\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx\) [100]

Optimal result

Integrand size = 29, antiderivative size = 353 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx=-\frac {\left (a d (3 d f g+4 d e h-7 c f h)-b \left (4 d^2 e g-c d f g-3 c^2 f h\right )\right ) \sqrt {e+f x}}{4 d^2 (b c-a d)^2 (c+d x)}+\frac {(d g-c h) (e+f x)^{3/2}}{2 d (b c-a d) (c+d x)^2}-\frac {2 (b e-a f)^{3/2} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d)^3}+\frac {\left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (8 d^3 e^2 g-4 c d^2 e f g-c^2 d f^2 g-3 c^3 f^2 h\right )+2 a b d \left (5 c^2 f^2 h-2 d^2 e (3 f g+2 e h)+c d f (3 f g+2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{5/2} (b c-a d)^3 \sqrt {d e-c f}} \] Output:

-1/4*(a*d*(-7*c*f*h+4*d*e*h+3*d*f*g)-b*(-3*c^2*f*h-c*d*f*g+4*d^2*e*g))*(f* 
x+e)^(1/2)/d^2/(-a*d+b*c)^2/(d*x+c)+1/2*(-c*h+d*g)*(f*x+e)^(3/2)/d/(-a*d+b 
*c)/(d*x+c)^2-2*(-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^(1/2)/(-a*d+b*c)^3+1/4*(3*a^2*d^2*f*(-5*c*f*h+4*d*e*h+ 
d*f*g)+b^2*(-3*c^3*f^2*h-c^2*d*f^2*g-4*c*d^2*e*f*g+8*d^3*e^2*g)+2*a*b*d*(5 
*c^2*f^2*h-2*d^2*e*(2*e*h+3*f*g)+c*d*f*(2*e*h+3*f*g)))*arctanh(d^(1/2)*(f* 
x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(5/2)/(-a*d+b*c)^3/(-c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 1.59 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.03 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\frac {1}{4} \left (\frac {\sqrt {e+f x} \left (-a d \left (-7 c^2 f h+d^2 (2 e g+5 f g x+4 e h x)+c d (3 f g+2 e h-9 f h x)\right )+b \left (-3 c^3 f h+4 d^3 e g x+c d^2 g (6 e+f x)-c^2 d (f g+2 e h+5 f h x)\right )\right )}{d^2 (b c-a d)^2 (c+d x)^2}+\frac {8 (-b e+a f)^{3/2} (b g-a h) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{\sqrt {b} (b c-a d)^3}+\frac {\left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (8 d^3 e^2 g-4 c d^2 e f g-c^2 d f^2 g-3 c^3 f^2 h\right )+2 a b d \left (5 c^2 f^2 h-2 d^2 e (3 f g+2 e h)+c d f (3 f g+2 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{5/2} (-b c+a d)^3 \sqrt {-d e+c f}}\right ) \] Input:

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

Output:

((Sqrt[e + f*x]*(-(a*d*(-7*c^2*f*h + d^2*(2*e*g + 5*f*g*x + 4*e*h*x) + c*d 
*(3*f*g + 2*e*h - 9*f*h*x))) + b*(-3*c^3*f*h + 4*d^3*e*g*x + c*d^2*g*(6*e 
+ f*x) - c^2*d*(f*g + 2*e*h + 5*f*h*x))))/(d^2*(b*c - a*d)^2*(c + d*x)^2) 
+ (8*(-(b*e) + a*f)^(3/2)*(b*g - a*h)*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[ 
-(b*e) + a*f]])/(Sqrt[b]*(b*c - a*d)^3) + ((3*a^2*d^2*f*(d*f*g + 4*d*e*h - 
 5*c*f*h) + b^2*(8*d^3*e^2*g - 4*c*d^2*e*f*g - c^2*d*f^2*g - 3*c^3*f^2*h) 
+ 2*a*b*d*(5*c^2*f^2*h - 2*d^2*e*(3*f*g + 2*e*h) + c*d*f*(3*f*g + 2*e*h))) 
*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(5/2)*(-(b*c) + a* 
d)^3*Sqrt[-(d*e) + c*f]))/4
 

Rubi [A] (verified)

Time = 0.73 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 166, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {-\frac {\int -\frac {-d f (7 c f h-3 d (f g+4 e h)) a^2+b \left (-4 e (3 f g+2 e h) d^2+c f^2 g d+3 c^2 f^2 h\right ) a+8 b^2 d^2 e^2 g+f \left (\left (3 f h c^2+d f g c+4 d^2 e g\right ) b^2-a d (5 d f g+4 d e h+7 c f h) b+8 a^2 d^2 f h\right ) x}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{d (b c-a d)}-\frac {\sqrt {e+f x} \left (a d (-7 c f h+4 d e h+3 d f g)-b \left (-3 c^2 f h-c d f g+4 d^2 e g\right )\right )}{d (c+d x) (b c-a d)}}{4 d (b c-a d)}+\frac {(e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {-d f (7 c f h-3 d (f g+4 e h)) a^2+b \left (-4 e (3 f g+2 e h) d^2+c f^2 g d+3 c^2 f^2 h\right ) a+8 b^2 d^2 e^2 g+f \left (\left (3 f h c^2+d f g c+4 d^2 e g\right ) b^2-a d (5 d f g+4 d e h+7 c f h) b+8 a^2 d^2 f h\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{2 d (b c-a d)}-\frac {\sqrt {e+f x} \left (a d (-7 c f h+4 d e h+3 d f g)-b \left (-3 c^2 f h-c d f g+4 d^2 e g\right )\right )}{d (c+d x) (b c-a d)}}{4 d (b c-a d)}+\frac {(e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {8 d^2 (b e-a f)^2 (b g-a h) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {\left (3 a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (5 c^2 f^2 h+c d f (2 e h+3 f g)-2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-3 c^3 f^2 h-c^2 d f^2 g-4 c d^2 e f g+8 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} \left (a d (-7 c f h+4 d e h+3 d f g)-b \left (-3 c^2 f h-c d f g+4 d^2 e g\right )\right )}{d (c+d x) (b c-a d)}}{4 d (b c-a d)}+\frac {(e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {\frac {16 d^2 (b e-a f)^2 (b g-a h) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 \left (3 a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (5 c^2 f^2 h+c d f (2 e h+3 f g)-2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-3 c^3 f^2 h-c^2 d f^2 g-4 c d^2 e f g+8 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} \left (a d (-7 c f h+4 d e h+3 d f g)-b \left (-3 c^2 f h-c d f g+4 d^2 e g\right )\right )}{d (c+d x) (b c-a d)}}{4 d (b c-a d)}+\frac {(e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (3 a^2 d^2 f (-5 c f h+4 d e h+d f g)+2 a b d \left (5 c^2 f^2 h+c d f (2 e h+3 f g)-2 d^2 e (2 e h+3 f g)\right )+b^2 \left (-3 c^3 f^2 h-c^2 d f^2 g-4 c d^2 e f g+8 d^3 e^2 g\right )\right )}{\sqrt {d} (b c-a d) \sqrt {d e-c f}}-\frac {16 d^2 (b e-a f)^{3/2} (b g-a h) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} \left (a d (-7 c f h+4 d e h+3 d f g)-b \left (-3 c^2 f h-c d f g+4 d^2 e g\right )\right )}{d (c+d x) (b c-a d)}}{4 d (b c-a d)}+\frac {(e+f x)^{3/2} (d g-c h)}{2 d (c+d x)^2 (b c-a d)}\)

Input:

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

Output:

((d*g - c*h)*(e + f*x)^(3/2))/(2*d*(b*c - a*d)*(c + d*x)^2) + (-(((a*d*(3* 
d*f*g + 4*d*e*h - 7*c*f*h) - b*(4*d^2*e*g - c*d*f*g - 3*c^2*f*h))*Sqrt[e + 
 f*x])/(d*(b*c - a*d)*(c + d*x))) + ((-16*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*(3*a^2*d^2*f*(d*f*g + 4*d*e*h - 5*c*f*h) + b^2*(8*d^3*e^2*g - 4*c*d^2 
*e*f*g - c^2*d*f^2*g - 3*c^3*f^2*h) + 2*a*b*d*(5*c^2*f^2*h - 2*d^2*e*(3*f* 
g + 2*e*h) + c*d*f*(3*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*d*(b*c - a*d)))/(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 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 = 0.81 (sec) , antiderivative size = 388, normalized size of antiderivative = 1.10

method	result	size
pseudoelliptic	\(\frac {-\frac {15 \sqrt {\left (a f -b e \right ) b}\, \left (\frac {4 \left (-\frac {2 b^{2} e^{2} g}{3}+\frac {2 a \left (e h +\frac {3 f g}{2}\right ) e b}{3}-a^{2} \left (e h +\frac {f g}{4}\right ) f \right ) d^{3}}{5}+c \left (\frac {4 b^{2} e g}{15}-\frac {4 a \left (e h +\frac {3 f g}{2}\right ) b}{15}+a^{2} f h \right ) f \,d^{2}-\frac {2 c^{2} \left (a h -\frac {b g}{10}\right ) b \,f^{2} d}{3}+\frac {b^{2} c^{3} f^{2} h}{5}\right ) \left (x d +c \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4}+2 \sqrt {\left (c f -d e \right ) d}\, \left (d^{2} \left (x d +c \right )^{2} \left (a f -b e \right )^{2} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\frac {7 \sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right ) \sqrt {f x +e}\, \left (\frac {2 \left (2 b e g x -a \left (\frac {5 f g x}{2}+e \left (2 h x +g \right )\right )\right ) d^{3}}{7}-\frac {2 c \left (-3 \left (\frac {f x}{6}+e \right ) g b +a \left (\frac {3 \left (-3 h x +g \right ) f}{2}+e h \right )\right ) d^{2}}{7}+c^{2} \left (\frac {\left (\left (-5 h x -g \right ) f -2 e h \right ) b}{7}+a f h \right ) d -\frac {3 c^{3} h b f}{7}\right )}{8}\right )}{\sqrt {\left (c f -d e \right ) d}\, \sqrt {\left (a f -b e \right ) b}\, \left (a d -b c \right )^{3} \left (x d +c \right )^{2} d^{2}}\)	\(388\)
derivativedivides	\(2 f^{2} \left (\frac {\left (a f -b e \right )^{2} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}}-\frac {\frac {-\frac {f \left (9 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -5 a^{2} d^{3} f g -14 a b \,c^{2} d f h +4 a b c \,d^{2} e h +6 a b c \,d^{2} f g +4 a b \,d^{3} e g +5 b^{2} c^{3} f h -b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 d}-\frac {f \left (7 a^{2} c^{2} d^{2} f^{2} h -11 a^{2} c \,d^{3} e f h -3 a^{2} c \,d^{3} f^{2} g +4 a^{2} d^{4} e^{2} h +3 a^{2} d^{4} e f g -10 a b \,c^{3} d \,f^{2} h +14 a b \,c^{2} d^{2} e f h +2 a b \,c^{2} d^{2} f^{2} g -4 a b c \,d^{3} e^{2} h +2 a b c \,d^{3} e f g -4 a b \,d^{4} e^{2} g +3 b^{2} c^{4} f^{2} h -3 b^{2} c^{3} d e f h +b^{2} c^{3} d \,f^{2} g -5 b^{2} c^{2} d^{2} e f g +4 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8 d^{2}}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a^{2} c \,d^{2} f^{2} h -12 a^{2} d^{3} e f h -3 a^{2} d^{3} f^{2} g -10 a b \,c^{2} d \,f^{2} h -4 a b c \,d^{2} e f h -6 a b c \,d^{2} f^{2} g +8 e^{2} h b a \,d^{3}+12 a b \,d^{3} e f g +3 b^{2} c^{3} f^{2} h +b^{2} c^{2} d \,f^{2} g +4 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 d^{2} \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{3} f^{2}}\right )\)	\(617\)
default	\(2 f^{2} \left (\frac {\left (a f -b e \right )^{2} \left (a h -b g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{f^{2} \left (a d -b c \right )^{3} \sqrt {\left (a f -b e \right ) b}}-\frac {\frac {-\frac {f \left (9 a^{2} c \,d^{2} f h -4 a^{2} d^{3} e h -5 a^{2} d^{3} f g -14 a b \,c^{2} d f h +4 a b c \,d^{2} e h +6 a b c \,d^{2} f g +4 a b \,d^{3} e g +5 b^{2} c^{3} f h -b^{2} c^{2} d f g -4 b^{2} c \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 d}-\frac {f \left (7 a^{2} c^{2} d^{2} f^{2} h -11 a^{2} c \,d^{3} e f h -3 a^{2} c \,d^{3} f^{2} g +4 a^{2} d^{4} e^{2} h +3 a^{2} d^{4} e f g -10 a b \,c^{3} d \,f^{2} h +14 a b \,c^{2} d^{2} e f h +2 a b \,c^{2} d^{2} f^{2} g -4 a b c \,d^{3} e^{2} h +2 a b c \,d^{3} e f g -4 a b \,d^{4} e^{2} g +3 b^{2} c^{4} f^{2} h -3 b^{2} c^{3} d e f h +b^{2} c^{3} d \,f^{2} g -5 b^{2} c^{2} d^{2} e f g +4 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8 d^{2}}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a^{2} c \,d^{2} f^{2} h -12 a^{2} d^{3} e f h -3 a^{2} d^{3} f^{2} g -10 a b \,c^{2} d \,f^{2} h -4 a b c \,d^{2} e f h -6 a b c \,d^{2} f^{2} g +8 e^{2} h b a \,d^{3}+12 a b \,d^{3} e f g +3 b^{2} c^{3} f^{2} h +b^{2} c^{2} d \,f^{2} g +4 b^{2} c \,d^{2} e f g -8 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 d^{2} \sqrt {\left (c f -d e \right ) d}}}{\left (a d -b c \right )^{3} f^{2}}\right )\)	\(617\)

Input:

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

Output:

2*(-15/8*((a*f-b*e)*b)^(1/2)*(4/5*(-2/3*b^2*e^2*g+2/3*a*(e*h+3/2*f*g)*e*b- 
a^2*(e*h+1/4*f*g)*f)*d^3+c*(4/15*b^2*e*g-4/15*a*(e*h+3/2*f*g)*b+a^2*f*h)*f 
*d^2-2/3*c^2*(a*h-1/10*b*g)*b*f^2*d+1/5*b^2*c^3*f^2*h)*(d*x+c)^2*arctan(d* 
(f*x+e)^(1/2)/((c*f-d*e)*d)^(1/2))+((c*f-d*e)*d)^(1/2)*(d^2*(d*x+c)^2*(a*f 
-b*e)^2*(a*h-b*g)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+7/8*((a*f-b* 
e)*b)^(1/2)*(a*d-b*c)*(f*x+e)^(1/2)*(2/7*(2*b*e*g*x-a*(5/2*f*g*x+e*(2*h*x+ 
g)))*d^3-2/7*c*(-3*(1/6*f*x+e)*g*b+a*(3/2*(-3*h*x+g)*f+e*h))*d^2+c^2*(1/7* 
((-5*h*x-g)*f-2*e*h)*b+a*f*h)*d-3/7*c^3*h*b*f)))/((a*f-b*e)*b)^(1/2)/((c*f 
-d*e)*d)^(1/2)/(a*d-b*c)^3/(d*x+c)^2/d^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1542 vs. \(2 (323) = 646\).

Time = 35.30 (sec) , antiderivative size = 6213, normalized size of antiderivative = 17.60 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(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. 687 vs. \(2 (323) = 646\).

Time = 0.16 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.95 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\frac {2 \, {\left (b^{3} e^{2} g - 2 \, a b^{2} e f g + a^{2} b f^{2} g - a b^{2} e^{2} h + 2 \, a^{2} b e f h - a^{3} f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt {-b^{2} e + a b f}} - \frac {{\left (8 \, b^{2} d^{3} e^{2} g - 4 \, b^{2} c d^{2} e f g - 12 \, a b d^{3} e f g - b^{2} c^{2} d f^{2} g + 6 \, a b c d^{2} f^{2} g + 3 \, a^{2} d^{3} f^{2} g - 8 \, a b d^{3} e^{2} h + 4 \, a b c d^{2} e f h + 12 \, a^{2} d^{3} e f h - 3 \, b^{2} c^{3} f^{2} h + 10 \, a b c^{2} d f^{2} h - 15 \, a^{2} c d^{2} f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{4 \, {\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} \sqrt {-d^{2} e + c d f}} + \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b d^{3} e f g - 4 \, \sqrt {f x + e} b d^{3} e^{2} f g + {\left (f x + e\right )}^{\frac {3}{2}} b c d^{2} f^{2} g - 5 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} f^{2} g + 5 \, \sqrt {f x + e} b c d^{2} e f^{2} g + 3 \, \sqrt {f x + e} a d^{3} e f^{2} g - \sqrt {f x + e} b c^{2} d f^{3} g - 3 \, \sqrt {f x + e} a c d^{2} f^{3} g - 4 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} e f h + 4 \, \sqrt {f x + e} a d^{3} e^{2} f h - 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b c^{2} d f^{2} h + 9 \, {\left (f x + e\right )}^{\frac {3}{2}} a c d^{2} f^{2} h + 3 \, \sqrt {f x + e} b c^{2} d e f^{2} h - 11 \, \sqrt {f x + e} a c d^{2} e f^{2} h - 3 \, \sqrt {f x + e} b c^{3} f^{3} h + 7 \, \sqrt {f x + e} a c^{2} d f^{3} h}{4 \, {\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}^{2}} \] Input:

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)/(d*x+c)^3,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^3 - 3*a*b 
^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sqrt(-b^2*e + a*b*f)) - 1/4*(8*b^2*d^3 
*e^2*g - 4*b^2*c*d^2*e*f*g - 12*a*b*d^3*e*f*g - b^2*c^2*d*f^2*g + 6*a*b*c* 
d^2*f^2*g + 3*a^2*d^3*f^2*g - 8*a*b*d^3*e^2*h + 4*a*b*c*d^2*e*f*h + 12*a^2 
*d^3*e*f*h - 3*b^2*c^3*f^2*h + 10*a*b*c^2*d*f^2*h - 15*a^2*c*d^2*f^2*h)*ar 
ctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^3*c^3*d^2 - 3*a*b^2*c^2*d^3 
 + 3*a^2*b*c*d^4 - a^3*d^5)*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 + (f*x + e)^(3/2)*b*c*d^2*f^2 
*g - 5*(f*x + e)^(3/2)*a*d^3*f^2*g + 5*sqrt(f*x + e)*b*c*d^2*e*f^2*g + 3*s 
qrt(f*x + e)*a*d^3*e*f^2*g - sqrt(f*x + e)*b*c^2*d*f^3*g - 3*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 - 5*(f*x + e)^(3/2)*b*c^2*d*f^2*h + 9*(f*x + e)^(3/2)*a*c*d^2*f^2*h + 
 3*sqrt(f*x + e)*b*c^2*d*e*f^2*h - 11*sqrt(f*x + e)*a*c*d^2*e*f^2*h - 3*sq 
rt(f*x + e)*b*c^3*f^3*h + 7*sqrt(f*x + e)*a*c^2*d*f^3*h)/((b^2*c^2*d^2 - 2 
*a*b*c*d^3 + a^2*d^4)*((f*x + e)*d - d*e + c*f)^2)
 

Mupad [B] (verification not implemented)

Time = 16.48 (sec) , antiderivative size = 283348, normalized size of antiderivative = 802.69 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

- (((e + f*x)^(3/2)*(5*a*d^2*f^2*g + 5*b*c^2*f^2*h - 9*a*c*d*f^2*h - b*c*d 
*f^2*g + 4*a*d^2*e*f*h - 4*b*d^2*e*f*g))/(4*d*(a^2*d^2 + b^2*c^2 - 2*a*b*c 
*d)) + ((e + f*x)^(1/2)*(3*b*c^3*f^3*h + 3*a*c*d^2*f^3*g - 7*a*c^2*d*f^3*h 
 + b*c^2*d*f^3*g - 3*a*d^3*e*f^2*g - 4*a*d^3*e^2*f*h + 4*b*d^3*e^2*f*g + 1 
1*a*c*d^2*e*f^2*h - 5*b*c*d^2*e*f^2*g - 3*b*c^2*d*e*f^2*h))/(4*d^2*(a^2*d^ 
2 + b^2*c^2 - 2*a*b*c*d)))/(d^2*(e + f*x)^2 - (e + f*x)*(2*d^2*e - 2*c*d*f 
) + c^2*f^2 + d^2*e^2 - 2*c*d*e*f) - atan(((((136*a^7*b^3*c*d^10*f^5*g - 8 
*a*b^9*c^7*d^4*f^5*g - 24*a^8*b^2*d^11*f^5*g - 24*a*b^9*c^8*d^3*f^5*h + 56 
*a^8*b^2*c*d^10*f^5*h + 56*a^7*b^3*d^11*e*f^4*g - 32*a^8*b^2*d^11*e*f^4*h 
+ 8*b^10*c^7*d^4*e*f^4*g + 24*b^10*c^8*d^3*e*f^4*h + 24*a^2*b^8*c^6*d^5*f^ 
5*g + 24*a^3*b^7*c^5*d^6*f^5*g - 200*a^4*b^6*c^4*d^7*f^5*g + 360*a^5*b^5*c 
^3*d^8*f^5*g - 312*a^6*b^4*c^2*d^9*f^5*g + 200*a^2*b^8*c^7*d^4*f^5*h - 696 
*a^3*b^7*c^6*d^5*f^5*h + 1320*a^4*b^6*c^5*d^6*f^5*h - 1480*a^5*b^5*c^4*d^7 
*f^5*h + 984*a^6*b^4*c^3*d^8*f^5*h - 360*a^7*b^3*c^2*d^9*f^5*h - 32*a^6*b^ 
4*d^11*e^2*f^3*g + 32*a^7*b^3*d^11*e^2*f^3*h - 32*b^10*c^6*d^5*e^2*f^3*g - 
 480*a^2*b^8*c^4*d^7*e^2*f^3*g + 640*a^3*b^7*c^3*d^8*e^2*f^3*g - 480*a^4*b 
^6*c^2*d^9*e^2*f^3*g - 192*a^2*b^8*c^5*d^6*e^2*f^3*h + 480*a^3*b^7*c^4*d^7 
*e^2*f^3*h - 640*a^4*b^6*c^3*d^8*e^2*f^3*h + 480*a^5*b^5*c^2*d^9*e^2*f^3*h 
 + 8*a*b^9*c^6*d^5*e*f^4*g - 328*a^6*b^4*c*d^10*e*f^4*g - 200*a*b^9*c^7*d^ 
4*e*f^4*h + 136*a^7*b^3*c*d^10*e*f^4*h + 192*a*b^9*c^5*d^6*e^2*f^3*g - ...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 4007, normalized size of antiderivative = 11.35 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x) (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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