\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx\) [91]

Optimal result

Integrand size = 29, antiderivative size = 247 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx=-\frac {f (b d g-3 b c h+2 a d h) \sqrt {e+f x}}{b d^2 (b c-a d)}+\frac {(d g-c h) (e+f x)^{3/2}}{d (b c-a d) (c+d x)}-\frac {2 (b e-a f)^{3/2} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2} (b c-a d)^2}-\frac {\sqrt {d e-c f} \left (a d (3 d f g+2 d e h-5 c f h)-b \left (2 d^2 e g+c d f g-3 c^2 f h\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{5/2} (b c-a d)^2} \] Output:

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

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\frac {\frac {(b c-a d) \sqrt {e+f x} \left (-2 a d f h (c+d x)+b \left (d^2 e g+3 c^2 f h-c d (f g+e h-2 f h x)\right )\right )}{b d^2 (c+d x)}+\frac {2 (-b e+a f)^{3/2} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{3/2}}+\frac {\sqrt {-d e+c f} \left (a d (-3 d f g-2 d e h+5 c f h)+b \left (2 d^2 e g+c d f g-3 c^2 f h\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2}}}{(b c-a d)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 171, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

((d*g - c*h)*(e + f*x)^(3/2))/(d*(b*c - a*d)*(c + d*x)) + ((-2*f*(b*d*g - 
3*b*c*h + 2*a*d*h)*Sqrt[e + f*x])/(b*d) + ((-4*d^2*(b*e - a*f)^(3/2)*(b*g 
- a*h)*ArcTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[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*(2*d^2* 
e*g + c*d*f*g - 3*c^2*f*h))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f 
]])/(Sqrt[d]*(b*c - a*d)))/(b*d))/(2*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 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && 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 = 0.78 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {5 \left (c f -d e \right ) \sqrt {\left (a f -b e \right ) b}\, \left (\frac {b \left (-3 c^{2} f h +c d f g +2 d^{2} e g \right )}{5}+a d \left (\frac {\left (-2 e h -3 f g \right ) d}{5}+c f h \right )\right ) \left (x d +c \right ) b \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+2 \sqrt {\left (c f -d e \right ) d}\, \left (-d^{2} \left (a f -b e \right )^{2} \left (a h -b g \right ) \left (x d +c \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\left (a d -b c \right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}\, \left (\frac {\left (-d^{2} e g +c \left (\left (-2 h x +g \right ) f +e h \right ) d -3 c^{2} f h \right ) b}{2}+a d f h \left (x d +c \right )\right )\right )}{b \,d^{2} \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}\, \left (x d +c \right ) \sqrt {\left (c f -d e \right ) d}}\)	\(281\)
derivativedivides	\(2 f \left (\frac {h \sqrt {f x +e}}{b \,d^{2}}+\frac {\frac {\left (-\frac {1}{2} a \,c^{2} d \,f^{2} h +\frac {1}{2} a c \,d^{2} e f h +\frac {1}{2} a c \,d^{2} f^{2} g -\frac {1}{2} a \,d^{3} e f g +\frac {1}{2} b \,c^{3} f^{2} h -\frac {1}{2} b \,c^{2} d e f h -\frac {1}{2} b \,c^{2} d \,f^{2} g +\frac {1}{2} b c \,d^{2} e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (5 a \,c^{2} d \,f^{2} h -7 a c \,d^{2} e f h -3 a c \,d^{2} f^{2} g +2 a \,d^{3} e^{2} h +3 a \,d^{3} e f g -3 b \,c^{3} f^{2} h +3 b \,c^{2} d e f h +b \,c^{2} d \,f^{2} g +b c \,d^{2} e f g -2 b \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}}{d^{2} \left (a d -b c \right )^{2} f}+\frac {\left (-f^{2} a^{3} h +2 a^{2} b e f h +a^{2} b \,f^{2} g -a \,b^{2} e^{2} h -2 a \,b^{2} e f g +b^{3} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{b f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}\right )\)	\(389\)
default	\(2 f \left (\frac {h \sqrt {f x +e}}{b \,d^{2}}+\frac {\frac {\left (-\frac {1}{2} a \,c^{2} d \,f^{2} h +\frac {1}{2} a c \,d^{2} e f h +\frac {1}{2} a c \,d^{2} f^{2} g -\frac {1}{2} a \,d^{3} e f g +\frac {1}{2} b \,c^{3} f^{2} h -\frac {1}{2} b \,c^{2} d e f h -\frac {1}{2} b \,c^{2} d \,f^{2} g +\frac {1}{2} b c \,d^{2} e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (5 a \,c^{2} d \,f^{2} h -7 a c \,d^{2} e f h -3 a c \,d^{2} f^{2} g +2 a \,d^{3} e^{2} h +3 a \,d^{3} e f g -3 b \,c^{3} f^{2} h +3 b \,c^{2} d e f h +b \,c^{2} d \,f^{2} g +b c \,d^{2} e f g -2 b \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}}{d^{2} \left (a d -b c \right )^{2} f}+\frac {\left (-f^{2} a^{3} h +2 a^{2} b e f h +a^{2} b \,f^{2} g -a \,b^{2} e^{2} h -2 a \,b^{2} e f g +b^{3} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{b f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}\right )\)	\(389\)
risch	\(\frac {2 h \sqrt {f x +e}\, f}{b \,d^{2}}-\frac {2 f \left (\frac {d^{2} \left (f^{2} a^{3} h -2 a^{2} b e f h -a^{2} b \,f^{2} g +a \,b^{2} e^{2} h +2 a \,b^{2} e f g -b^{3} e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}}-\frac {b \left (\frac {\left (-\frac {1}{2} a \,c^{2} d \,f^{2} h +\frac {1}{2} a c \,d^{2} e f h +\frac {1}{2} a c \,d^{2} f^{2} g -\frac {1}{2} a \,d^{3} e f g +\frac {1}{2} b \,c^{3} f^{2} h -\frac {1}{2} b \,c^{2} d e f h -\frac {1}{2} b \,c^{2} d \,f^{2} g +\frac {1}{2} b c \,d^{2} e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (5 a \,c^{2} d \,f^{2} h -7 a c \,d^{2} e f h -3 a c \,d^{2} f^{2} g +2 a \,d^{3} e^{2} h +3 a \,d^{3} e f g -3 b \,c^{3} f^{2} h +3 b \,c^{2} d e f h +b \,c^{2} d \,f^{2} g +b c \,d^{2} e f g -2 b \,d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (a d -b c \right )^{2} f}\right )}{b \,d^{2}}\)	\(397\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 522 vs. \(2 (223) = 446\).

Time = 14.10 (sec) , antiderivative size = 2120, normalized size of antiderivative = 8.58 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(2*((b^2*c*d^2*e - a*b*c*d^2*f)*g - (a*b*c*d^2*e - a^2*c*d^2*f)*h + ( 
(b^2*d^3*e - a*b*d^3*f)*g - (a*b*d^3*e - a^2*d^3*f)*h)*x)*sqrt((b*e - a*f) 
/b)*log((b*f*x + 2*b*e - a*f - 2*sqrt(f*x + e)*b*sqrt((b*e - a*f)/b))/(b*x 
 + a)) + ((2*b^2*c*d^2*e + (b^2*c^2*d - 3*a*b*c*d^2)*f)*g - (2*a*b*c*d^2*e 
 + (3*b^2*c^3 - 5*a*b*c^2*d)*f)*h + ((2*b^2*d^3*e + (b^2*c*d^2 - 3*a*b*d^3 
)*f)*g - (2*a*b*d^3*e + (3*b^2*c^2*d - 5*a*b*c*d^2)*f)*h)*x)*sqrt((d*e - c 
*f)/d)*log((d*f*x + 2*d*e - c*f + 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/( 
d*x + c)) + 2*(2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f*h*x + ((b^2*c*d^2 - 
 a*b*d^3)*e - (b^2*c^2*d - a*b*c*d^2)*f)*g - ((b^2*c^2*d - a*b*c*d^2)*e - 
(3*b^2*c^3 - 5*a*b*c^2*d + 2*a^2*c*d^2)*f)*h)*sqrt(f*x + e))/(b^3*c^3*d^2 
- 2*a*b^2*c^2*d^3 + a^2*b*c*d^4 + (b^3*c^2*d^3 - 2*a*b^2*c*d^4 + a^2*b*d^5 
)*x), -1/2*(4*((b^2*c*d^2*e - a*b*c*d^2*f)*g - (a*b*c*d^2*e - a^2*c*d^2*f) 
*h + ((b^2*d^3*e - a*b*d^3*f)*g - (a*b*d^3*e - a^2*d^3*f)*h)*x)*sqrt(-(b*e 
 - a*f)/b)*arctan(-sqrt(f*x + e)*b*sqrt(-(b*e - a*f)/b)/(b*e - a*f)) - ((2 
*b^2*c*d^2*e + (b^2*c^2*d - 3*a*b*c*d^2)*f)*g - (2*a*b*c*d^2*e + (3*b^2*c^ 
3 - 5*a*b*c^2*d)*f)*h + ((2*b^2*d^3*e + (b^2*c*d^2 - 3*a*b*d^3)*f)*g - (2* 
a*b*d^3*e + (3*b^2*c^2*d - 5*a*b*c*d^2)*f)*h)*x)*sqrt((d*e - c*f)/d)*log(( 
d*f*x + 2*d*e - c*f + 2*sqrt(f*x + e)*d*sqrt((d*e - c*f)/d))/(d*x + c)) - 
2*(2*(b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*f*h*x + ((b^2*c*d^2 - a*b*d^3)*e 
- (b^2*c^2*d - a*b*c*d^2)*f)*g - ((b^2*c^2*d - a*b*c*d^2)*e - (3*b^2*c^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)/(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(a*f-b*e>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.61 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\frac {2 \, {\left (b^{3} e^{2} g - 2 \, a b^{2} e f g + a^{2} b f^{2} g - a b^{2} e^{2} h + 2 \, a^{2} b e f h - a^{3} f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {-b^{2} e + a b f}} - \frac {{\left (2 \, b d^{3} e^{2} g - b c d^{2} e f g - 3 \, a d^{3} e f g - b c^{2} d f^{2} g + 3 \, a c d^{2} f^{2} g - 2 \, a d^{3} e^{2} h - 3 \, b c^{2} d e f h + 7 \, a c d^{2} e f h + 3 \, b c^{3} f^{2} h - 5 \, a c^{2} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} \sqrt {-d^{2} e + c d f}} + \frac {2 \, \sqrt {f x + e} f h}{b d^{2}} + \frac {\sqrt {f x + e} d^{2} e f g - \sqrt {f x + e} c d f^{2} g - \sqrt {f x + e} c d e f h + \sqrt {f x + e} c^{2} f^{2} h}{{\left (b c d^{2} - a d^{3}\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 8.59 (sec) , antiderivative size = 18557, normalized size of antiderivative = 75.13 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((2*(2*a*b^7*c^5*d^4*f^5*g + 2*a^5*b^3*c*d^8*f^5*g - 6*a*b^7*c^6*d^ 
3*f^5*h + 4*a^6*b^2*c*d^8*f^5*h - 2*a^5*b^3*d^9*e*f^4*g - 4*a^6*b^2*d^9*e* 
f^4*h - 2*b^8*c^5*d^4*e*f^4*g + 6*b^8*c^6*d^3*e*f^4*h - 8*a^2*b^6*c^4*d^5* 
f^5*g + 12*a^3*b^5*c^3*d^6*f^5*g - 8*a^4*b^4*c^2*d^7*f^5*g + 28*a^2*b^6*c^ 
5*d^4*f^5*h - 52*a^3*b^5*c^4*d^5*f^5*h + 48*a^4*b^4*c^3*d^6*f^5*h - 22*a^5 
*b^3*c^2*d^7*f^5*h + 2*a^4*b^4*d^9*e^2*f^3*g + 4*a^5*b^3*d^9*e^2*f^3*h + 2 
*b^8*c^4*d^5*e^2*f^3*g - 6*b^8*c^5*d^4*e^2*f^3*h + 12*a^2*b^6*c^2*d^7*e^2* 
f^3*g - 52*a^2*b^6*c^3*d^6*e^2*f^3*h + 48*a^3*b^5*c^2*d^7*e^2*f^3*h + 6*a* 
b^7*c^4*d^5*e*f^4*g + 6*a^4*b^4*c*d^8*e*f^4*g - 22*a*b^7*c^5*d^4*e*f^4*h + 
 18*a^5*b^3*c*d^8*e*f^4*h - 8*a*b^7*c^3*d^6*e^2*f^3*g - 4*a^2*b^6*c^3*d^6* 
e*f^4*g - 8*a^3*b^5*c*d^8*e^2*f^3*g - 4*a^3*b^5*c^2*d^7*e*f^4*g + 28*a*b^7 
*c^4*d^5*e^2*f^3*h + 24*a^2*b^6*c^4*d^5*e*f^4*h + 4*a^3*b^5*c^3*d^6*e*f^4* 
h - 22*a^4*b^4*c*d^8*e^2*f^3*h - 26*a^4*b^4*c^2*d^7*e*f^4*h))/(a^3*b*d^6 - 
 b^4*c^3*d^3 + 3*a*b^3*c^2*d^4 - 3*a^2*b^2*c*d^5) - (2*(e + f*x)^(1/2)*((4 
*a^2*d^5*e^3*h^2 + 4*b^2*d^5*e^3*g^2 - 9*b^2*c^5*f^3*h^2 - 25*a^2*c^3*d^2* 
f^3*h^2 - b^2*c^3*d^2*f^3*g^2 - 9*a^2*c*d^4*f^3*g^2 + 9*a^2*d^5*e*f^2*g^2 
+ 6*a*b*c^2*d^3*f^3*g^2 - 24*a^2*c*d^4*e^2*f*h^2 + 9*b^2*c^4*d*e*f^2*h^2 + 
 30*a^2*c^2*d^3*f^3*g*h - 8*a*b*d^5*e^3*g*h + 45*a^2*c^2*d^3*e*f^2*h^2 - 3 
*b^2*c^2*d^3*e*f^2*g^2 + 30*a*b*c^4*d*f^3*h^2 - 12*a*b*d^5*e^2*f*g^2 + 6*b 
^2*c^4*d*f^3*g*h + 12*a^2*d^5*e^2*f*g*h + 6*a*b*c*d^4*e*f^2*g^2 - 28*a*...
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 1199, normalized size of antiderivative = 4.85 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b 
*e)))*a**2*c*d**3*f*h - 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/( 
sqrt(b)*sqrt(a*f - b*e)))*a**2*d**4*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*b*c*d**3*e*h + 2*sqrt(b)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*c*d** 
3*f*g + 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
 - b*e)))*a*b*d**4*e*h*x + 2*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b 
)/(sqrt(b)*sqrt(a*f - b*e)))*a*b*d**4*f*g*x - 2*sqrt(b)*sqrt(a*f - b*e)*at 
an((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2*c*d**3*e*g - 2*sqrt(b 
)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*b**2*d 
**4*e*g*x + 5*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt 
(c*f - d*e)))*a*b**2*c**2*d*f*h - 2*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + 
 f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c*d**2*e*h - 3*sqrt(d)*sqrt(c*f 
 - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*c*d**2*f* 
g + 5*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d 
*e)))*a*b**2*c*d**2*f*h*x - 2*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)* 
d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*d**3*e*h*x - 3*sqrt(d)*sqrt(c*f - d*e 
)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*b**2*d**3*f*g*x - 3* 
sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))* 
b**3*c**3*f*h + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)...