\(\int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx\) [177]

Optimal result

Integrand size = 29, antiderivative size = 265 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {2 (b e-a f)^3 (f g-e h)}{f^3 (d e-c f)^2 \sqrt {e+f x}}+\frac {2 b^2 (3 a d f h+b (d f g-2 d e h-2 c f h)) \sqrt {e+f x}}{d^3 f^3}+\frac {(b c-a d)^3 (d g-c h) \sqrt {e+f x}}{d^3 (d e-c f)^2 (c+d x)}+\frac {2 b^3 h (e+f x)^{3/2}}{3 d^2 f^3}+\frac {(b c-a d)^2 \left (a d (3 d f g-2 d e h-c f h)-b \left (6 d^2 e g+5 c^2 f h-c d (3 f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{7/2} (d e-c f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.42 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {-9 a^2 b d^2 f^2 \left (2 d^2 e (-f g+e h) x+c^2 f h (e+f x)+c d \left (-3 e f g+2 e^2 h-f^2 g x\right )\right )+3 a^3 d^3 f^3 (c (-2 f g+3 e h+f h x)-d (3 f g x+e (g-2 h x)))-9 a b^2 d f \left (-3 c^3 f^2 h (e+f x)-2 d^3 e^2 x (-f g+2 e h+f h x)+c^2 d f (e+f x) (4 e h+f (g-2 h x))+2 c d^2 e \left (-2 e^2 h+2 f^2 h x^2+e f (g+h x)\right )\right )+b^3 \left (-15 c^4 f^3 h (e+f x)+c^3 d f^2 (e+f x) (9 f g+14 e h-10 f h x)+2 d^4 e^2 x \left (-8 e^2 h+e f (6 g-4 h x)+f^2 x (3 g+h x)\right )-2 c d^3 e \left (-6 e^2 f g+8 e^3 h+3 e f^2 x (g-h x)+2 f^3 x^2 (3 g+h x)\right )+2 c^2 d^2 f (e+f x) \left (4 e^2 h+f^2 x (3 g+h x)+e f (-6 g+5 h x)\right )\right )}{3 d^3 f^3 (d e-c f)^2 (c+d x) \sqrt {e+f x}}+\frac {(b c-a d)^2 \left (a d (-3 d f g+2 d e h+c f h)+b \left (6 d^2 e g+5 c^2 f h-c d (3 f g+8 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{7/2} (-d e+c f)^{5/2}} \] Input:

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

Output:

(-9*a^2*b*d^2*f^2*(2*d^2*e*(-(f*g) + e*h)*x + c^2*f*h*(e + f*x) + c*d*(-3* 
e*f*g + 2*e^2*h - f^2*g*x)) + 3*a^3*d^3*f^3*(c*(-2*f*g + 3*e*h + f*h*x) - 
d*(3*f*g*x + e*(g - 2*h*x))) - 9*a*b^2*d*f*(-3*c^3*f^2*h*(e + f*x) - 2*d^3 
*e^2*x*(-(f*g) + 2*e*h + f*h*x) + c^2*d*f*(e + f*x)*(4*e*h + f*(g - 2*h*x) 
) + 2*c*d^2*e*(-2*e^2*h + 2*f^2*h*x^2 + e*f*(g + h*x))) + b^3*(-15*c^4*f^3 
*h*(e + f*x) + c^3*d*f^2*(e + f*x)*(9*f*g + 14*e*h - 10*f*h*x) + 2*d^4*e^2 
*x*(-8*e^2*h + e*f*(6*g - 4*h*x) + f^2*x*(3*g + h*x)) - 2*c*d^3*e*(-6*e^2* 
f*g + 8*e^3*h + 3*e*f^2*x*(g - h*x) + 2*f^3*x^2*(3*g + h*x)) + 2*c^2*d^2*f 
*(e + f*x)*(4*e^2*h + f^2*x*(3*g + h*x) + e*f*(-6*g + 5*h*x))))/(3*d^3*f^3 
*(d*e - c*f)^2*(c + d*x)*Sqrt[e + f*x]) + ((b*c - a*d)^2*(a*d*(-3*d*f*g + 
2*d*e*h + c*f*h) + b*(6*d^2*e*g + 5*c^2*f*h - c*d*(3*f*g + 8*e*h)))*ArcTan 
[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(7/2)*(-(d*e) + c*f)^(5/2 
))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 471, normalized size of antiderivative = 1.78, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {166, 27, 167, 27, 164, 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.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int \frac {(a+b x)^2 (6 b e (d g-c h)-a (3 d f g-2 d e h-c f h)+b (3 d f g+2 d e h-5 c f h) x)}{2 (c+d x) (e+f x)^{3/2}}dx}{d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b x)^2 (6 b e (d g-c h)-a (3 d f g-2 d e h-c f h)+b (3 d f g+2 d e h-5 c f h) x)}{(c+d x) (e+f x)^{3/2}}dx}{2 d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {\frac {2 (a+b x)^2 (b e-a f) (-c f h-2 d e h+3 d f g)}{f \sqrt {e+f x} (d e-c f)}-\frac {2 \int \frac {(a+b x) \left (d f (3 d f g-2 d e h-c f h) a^2-b f \left (f h c^2+d (9 f g-16 e h) c+6 d^2 e g\right ) a+4 b^2 c e (3 d f g-2 d e h-c f h)-b \left (3 a d f (3 d f g-2 d e h-c f h)+b \left (-2 e (3 f g-4 e h) d^2-c f (3 f g+4 e h) d+5 c^2 f^2 h\right )\right ) x\right )}{2 (c+d x) \sqrt {e+f x}}dx}{f (d e-c f)}}{2 d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 (a+b x)^2 (b e-a f) (-c f h-2 d e h+3 d f g)}{f \sqrt {e+f x} (d e-c f)}-\frac {\int \frac {(a+b x) \left (d f (3 d f g-2 d e h-c f h) a^2-b f \left (f h c^2+d (9 f g-16 e h) c+6 d^2 e g\right ) a+4 b^2 c e (3 d f g-2 d e h-c f h)-b \left (3 a d f (3 d f g-2 d e h-c f h)+b \left (-2 e (3 f g-4 e h) d^2-c f (3 f g+4 e h) d+5 c^2 f^2 h\right )\right ) x\right )}{(c+d x) \sqrt {e+f x}}dx}{f (d e-c f)}}{2 d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

\(\Big \downarrow \) 164

\(\displaystyle \frac {\frac {2 (a+b x)^2 (b e-a f) (-c f h-2 d e h+3 d f g)}{f \sqrt {e+f x} (d e-c f)}-\frac {\frac {f (b c-a d)^2 \left (a d (-c f h-2 d e h+3 d f g)-b \left (5 c^2 f h-c d (8 e h+3 f g)+6 d^2 e g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d^2}-\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (-c f h-2 d e h+3 d f g)+b d f x \left (3 a d f (-c f h-2 d e h+3 d f g)+b \left (5 c^2 f^2 h-c d f (4 e h+3 f g)-2 d^2 e (3 f g-4 e h)\right )\right )+9 a b d f \left (3 c^2 f^2 h-c d f (4 e h+f g)-2 d^2 e (f g-2 e h)\right )-\left (b^2 \left (15 c^3 f^3 h-c^2 d f^2 (14 e h+9 f g)+4 c d^2 e f (3 f g-2 e h)-4 d^3 e^2 (3 f g-4 e h)\right )\right )\right )}{3 d^2 f^2}}{f (d e-c f)}}{2 d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\frac {2 (a+b x)^2 (b e-a f) (-c f h-2 d e h+3 d f g)}{f \sqrt {e+f x} (d e-c f)}-\frac {\frac {2 (b c-a d)^2 \left (a d (-c f h-2 d e h+3 d f g)-b \left (5 c^2 f h-c d (8 e h+3 f g)+6 d^2 e g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d^2}-\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (-c f h-2 d e h+3 d f g)+b d f x \left (3 a d f (-c f h-2 d e h+3 d f g)+b \left (5 c^2 f^2 h-c d f (4 e h+3 f g)-2 d^2 e (3 f g-4 e h)\right )\right )+9 a b d f \left (3 c^2 f^2 h-c d f (4 e h+f g)-2 d^2 e (f g-2 e h)\right )-\left (b^2 \left (15 c^3 f^3 h-c^2 d f^2 (14 e h+9 f g)+4 c d^2 e f (3 f g-2 e h)-4 d^3 e^2 (3 f g-4 e h)\right )\right )\right )}{3 d^2 f^2}}{f (d e-c f)}}{2 d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {2 (a+b x)^2 (b e-a f) (-c f h-2 d e h+3 d f g)}{f \sqrt {e+f x} (d e-c f)}-\frac {-\frac {2 b \sqrt {e+f x} \left (6 a^2 d^2 f^2 (-c f h-2 d e h+3 d f g)+b d f x \left (3 a d f (-c f h-2 d e h+3 d f g)+b \left (5 c^2 f^2 h-c d f (4 e h+3 f g)-2 d^2 e (3 f g-4 e h)\right )\right )+9 a b d f \left (3 c^2 f^2 h-c d f (4 e h+f g)-2 d^2 e (f g-2 e h)\right )-\left (b^2 \left (15 c^3 f^3 h-c^2 d f^2 (14 e h+9 f g)+4 c d^2 e f (3 f g-2 e h)-4 d^3 e^2 (3 f g-4 e h)\right )\right )\right )}{3 d^2 f^2}-\frac {2 f (b c-a d)^2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (a d (-c f h-2 d e h+3 d f g)-b \left (5 c^2 f h-c d (8 e h+3 f g)+6 d^2 e g\right )\right )}{d^{5/2} \sqrt {d e-c f}}}{f (d e-c f)}}{2 d (d e-c f)}-\frac {(a+b x)^3 (d g-c h)}{d (c+d x) \sqrt {e+f x} (d e-c f)}\)

Input:

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

Output:

-(((d*g - c*h)*(a + b*x)^3)/(d*(d*e - c*f)*(c + d*x)*Sqrt[e + f*x])) + ((2 
*(b*e - a*f)*(3*d*f*g - 2*d*e*h - c*f*h)*(a + b*x)^2)/(f*(d*e - c*f)*Sqrt[ 
e + f*x]) - ((-2*b*Sqrt[e + f*x]*(6*a^2*d^2*f^2*(3*d*f*g - 2*d*e*h - c*f*h 
) + 9*a*b*d*f*(3*c^2*f^2*h - 2*d^2*e*(f*g - 2*e*h) - c*d*f*(f*g + 4*e*h)) 
- b^2*(15*c^3*f^3*h - 4*d^3*e^2*(3*f*g - 4*e*h) + 4*c*d^2*e*f*(3*f*g - 2*e 
*h) - c^2*d*f^2*(9*f*g + 14*e*h)) + b*d*f*(3*a*d*f*(3*d*f*g - 2*d*e*h - c* 
f*h) + b*(5*c^2*f^2*h - 2*d^2*e*(3*f*g - 4*e*h) - c*d*f*(3*f*g + 4*e*h)))* 
x))/(3*d^2*f^2) - (2*(b*c - a*d)^2*f*(a*d*(3*d*f*g - 2*d*e*h - c*f*h) - b* 
(6*d^2*e*g + 5*c^2*f*h - c*d*(3*f*g + 8*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f 
*x])/Sqrt[d*e - c*f]])/(d^(5/2)*Sqrt[d*e - c*f]))/(f*(d*e - c*f)))/(2*d*(d 
*e - c*f))
 

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 164

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ 
))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - 
 b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( 
c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h 
*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3))   Int[( 
a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] 
&& NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
 

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 167

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] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(513\) vs. \(2(245)=490\).

Time = 0.73 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.94

method	result	size
risch	\(\frac {2 b^{2} \left (h b d f x +9 a d f h -6 b c f h -5 b d e h +3 b g d f \right ) \sqrt {f x +e}}{3 f^{3} d^{3}}+\frac {\frac {2 d^{3} \left (a^{3} e \,f^{3} h -a^{3} f^{4} g -3 a^{2} b \,e^{2} f^{2} h +3 a^{2} b e \,f^{3} g +3 a \,b^{2} e^{3} f h -3 a \,b^{2} e^{2} f^{2} g -b^{3} e^{4} h +b^{3} e^{3} f g \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {2 f^{3} \left (\frac {\left (\frac {1}{2} a^{3} c \,d^{3} f h -\frac {1}{2} a^{3} d^{4} f g -\frac {3}{2} a^{2} b \,c^{2} d^{2} f h +\frac {3}{2} a^{2} b c \,d^{3} f g +\frac {3}{2} a \,b^{2} c^{3} d f h -\frac {3}{2} a \,b^{2} c^{2} d^{2} f g -\frac {1}{2} c^{4} f h \,b^{3}+\frac {1}{2} b^{3} c^{3} d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (a^{3} c \,d^{3} f h +2 a^{3} d^{4} e h -3 a^{3} d^{4} f g +3 a^{2} b \,c^{2} d^{2} f h -12 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g +6 a^{2} b \,d^{4} e g -9 a \,b^{2} c^{3} d f h +18 a \,b^{2} c^{2} d^{2} e h +3 a \,b^{2} c^{2} d^{2} f g -12 a \,b^{2} c \,d^{3} e g +5 c^{4} f h \,b^{3}-8 b^{3} c^{3} d e h -3 b^{3} c^{3} d f g +6 b^{3} c^{2} d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2}}}{d^{3} f^{3}}\)	\(514\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-2 b c f h \sqrt {f x +e}-2 b d e h \sqrt {f x +e}+b d f g \sqrt {f x +e}\right )}{d^{3}}-\frac {2 \left (-a^{3} e \,f^{3} h +a^{3} f^{4} g +3 a^{2} b \,e^{2} f^{2} h -3 a^{2} b e \,f^{3} g -3 a \,b^{2} e^{3} f h +3 a \,b^{2} e^{2} f^{2} g +b^{3} e^{4} h -b^{3} e^{3} f g \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {2 f^{3} \left (\frac {\left (\frac {1}{2} a^{3} c \,d^{3} f h -\frac {1}{2} a^{3} d^{4} f g -\frac {3}{2} a^{2} b \,c^{2} d^{2} f h +\frac {3}{2} a^{2} b c \,d^{3} f g +\frac {3}{2} a \,b^{2} c^{3} d f h -\frac {3}{2} a \,b^{2} c^{2} d^{2} f g -\frac {1}{2} c^{4} f h \,b^{3}+\frac {1}{2} b^{3} c^{3} d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (a^{3} c \,d^{3} f h +2 a^{3} d^{4} e h -3 a^{3} d^{4} f g +3 a^{2} b \,c^{2} d^{2} f h -12 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g +6 a^{2} b \,d^{4} e g -9 a \,b^{2} c^{3} d f h +18 a \,b^{2} c^{2} d^{2} e h +3 a \,b^{2} c^{2} d^{2} f g -12 a \,b^{2} c \,d^{3} e g +5 c^{4} f h \,b^{3}-8 b^{3} c^{3} d e h -3 b^{3} c^{3} d f g +6 b^{3} c^{2} d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} d^{3}}}{f^{3}}\)	\(533\)
default	\(\frac {\frac {2 b^{2} \left (\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-2 b c f h \sqrt {f x +e}-2 b d e h \sqrt {f x +e}+b d f g \sqrt {f x +e}\right )}{d^{3}}-\frac {2 \left (-a^{3} e \,f^{3} h +a^{3} f^{4} g +3 a^{2} b \,e^{2} f^{2} h -3 a^{2} b e \,f^{3} g -3 a \,b^{2} e^{3} f h +3 a \,b^{2} e^{2} f^{2} g +b^{3} e^{4} h -b^{3} e^{3} f g \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {2 f^{3} \left (\frac {\left (\frac {1}{2} a^{3} c \,d^{3} f h -\frac {1}{2} a^{3} d^{4} f g -\frac {3}{2} a^{2} b \,c^{2} d^{2} f h +\frac {3}{2} a^{2} b c \,d^{3} f g +\frac {3}{2} a \,b^{2} c^{3} d f h -\frac {3}{2} a \,b^{2} c^{2} d^{2} f g -\frac {1}{2} c^{4} f h \,b^{3}+\frac {1}{2} b^{3} c^{3} d f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) d +c f -d e}+\frac {\left (a^{3} c \,d^{3} f h +2 a^{3} d^{4} e h -3 a^{3} d^{4} f g +3 a^{2} b \,c^{2} d^{2} f h -12 a^{2} b c \,d^{3} e h +3 a^{2} b c \,d^{3} f g +6 a^{2} b \,d^{4} e g -9 a \,b^{2} c^{3} d f h +18 a \,b^{2} c^{2} d^{2} e h +3 a \,b^{2} c^{2} d^{2} f g -12 a \,b^{2} c \,d^{3} e g +5 c^{4} f h \,b^{3}-8 b^{3} c^{3} d e h -3 b^{3} c^{3} d f g +6 b^{3} c^{2} d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 \sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2} d^{3}}}{f^{3}}\)	\(533\)
pseudoelliptic	\(\frac {\left (x d +c \right ) \sqrt {f x +e}\, \left (\left (-3 a f g +2 e \left (a h +3 b g \right )\right ) d^{2}+c \left (\left (a h -3 b g \right ) f -8 e h b \right ) d +5 b \,c^{2} f h \right ) f^{3} \left (a d -b c \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+3 \sqrt {\left (c f -d e \right ) d}\, \left (\left (-a^{3} f^{4} g x -\frac {a^{2} \left (\left (-2 a h -6 b g \right ) x +g a \right ) e \,f^{3}}{3}-2 \left (-\frac {h \,b^{2} x^{2}}{9}+\left (-b h a -\frac {1}{3} b^{2} g \right ) x +a^{2} h +g a b \right ) x b \,e^{2} f^{2}+4 \left (-\frac {2}{9} b h x +a h +\frac {1}{3} b g \right ) x \,b^{2} e^{3} f -\frac {16 b^{3} e^{4} h x}{9}\right ) d^{4}+c \left (\left (\left (\frac {1}{3} h \,a^{3}+a^{2} b g \right ) x -\frac {2 g \,a^{3}}{3}\right ) f^{4}+\left (-\frac {4 x^{3} h \,b^{3}}{9}+\left (-\frac {4}{3} g \,b^{3}-4 h a \,b^{2}\right ) x^{2}+h \,a^{3}+3 a^{2} b g \right ) e \,f^{3}-2 \left (-\frac {h \,b^{2} x^{2}}{3}+\left (a h +\frac {b g}{3}\right ) b x +a^{2} h +g a b \right ) b \,e^{2} f^{2}+4 \left (a h +\frac {b g}{3}\right ) b^{2} e^{3} f -\frac {16 b^{3} e^{4} h}{9}\right ) d^{3}-\left (\left (-\frac {2 h \,b^{2} x^{2}}{9}+\left (-2 b h a -\frac {2}{3} b^{2} g \right ) x +a^{2} h +g a b \right ) f^{2}+4 \left (-\frac {5}{18} b h x +a h +\frac {1}{3} b g \right ) b e f -\frac {8 b^{2} e^{2} h}{9}\right ) c^{2} \left (f x +e \right ) b f \,d^{2}+3 c^{3} \left (\left (-\frac {10}{27} b h x +a h +\frac {1}{3} b g \right ) f +\frac {14 e h b}{27}\right ) \left (f x +e \right ) b^{2} f^{2} d -\frac {5 b^{3} c^{4} f^{3} h \left (f x +e \right )}{3}\right )}{f^{3} d^{3} \left (c f -d e \right )^{2} \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right ) \sqrt {f x +e}}\)	\(550\)

Input:

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

Output:

2/3*b^2*(b*d*f*h*x+9*a*d*f*h-6*b*c*f*h-5*b*d*e*h+3*b*d*f*g)*(f*x+e)^(1/2)/ 
f^3/d^3+2/d^3/f^3*(d^3*(a^3*e*f^3*h-a^3*f^4*g-3*a^2*b*e^2*f^2*h+3*a^2*b*e* 
f^3*g+3*a*b^2*e^3*f*h-3*a*b^2*e^2*f^2*g-b^3*e^4*h+b^3*e^3*f*g)/(c*f-d*e)^2 
/(f*x+e)^(1/2)+f^3/(c*f-d*e)^2*((1/2*a^3*c*d^3*f*h-1/2*a^3*d^4*f*g-3/2*a^2 
*b*c^2*d^2*f*h+3/2*a^2*b*c*d^3*f*g+3/2*a*b^2*c^3*d*f*h-3/2*a*b^2*c^2*d^2*f 
*g-1/2*c^4*f*h*b^3+1/2*b^3*c^3*d*f*g)*(f*x+e)^(1/2)/((f*x+e)*d+c*f-d*e)+1/ 
2*(a^3*c*d^3*f*h+2*a^3*d^4*e*h-3*a^3*d^4*f*g+3*a^2*b*c^2*d^2*f*h-12*a^2*b* 
c*d^3*e*h+3*a^2*b*c*d^3*f*g+6*a^2*b*d^4*e*g-9*a*b^2*c^3*d*f*h+18*a*b^2*c^2 
*d^2*e*h+3*a*b^2*c^2*d^2*f*g-12*a*b^2*c*d^3*e*g+5*b^3*c^4*f*h-8*b^3*c^3*d* 
e*h-3*b^3*c^3*d*f*g+6*b^3*c^2*d^2*e*g)/((c*f-d*e)*d)^(1/2)*arctan(d*(f*x+e 
)^(1/2)/((c*f-d*e)*d)^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1867 vs. \(2 (245) = 490\).

Time = 0.28 (sec) , antiderivative size = 3748, normalized size of antiderivative = 14.14 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)^3*(h*x+g)/(d*x+c)^2/(f*x+e)^(3/2),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. 936 vs. \(2 (245) = 490\).

Time = 0.15 (sec) , antiderivative size = 936, normalized size of antiderivative = 3.53 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(6*b^3*c^2*d^2*e*g - 12*a*b^2*c*d^3*e*g + 6*a^2*b*d^4*e*g - 3*b^3*c^3*d*f* 
g + 3*a*b^2*c^2*d^2*f*g + 3*a^2*b*c*d^3*f*g - 3*a^3*d^4*f*g - 8*b^3*c^3*d* 
e*h + 18*a*b^2*c^2*d^2*e*h - 12*a^2*b*c*d^3*e*h + 2*a^3*d^4*e*h + 5*b^3*c^ 
4*f*h - 9*a*b^2*c^3*d*f*h + 3*a^2*b*c^2*d^2*f*h + a^3*c*d^3*f*h)*arctan(sq 
rt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^5*e^2 - 2*c*d^4*e*f + c^2*d^3*f^2) 
*sqrt(-d^2*e + c*d*f)) + (2*(f*x + e)*b^3*d^4*e^3*f*g - 2*b^3*d^4*e^4*f*g 
- 6*(f*x + e)*a*b^2*d^4*e^2*f^2*g + 2*b^3*c*d^3*e^3*f^2*g + 6*a*b^2*d^4*e^ 
3*f^2*g + 6*(f*x + e)*a^2*b*d^4*e*f^3*g - 6*a*b^2*c*d^3*e^2*f^3*g - 6*a^2* 
b*d^4*e^2*f^3*g + (f*x + e)*b^3*c^3*d*f^4*g - 3*(f*x + e)*a*b^2*c^2*d^2*f^ 
4*g + 3*(f*x + e)*a^2*b*c*d^3*f^4*g - 3*(f*x + e)*a^3*d^4*f^4*g + 6*a^2*b* 
c*d^3*e*f^4*g + 2*a^3*d^4*e*f^4*g - 2*a^3*c*d^3*f^5*g - 2*(f*x + e)*b^3*d^ 
4*e^4*h + 2*b^3*d^4*e^5*h + 6*(f*x + e)*a*b^2*d^4*e^3*f*h - 2*b^3*c*d^3*e^ 
4*f*h - 6*a*b^2*d^4*e^4*f*h - 6*(f*x + e)*a^2*b*d^4*e^2*f^2*h + 6*a*b^2*c* 
d^3*e^3*f^2*h + 6*a^2*b*d^4*e^3*f^2*h + 2*(f*x + e)*a^3*d^4*e*f^3*h - 6*a^ 
2*b*c*d^3*e^2*f^3*h - 2*a^3*d^4*e^2*f^3*h - (f*x + e)*b^3*c^4*f^4*h + 3*(f 
*x + e)*a*b^2*c^3*d*f^4*h - 3*(f*x + e)*a^2*b*c^2*d^2*f^4*h + (f*x + e)*a^ 
3*c*d^3*f^4*h + 2*a^3*c*d^3*e*f^4*h)/((d^5*e^2*f^3 - 2*c*d^4*e*f^4 + c^2*d 
^3*f^5)*((f*x + e)^(3/2)*d - sqrt(f*x + e)*d*e + sqrt(f*x + e)*c*f)) + 2/3 
*(3*sqrt(f*x + e)*b^3*d^4*f^7*g + (f*x + e)^(3/2)*b^3*d^4*f^6*h - 6*sqrt(f 
*x + e)*b^3*d^4*e*f^6*h - 6*sqrt(f*x + e)*b^3*c*d^3*f^7*h + 9*sqrt(f*x ...
 

Mupad [B] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 837, normalized size of antiderivative = 3.16 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\sqrt {e+f\,x}\,\left (\frac {2\,b^3\,f\,g-8\,b^3\,e\,h+6\,a\,b^2\,f\,h}{d^2\,f^3}-\frac {4\,b^3\,h\,\left (c\,f-d\,e\right )}{d^3\,f^3}\right )-\frac {\frac {2\,\left (-h\,a^3\,d^3\,e\,f^3+g\,a^3\,d^3\,f^4+3\,h\,a^2\,b\,d^3\,e^2\,f^2-3\,g\,a^2\,b\,d^3\,e\,f^3-3\,h\,a\,b^2\,d^3\,e^3\,f+3\,g\,a\,b^2\,d^3\,e^2\,f^2+h\,b^3\,d^3\,e^4-g\,b^3\,d^3\,e^3\,f\right )}{c\,f-d\,e}-\frac {\left (e+f\,x\right )\,\left (h\,a^3\,c\,d^3\,f^4+2\,h\,a^3\,d^4\,e\,f^3-3\,g\,a^3\,d^4\,f^4-3\,h\,a^2\,b\,c^2\,d^2\,f^4+3\,g\,a^2\,b\,c\,d^3\,f^4-6\,h\,a^2\,b\,d^4\,e^2\,f^2+6\,g\,a^2\,b\,d^4\,e\,f^3+3\,h\,a\,b^2\,c^3\,d\,f^4-3\,g\,a\,b^2\,c^2\,d^2\,f^4+6\,h\,a\,b^2\,d^4\,e^3\,f-6\,g\,a\,b^2\,d^4\,e^2\,f^2-h\,b^3\,c^4\,f^4+g\,b^3\,c^3\,d\,f^4-2\,h\,b^3\,d^4\,e^4+2\,g\,b^3\,d^4\,e^3\,f\right )}{{\left (c\,f-d\,e\right )}^2}}{\sqrt {e+f\,x}\,\left (c\,d^3\,f^4-d^4\,e\,f^3\right )+d^4\,f^3\,{\left (e+f\,x\right )}^{3/2}}+\frac {2\,b^3\,h\,{\left (e+f\,x\right )}^{3/2}}{3\,d^2\,f^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {e+f\,x}\,{\left (a\,d-b\,c\right )}^2\,\left (c^2\,d^3\,f^2-2\,c\,d^4\,e\,f+d^5\,e^2\right )\,\left (2\,a\,d^2\,e\,h-3\,a\,d^2\,f\,g+6\,b\,d^2\,e\,g+5\,b\,c^2\,f\,h+a\,c\,d\,f\,h-8\,b\,c\,d\,e\,h-3\,b\,c\,d\,f\,g\right )}{d^{5/2}\,{\left (c\,f-d\,e\right )}^{5/2}\,\left (2\,a^3\,d^4\,e\,h-3\,a^3\,d^4\,f\,g+5\,b^3\,c^4\,f\,h+6\,a^2\,b\,d^4\,e\,g+a^3\,c\,d^3\,f\,h-8\,b^3\,c^3\,d\,e\,h-3\,b^3\,c^3\,d\,f\,g+6\,b^3\,c^2\,d^2\,e\,g-12\,a\,b^2\,c\,d^3\,e\,g-12\,a^2\,b\,c\,d^3\,e\,h+3\,a^2\,b\,c\,d^3\,f\,g-9\,a\,b^2\,c^3\,d\,f\,h+18\,a\,b^2\,c^2\,d^2\,e\,h+3\,a\,b^2\,c^2\,d^2\,f\,g+3\,a^2\,b\,c^2\,d^2\,f\,h\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (2\,a\,d^2\,e\,h-3\,a\,d^2\,f\,g+6\,b\,d^2\,e\,g+5\,b\,c^2\,f\,h+a\,c\,d\,f\,h-8\,b\,c\,d\,e\,h-3\,b\,c\,d\,f\,g\right )}{d^{7/2}\,{\left (c\,f-d\,e\right )}^{5/2}} \] Input:

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

Output:

(e + f*x)^(1/2)*((2*b^3*f*g - 8*b^3*e*h + 6*a*b^2*f*h)/(d^2*f^3) - (4*b^3* 
h*(c*f - d*e))/(d^3*f^3)) - ((2*(a^3*d^3*f^4*g + b^3*d^3*e^4*h - a^3*d^3*e 
*f^3*h - b^3*d^3*e^3*f*g + 3*a*b^2*d^3*e^2*f^2*g + 3*a^2*b*d^3*e^2*f^2*h - 
 3*a^2*b*d^3*e*f^3*g - 3*a*b^2*d^3*e^3*f*h))/(c*f - d*e) - ((e + f*x)*(a^3 
*c*d^3*f^4*h - b^3*c^4*f^4*h - 2*b^3*d^4*e^4*h - 3*a^3*d^4*f^4*g + b^3*c^3 
*d*f^4*g + 2*a^3*d^4*e*f^3*h + 2*b^3*d^4*e^3*f*g - 3*a*b^2*c^2*d^2*f^4*g - 
 3*a^2*b*c^2*d^2*f^4*h - 6*a*b^2*d^4*e^2*f^2*g - 6*a^2*b*d^4*e^2*f^2*h + 3 
*a^2*b*c*d^3*f^4*g + 3*a*b^2*c^3*d*f^4*h + 6*a^2*b*d^4*e*f^3*g + 6*a*b^2*d 
^4*e^3*f*h))/(c*f - d*e)^2)/((e + f*x)^(1/2)*(c*d^3*f^4 - d^4*e*f^3) + d^4 
*f^3*(e + f*x)^(3/2)) + (2*b^3*h*(e + f*x)^(3/2))/(3*d^2*f^3) + (atan(((e 
+ f*x)^(1/2)*(a*d - b*c)^2*(d^5*e^2 + c^2*d^3*f^2 - 2*c*d^4*e*f)*(2*a*d^2* 
e*h - 3*a*d^2*f*g + 6*b*d^2*e*g + 5*b*c^2*f*h + a*c*d*f*h - 8*b*c*d*e*h - 
3*b*c*d*f*g))/(d^(5/2)*(c*f - d*e)^(5/2)*(2*a^3*d^4*e*h - 3*a^3*d^4*f*g + 
5*b^3*c^4*f*h + 6*a^2*b*d^4*e*g + a^3*c*d^3*f*h - 8*b^3*c^3*d*e*h - 3*b^3* 
c^3*d*f*g + 6*b^3*c^2*d^2*e*g - 12*a*b^2*c*d^3*e*g - 12*a^2*b*c*d^3*e*h + 
3*a^2*b*c*d^3*f*g - 9*a*b^2*c^3*d*f*h + 18*a*b^2*c^2*d^2*e*h + 3*a*b^2*c^2 
*d^2*f*g + 3*a^2*b*c^2*d^2*f*h)))*(a*d - b*c)^2*(2*a*d^2*e*h - 3*a*d^2*f*g 
 + 6*b*d^2*e*g + 5*b*c^2*f*h + a*c*d*f*h - 8*b*c*d*e*h - 3*b*c*d*f*g))/(d^ 
(7/2)*(c*f - d*e)^(5/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 3035, normalized size of antiderivative = 11.45 \[ \int \frac {(a+b x)^3 (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(3*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*s 
qrt(c*f - d*e)))*a**3*c**2*d**3*f**4*h + 6*sqrt(d)*sqrt(e + f*x)*sqrt(c*f 
- d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**4*e*f** 
3*h - 9*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt 
(d)*sqrt(c*f - d*e)))*a**3*c*d**4*f**4*g + 3*sqrt(d)*sqrt(e + f*x)*sqrt(c* 
f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*c*d**4*f** 
4*h*x + 6*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sq 
rt(d)*sqrt(c*f - d*e)))*a**3*d**5*e*f**3*h*x - 9*sqrt(d)*sqrt(e + f*x)*sqr 
t(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**3*d**5*f 
**4*g*x + 9*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/( 
sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**3*d**2*f**4*h - 36*sqrt(d)*sqrt(e + f* 
x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2* 
b*c**2*d**3*e*f**3*h + 9*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt( 
e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c**2*d**3*f**4*g + 9*sqrt(d) 
*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*a**2*b*c**2*d**3*f**4*h*x + 18*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e 
)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c*d**4*e*f**3*g 
 - 36*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d 
)*sqrt(c*f - d*e)))*a**2*b*c*d**4*e*f**3*h*x + 9*sqrt(d)*sqrt(e + f*x)*sqr 
t(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a**2*b*c...