\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)} \, dx\) [83]

Optimal result

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

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

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.94 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=\frac {\frac {(b c-a d) \sqrt {e+f x} \left (-3 a^2 d f h+b^2 (-d e g+2 c f h x)+a b (d e h+2 c f h+d f (g-2 h x))\right )}{b^2 d (a+b x)}+\frac {\sqrt {-b e+a f} \left (-3 a^2 d f h+a b f (d g+5 c h)+b^2 (2 d e g-3 c f g-2 c e h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{b^{5/2}}+\frac {2 (-d e+c f)^{3/2} (d g-c h) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2}}}{(b c-a d)^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.09, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-(((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x))) - ((-2*f*(b*d*g 
 + 2*b*c*h - 3*a*d*h)*Sqrt[e + f*x])/(b*d) + ((2*d*Sqrt[b*e - a*f]*(3*a^2* 
d*f*h - a*b*f*(d*g + 5*c*h) - b^2*(2*d*e*g - 3*c*f*g - 2*c*e*h))*ArcTanh[( 
Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)) + (4*b^2*(d 
*e - c*f)^(3/2)*(d*g - c*h)*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f 
]])/(Sqrt[d]*(b*c - a*d)))/(b*d))/(2*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 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 = 0.71 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {-3 \left (\left (a^{2} f h -\frac {1}{3} b f g a -\frac {2}{3} b^{2} e g \right ) d -\frac {5 c \left (\frac {\left (-2 e h -3 f g \right ) b}{5}+a f h \right ) b}{3}\right ) d \left (b x +a \right ) \left (a f -b e \right ) \sqrt {\left (c f -d e \right ) d}\, \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+3 \sqrt {\left (a f -b e \right ) b}\, \left (-\frac {2 b^{2} \left (c f -d e \right )^{2} \left (c h -d g \right ) \left (b x +a \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{3}+\left (a d -b c \right ) \left (\left (\frac {b^{2} e g}{3}-\frac {a \left (\left (-2 h x +g \right ) f +e h \right ) b}{3}+a^{2} f h \right ) d -\frac {2 b c f h \left (b x +a \right )}{3}\right ) \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\right )}{d \,b^{2} \left (b x +a \right ) \left (a d -b c \right )^{2} \sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}}\)	\(281\)
derivativedivides	\(2 f \left (\frac {h \sqrt {f x +e}}{d \,b^{2}}-\frac {\frac {\left (-\frac {1}{2} a^{3} d \,f^{2} h +\frac {1}{2} a^{2} b c \,f^{2} h +\frac {1}{2} a^{2} b d e f h +\frac {1}{2} a^{2} b d \,f^{2} g -\frac {1}{2} a \,b^{2} c e f h -\frac {1}{2} a \,b^{2} c \,f^{2} g -\frac {1}{2} a \,b^{2} d e f g +\frac {1}{2} b^{3} c e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (3 a^{3} d \,f^{2} h -5 a^{2} b c \,f^{2} h -3 a^{2} b d e f h -a^{2} b d \,f^{2} g +7 a \,b^{2} c e f h +3 a \,b^{2} c \,f^{2} g -a \,b^{2} d e f g -2 b^{3} c \,e^{2} h -3 b^{3} c e f g +2 b^{3} d \,e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}}{b^{2} \left (a d -b c \right )^{2} f}+\frac {\left (-c^{3} f^{2} h +2 c^{2} d e f h +c^{2} d \,f^{2} g -c \,d^{2} e^{2} h -2 c \,d^{2} e f g +d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d f \left (a d -b c \right )^{2} \sqrt {\left (c f -d e \right ) d}}\right )\)	\(392\)
default	\(2 f \left (\frac {h \sqrt {f x +e}}{d \,b^{2}}-\frac {\frac {\left (-\frac {1}{2} a^{3} d \,f^{2} h +\frac {1}{2} a^{2} b c \,f^{2} h +\frac {1}{2} a^{2} b d e f h +\frac {1}{2} a^{2} b d \,f^{2} g -\frac {1}{2} a \,b^{2} c e f h -\frac {1}{2} a \,b^{2} c \,f^{2} g -\frac {1}{2} a \,b^{2} d e f g +\frac {1}{2} b^{3} c e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (3 a^{3} d \,f^{2} h -5 a^{2} b c \,f^{2} h -3 a^{2} b d e f h -a^{2} b d \,f^{2} g +7 a \,b^{2} c e f h +3 a \,b^{2} c \,f^{2} g -a \,b^{2} d e f g -2 b^{3} c \,e^{2} h -3 b^{3} c e f g +2 b^{3} d \,e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}}{b^{2} \left (a d -b c \right )^{2} f}+\frac {\left (-c^{3} f^{2} h +2 c^{2} d e f h +c^{2} d \,f^{2} g -c \,d^{2} e^{2} h -2 c \,d^{2} e f g +d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{d f \left (a d -b c \right )^{2} \sqrt {\left (c f -d e \right ) d}}\right )\)	\(392\)
risch	\(\frac {2 h \sqrt {f x +e}\, f}{d \,b^{2}}-\frac {2 f \left (\frac {d \left (\frac {\left (-\frac {1}{2} a^{3} d \,f^{2} h +\frac {1}{2} a^{2} b c \,f^{2} h +\frac {1}{2} a^{2} b d e f h +\frac {1}{2} a^{2} b d \,f^{2} g -\frac {1}{2} a \,b^{2} c e f h -\frac {1}{2} a \,b^{2} c \,f^{2} g -\frac {1}{2} a \,b^{2} d e f g +\frac {1}{2} b^{3} c e f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (3 a^{3} d \,f^{2} h -5 a^{2} b c \,f^{2} h -3 a^{2} b d e f h -a^{2} b d \,f^{2} g +7 a \,b^{2} c e f h +3 a \,b^{2} c \,f^{2} g -a \,b^{2} d e f g -2 b^{3} c \,e^{2} h -3 b^{3} c e f g +2 b^{3} d \,e^{2} g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{2} f}+\frac {b^{2} \left (c^{3} f^{2} h -2 c^{2} d e f h -c^{2} d \,f^{2} g +c \,d^{2} e^{2} h +2 c \,d^{2} e f g -d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{f \left (a d -b c \right )^{2} \sqrt {\left (c f -d e \right ) d}}\right )}{d \,b^{2}}\)	\(398\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (223) = 446\).

Time = 12.39 (sec) , antiderivative size = 2136, normalized size of antiderivative = 8.65 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2/(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 [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.62 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=-\frac {{\left (2 \, b^{3} d e^{2} g - 3 \, b^{3} c e f g - a b^{2} d e f g + 3 \, a b^{2} c f^{2} g - a^{2} b d f^{2} g - 2 \, b^{3} c e^{2} h + 7 \, a b^{2} c e f h - 3 \, a^{2} b d e f h - 5 \, a^{2} b c f^{2} h + 3 \, a^{3} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt {-b^{2} e + a b f}} + \frac {2 \, {\left (d^{3} e^{2} g - 2 \, c d^{2} e f g + c^{2} d f^{2} g - c d^{2} e^{2} h + 2 \, c^{2} d e f h - c^{3} f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \sqrt {-d^{2} e + c d f}} + \frac {2 \, \sqrt {f x + e} f h}{b^{2} d} - \frac {\sqrt {f x + e} b^{2} e f g - \sqrt {f x + e} a b f^{2} g - \sqrt {f x + e} a b e f h + \sqrt {f x + e} a^{2} f^{2} h}{{\left (b^{3} c - a b^{2} d\right )} {\left ({\left (f x + e\right )} b - b e + a f\right )}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 9.21 (sec) , antiderivative size = 18554, normalized size of antiderivative = 75.12 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

atan(((((2*(2*a*b^8*c^5*d^3*f^5*g + 2*a^5*b^4*c*d^7*f^5*g + 4*a*b^8*c^6*d^ 
2*f^5*h - 6*a^6*b^3*c*d^7*f^5*h - 2*a^5*b^4*d^8*e*f^4*g + 6*a^6*b^3*d^8*e* 
f^4*h - 2*b^9*c^5*d^3*e*f^4*g - 4*b^9*c^6*d^2*e*f^4*h - 8*a^2*b^7*c^4*d^4* 
f^5*g + 12*a^3*b^6*c^3*d^5*f^5*g - 8*a^4*b^5*c^2*d^6*f^5*g - 22*a^2*b^7*c^ 
5*d^3*f^5*h + 48*a^3*b^6*c^4*d^4*f^5*h - 52*a^4*b^5*c^3*d^5*f^5*h + 28*a^5 
*b^4*c^2*d^6*f^5*h + 2*a^4*b^5*d^8*e^2*f^3*g - 6*a^5*b^4*d^8*e^2*f^3*h + 2 
*b^9*c^4*d^4*e^2*f^3*g + 4*b^9*c^5*d^3*e^2*f^3*h + 12*a^2*b^7*c^2*d^6*e^2* 
f^3*g + 48*a^2*b^7*c^3*d^5*e^2*f^3*h - 52*a^3*b^6*c^2*d^6*e^2*f^3*h + 6*a* 
b^8*c^4*d^4*e*f^4*g + 6*a^4*b^5*c*d^7*e*f^4*g + 18*a*b^8*c^5*d^3*e*f^4*h - 
 22*a^5*b^4*c*d^7*e*f^4*h - 8*a*b^8*c^3*d^5*e^2*f^3*g - 4*a^2*b^7*c^3*d^5* 
e*f^4*g - 8*a^3*b^6*c*d^7*e^2*f^3*g - 4*a^3*b^6*c^2*d^6*e*f^4*g - 22*a*b^8 
*c^4*d^4*e^2*f^3*h - 26*a^2*b^7*c^4*d^4*e*f^4*h + 4*a^3*b^6*c^3*d^5*e*f^4* 
h + 28*a^4*b^5*c*d^7*e^2*f^3*h + 24*a^4*b^5*c^2*d^6*e*f^4*h))/(b^6*c^3*d - 
 a^3*b^3*d^4 - 3*a*b^5*c^2*d^2 + 3*a^2*b^4*c*d^3) - (2*(e + f*x)^(1/2)*((4 
*b^5*c^2*e^3*h^2 + 4*b^5*d^2*e^3*g^2 - 9*a^5*d^2*f^3*h^2 - 25*a^3*b^2*c^2* 
f^3*h^2 - a^3*b^2*d^2*f^3*g^2 - 9*a*b^4*c^2*f^3*g^2 + 9*b^5*c^2*e*f^2*g^2 
+ 6*a^2*b^3*c*d*f^3*g^2 - 24*a*b^4*c^2*e^2*f*h^2 + 9*a^4*b*d^2*e*f^2*h^2 + 
 30*a^2*b^3*c^2*f^3*g*h - 8*b^5*c*d*e^3*g*h + 45*a^2*b^3*c^2*e*f^2*h^2 - 3 
*a^2*b^3*d^2*e*f^2*g^2 + 30*a^4*b*c*d*f^3*h^2 - 12*b^5*c*d*e^2*f*g^2 + 6*a 
^4*b*d^2*f^3*g*h + 12*b^5*c^2*e^2*f*g*h + 6*a*b^4*c*d*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 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)} \, dx =\text {Too large to display} \] Input:

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

Output:

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