sqrt {b e-a f} \left (a^2 d f h-b^2 (4 d e g-3 c f g-2 c e h)+a b (d f g+2 d e h-5 c f h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2} (b c-a d)^3}+\frac {\sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)-b \left (4 d^2 e g-c^2 f h-c d (f g+2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (b c-a d)^3} \] Output:

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

Mathematica [A] (verified)

Time = 2.71 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.08 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {\sqrt {e+f x} \left (a^2 d f h (c+d x)+b^2 \left (2 d^2 e g x+c^2 f h x+c d (e g-f g x-e h x)\right )+a b \left (c^2 f h-2 c d (f g+e h)+d^2 (e g-f g x-e h x)\right )\right )}{b d (b c-a d)^2 (a+b x) (c+d x)}-\frac {\sqrt {-b e+a f} \left (a^2 d f h+b^2 (-4 d e g+3 c f g+2 c e h)+a b (d f g+2 d e h-5 c f h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{3/2} (b c-a d)^3}+\frac {\sqrt {-d e+c f} \left (a d (-3 d f g-2 d e h+5 c f h)+b \left (4 d^2 e g-c^2 f h-c d (f g+2 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (-b c+a d)^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {166, 27, 25, 166, 25, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x)*(c + d*x))) + ((- 
2*(d*e - c*f)*(2*b*d*g - b*c*h - a*d*h)*Sqrt[e + f*x])/(d*(b*c - a*d)*(c + 
 d*x)) + ((-2*d*Sqrt[b*e - a*f]*(a^2*d*f*h - b^2*(4*d*e*g - 3*c*f*g - 2*c* 
e*h) + a*b*(d*f*g + 2*d*e*h - 5*c*f*h))*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sq 
rt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)) + (2*b*Sqrt[d*e - c*f]*(a*d*(3*d*f*g 
 + 2*d*e*h - 5*c*f*h) - b*(4*d^2*e*g - c^2*f*h - c*d*(f*g + 2*e*h)))*ArcTa 
nh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/(Sqrt[d]*(b*c - a*d)))/(d*(b* 
c - a*d)))/(2*b*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 = 0.85 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(2 f^{2} \left (\frac {\left (c f -d e \right ) \left (-\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 d \left (\left (f x +e \right ) d +c f -d e \right )}+\frac {\left (5 a c d f h -2 a \,d^{2} e h -3 a \,d^{2} f g -b \,c^{2} f h -2 b c d e h -b c d f g +4 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 d \sqrt {\left (c f -d e \right ) d}}\right )}{f^{2} \left (a d -b c \right )^{3}}-\frac {\left (a f -b e \right ) \left (\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{2 b \left (\left (f x +e \right ) b +a f -b e \right )}-\frac {\left (a^{2} d f h -5 a b c f h +2 a b d e h +a b d f g +2 b^{2} c e h +3 b^{2} c f g -4 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 b \sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3}}\right )\)	\(348\)
default	\(2 f^{2} \left (\frac {\left (c f -d e \right ) \left (-\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 d \left (\left (f x +e \right ) d +c f -d e \right )}+\frac {\left (5 a c d f h -2 a \,d^{2} e h -3 a \,d^{2} f g -b \,c^{2} f h -2 b c d e h -b c d f g +4 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 d \sqrt {\left (c f -d e \right ) d}}\right )}{f^{2} \left (a d -b c \right )^{3}}-\frac {\left (a f -b e \right ) \left (\frac {f \left (a^{2} d h -a b c h -a b d g +b^{2} c g \right ) \sqrt {f x +e}}{2 b \left (\left (f x +e \right ) b +a f -b e \right )}-\frac {\left (a^{2} d f h -5 a b c f h +2 a b d e h +a b d f g +2 b^{2} c e h +3 b^{2} c f g -4 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 b \sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3}}\right )\)	\(348\)
pseudoelliptic	\(-\frac {-d \sqrt {\left (c f -d e \right ) d}\, \left (\left (2 c e h +3 c f g -4 d e g \right ) b^{2}-5 a \left (\frac {\left (-2 e h -f g \right ) d}{5}+c f h \right ) b +a^{2} d f h \right ) \left (b x +a \right ) \left (x d +c \right ) \left (a f -b e \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (-5 \left (\frac {\left (4 d^{2} e g -2 c \left (e h +\frac {f g}{2}\right ) d -c^{2} f h \right ) b}{5}+a d \left (\frac {\left (-2 e h -3 f g \right ) d}{5}+c f h \right )\right ) \left (c f -d e \right ) \left (x d +c \right ) \left (b x +a \right ) b \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\left (a d -b c \right ) \left (\left (2 d^{2} e g x +c \left (-f g x +e \left (-h x +g \right )\right ) d +c^{2} f h x \right ) b^{2}+a \left (\left (-f g x +e \left (-h x +g \right )\right ) d^{2}-2 c d \left (e h +f g \right )+c^{2} f h \right ) b +a^{2} d f h \left (x d +c \right )\right ) \sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\right )}{\sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, b \left (b x +a \right ) \left (a d -b c \right )^{3} \left (x d +c \right ) d}\)	\(387\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1049 vs. \(2 (291) = 582\).

Time = 13.13 (sec) , antiderivative size = 4238, normalized size of antiderivative = 13.45 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2/(d*x+c)^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. 804 vs. \(2 (291) = 582\).

Time = 0.21 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.55 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx=-\frac {{\left (4 \, b^{3} d e^{2} g - 3 \, b^{3} c e f g - 5 \, a b^{2} d e f g + 3 \, a b^{2} c f^{2} g + a^{2} b d f^{2} g - 2 \, b^{3} c e^{2} h - 2 \, a b^{2} d e^{2} h + 7 \, a b^{2} c e f h + a^{2} b d e f h - 5 \, a^{2} b c f^{2} h + a^{3} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \sqrt {-b^{2} e + a b f}} + \frac {{\left (4 \, b d^{3} e^{2} g - 5 \, b c d^{2} e f g - 3 \, a d^{3} e f g + b c^{2} d f^{2} g + 3 \, a c d^{2} f^{2} g - 2 \, b c d^{2} e^{2} h - 2 \, a d^{3} e^{2} h + b c^{2} d e f h + 7 \, a c d^{2} e f h + b c^{3} f^{2} h - 5 \, a c^{2} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{3} c^{3} d - 3 \, a b^{2} c^{2} d^{2} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \sqrt {-d^{2} e + c d f}} - \frac {2 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{2} d^{2} e f g - 2 \, \sqrt {f x + e} b^{2} d^{2} e^{2} f g - {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c d f^{2} g - {\left (f x + e\right )}^{\frac {3}{2}} a b d^{2} f^{2} g + 2 \, \sqrt {f x + e} b^{2} c d e f^{2} g + 2 \, \sqrt {f x + e} a b d^{2} e f^{2} g - 2 \, \sqrt {f x + e} a b c d f^{3} g - {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c d e f h - {\left (f x + e\right )}^{\frac {3}{2}} a b d^{2} e f h + \sqrt {f x + e} b^{2} c d e^{2} f h + \sqrt {f x + e} a b d^{2} e^{2} f h + {\left (f x + e\right )}^{\frac {3}{2}} b^{2} c^{2} f^{2} h + {\left (f x + e\right )}^{\frac {3}{2}} a^{2} d^{2} f^{2} h - \sqrt {f x + e} b^{2} c^{2} e f^{2} h - 2 \, \sqrt {f x + e} a b c d e f^{2} h - \sqrt {f x + e} a^{2} d^{2} e f^{2} h + \sqrt {f x + e} a b c^{2} f^{3} h + \sqrt {f x + e} a^{2} c d f^{3} h}{{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} {\left ({\left (f x + e\right )}^{2} b d - 2 \, {\left (f x + e\right )} b d e + b d e^{2} + {\left (f x + e\right )} b c f + {\left (f x + e\right )} a d f - b c e f - a d e f + a c f^{2}\right )}} \] Input:

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

Output:

-(4*b^3*d*e^2*g - 3*b^3*c*e*f*g - 5*a*b^2*d*e*f*g + 3*a*b^2*c*f^2*g + a^2* 
b*d*f^2*g - 2*b^3*c*e^2*h - 2*a*b^2*d*e^2*h + 7*a*b^2*c*e*f*h + a^2*b*d*e* 
f*h - 5*a^2*b*c*f^2*h + a^3*d*f^2*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + 
a*b*f))/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*sqrt(-b^2 
*e + a*b*f)) + (4*b*d^3*e^2*g - 5*b*c*d^2*e*f*g - 3*a*d^3*e*f*g + b*c^2*d* 
f^2*g + 3*a*c*d^2*f^2*g - 2*b*c*d^2*e^2*h - 2*a*d^3*e^2*h + b*c^2*d*e*f*h 
+ 7*a*c*d^2*e*f*h + b*c^3*f^2*h - 5*a*c^2*d*f^2*h)*arctan(sqrt(f*x + e)*d/ 
sqrt(-d^2*e + c*d*f))/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3* 
d^4)*sqrt(-d^2*e + c*d*f)) - (2*(f*x + e)^(3/2)*b^2*d^2*e*f*g - 2*sqrt(f*x 
 + e)*b^2*d^2*e^2*f*g - (f*x + e)^(3/2)*b^2*c*d*f^2*g - (f*x + e)^(3/2)*a* 
b*d^2*f^2*g + 2*sqrt(f*x + e)*b^2*c*d*e*f^2*g + 2*sqrt(f*x + e)*a*b*d^2*e* 
f^2*g - 2*sqrt(f*x + e)*a*b*c*d*f^3*g - (f*x + e)^(3/2)*b^2*c*d*e*f*h - (f 
*x + e)^(3/2)*a*b*d^2*e*f*h + sqrt(f*x + e)*b^2*c*d*e^2*f*h + sqrt(f*x + e 
)*a*b*d^2*e^2*f*h + (f*x + e)^(3/2)*b^2*c^2*f^2*h + (f*x + e)^(3/2)*a^2*d^ 
2*f^2*h - sqrt(f*x + e)*b^2*c^2*e*f^2*h - 2*sqrt(f*x + e)*a*b*c*d*e*f^2*h 
- sqrt(f*x + e)*a^2*d^2*e*f^2*h + sqrt(f*x + e)*a*b*c^2*f^3*h + sqrt(f*x + 
 e)*a^2*c*d*f^3*h)/((b^3*c^2*d - 2*a*b^2*c*d^2 + a^2*b*d^3)*((f*x + e)^2*b 
*d - 2*(f*x + e)*b*d*e + b*d*e^2 + (f*x + e)*b*c*f + (f*x + e)*a*d*f - b*c 
*e*f - a*d*e*f + a*c*f^2))
 

Mupad [B] (verification not implemented)

Time = 8.69 (sec) , antiderivative size = 38955, normalized size of antiderivative = 123.67 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

- atan(((((8*a*b^9*c^7*d^3*f^5*g + 8*a^7*b^3*c*d^9*f^5*g - 4*a*b^9*c^8*d^2 
*f^5*h - 4*a^8*b^2*c*d^9*f^5*h - 8*a^7*b^3*d^10*e*f^4*g + 4*a^8*b^2*d^10*e 
*f^4*h - 8*b^10*c^7*d^3*e*f^4*g + 4*b^10*c^8*d^2*e*f^4*h - 48*a^2*b^8*c^6* 
d^4*f^5*g + 120*a^3*b^7*c^5*d^5*f^5*g - 160*a^4*b^6*c^4*d^6*f^5*g + 120*a^ 
5*b^5*c^3*d^7*f^5*g - 48*a^6*b^4*c^2*d^8*f^5*g + 20*a^2*b^8*c^7*d^3*f^5*h 
- 36*a^3*b^7*c^6*d^4*f^5*h + 20*a^4*b^6*c^5*d^5*f^5*h + 20*a^5*b^5*c^4*d^6 
*f^5*h - 36*a^6*b^4*c^3*d^7*f^5*h + 20*a^7*b^3*c^2*d^8*f^5*h + 8*a^6*b^4*d 
^10*e^2*f^3*g - 4*a^7*b^3*d^10*e^2*f^3*h + 8*b^10*c^6*d^4*e^2*f^3*g - 4*b^ 
10*c^7*d^3*e^2*f^3*h + 120*a^2*b^8*c^4*d^6*e^2*f^3*g - 160*a^3*b^7*c^3*d^7 
*e^2*f^3*g + 120*a^4*b^6*c^2*d^8*e^2*f^3*g - 36*a^2*b^8*c^5*d^5*e^2*f^3*h 
+ 20*a^3*b^7*c^4*d^6*e^2*f^3*h + 20*a^4*b^6*c^3*d^7*e^2*f^3*h - 36*a^5*b^5 
*c^2*d^8*e^2*f^3*h + 40*a*b^9*c^6*d^4*e*f^4*g + 40*a^6*b^4*c*d^9*e*f^4*g - 
 16*a*b^9*c^7*d^3*e*f^4*h - 16*a^7*b^3*c*d^9*e*f^4*h - 48*a*b^9*c^5*d^5*e^ 
2*f^3*g - 72*a^2*b^8*c^5*d^5*e*f^4*g + 40*a^3*b^7*c^4*d^6*e*f^4*g + 40*a^4 
*b^6*c^3*d^7*e*f^4*g - 48*a^5*b^5*c*d^9*e^2*f^3*g - 72*a^5*b^5*c^2*d^8*e*f 
^4*g + 20*a*b^9*c^6*d^4*e^2*f^3*h + 16*a^2*b^8*c^6*d^4*e*f^4*h + 16*a^3*b^ 
7*c^5*d^5*e*f^4*h - 40*a^4*b^6*c^4*d^6*e*f^4*h + 16*a^5*b^5*c^3*d^7*e*f^4* 
h + 20*a^6*b^4*c*d^9*e^2*f^3*h + 16*a^6*b^4*c^2*d^8*e*f^4*h)/(a^6*b*d^7 + 
b^7*c^6*d - 6*a*b^6*c^5*d^2 - 6*a^5*b^2*c*d^6 + 15*a^2*b^5*c^4*d^3 - 20*a^ 
3*b^4*c^3*d^4 + 15*a^4*b^3*c^2*d^5) - (2*(e + f*x)^(1/2)*((4*a^2*d^5*e^...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 2886, normalized size of antiderivative = 9.16 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

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