\(\int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx\) [111]

Optimal result

Integrand size = 27, antiderivative size = 237 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx=-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{2 b^2 (b e-a f) (a+b x)^2}-\frac {\left (5 a^2 d f h+b^2 (4 d e g-3 c f g+4 c e h)-a b (d f g+8 d e h+c f h)\right ) \sqrt {e+f x}}{4 b^2 (b e-a f)^2 (a+b x)}-\frac {\left (3 a^2 d f^2 h+a b f (d f g-8 d e h+c f h)+b^2 (c f (3 f g-4 e h)-4 d e (f g-2 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 b^{5/2} (b e-a f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.78 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.01 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx=-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b^3 (4 d e g x-3 c f g x+2 c e (g+2 h x))+a^2 b (-6 d e h+c f h+d f (g+5 h x))-a b^2 (d (-2 e g+f g x+8 e h x)+c (5 f g-2 e h+f h x))\right )}{4 b^2 (b e-a f)^2 (a+b x)^2}+\frac {\left (3 a^2 d f^2 h+a b f (d f g-8 d e h+c f h)+b^2 (c f (3 f g-4 e h)+4 d e (-f g+2 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{4 b^{5/2} (-b e+a f)^{5/2}} \] Input:

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

Output:

-1/4*(Sqrt[e + f*x]*(3*a^3*d*f*h + b^3*(4*d*e*g*x - 3*c*f*g*x + 2*c*e*(g + 
 2*h*x)) + a^2*b*(-6*d*e*h + c*f*h + d*f*(g + 5*h*x)) - a*b^2*(d*(-2*e*g + 
 f*g*x + 8*e*h*x) + c*(5*f*g - 2*e*h + f*h*x))))/(b^2*(b*e - a*f)^2*(a + b 
*x)^2) + ((3*a^2*d*f^2*h + a*b*f*(d*f*g - 8*d*e*h + c*f*h) + b^2*(c*f*(3*f 
*g - 4*e*h) + 4*d*e*(-(f*g) + 2*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt 
[-(b*e) + a*f]])/(4*b^(5/2)*(-(b*e) + a*f)^(5/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {162, 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.

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

\(\Big \downarrow \) 162

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/4*(Sqrt[e + f*x]*(2*b^3*c*e*g + 3*a^3*d*f*h + 2*a*b^2*(d*e*g - (5*c*f*g 
)/2 + c*e*h) + a^2*b*(d*f*g - 6*d*e*h + c*f*h) + b*(5*a^2*d*f*h + b^2*(4*d 
*e*g - 3*c*f*g + 4*c*e*h) - a*b*(d*f*g + 8*d*e*h + c*f*h))*x))/(b^2*(b*e - 
 a*f)^2*(a + b*x)^2) - ((3*a^2*d*f^2*h + a*b*f*(d*f*g - 8*d*e*h + c*f*h) + 
 b^2*(c*f*(3*f*g - 4*e*h) - 4*d*e*(f*g - 2*e*h)))*ArcTanh[(Sqrt[b]*Sqrt[e 
+ f*x])/Sqrt[b*e - a*f]])/(4*b^(5/2)*(b*e - a*f)^(5/2))
 

Defintions of rubi rules used

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 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

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 = 0.47 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.05

method	result	size
pseudoelliptic	\(-\frac {3 \left (-\left (\left (c g \,f^{2}-\frac {4 e \left (c h +d g \right ) f}{3}+\frac {8 d \,e^{2} h}{3}\right ) b^{2}+\frac {a \left (f \left (c h +d g \right )-8 d e h \right ) f b}{3}+a^{2} d \,f^{2} h \right ) \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (\left (-c f g x +\frac {2 e \left (2 \left (c h +d g \right ) x +c g \right )}{3}\right ) b^{3}+\frac {2 a \left (\frac {\left (\left (-c h -d g \right ) x -5 c g \right ) f}{2}+e \left (-4 d h x +c h +d g \right )\right ) b^{2}}{3}+\frac {a^{2} \left (\left (5 d h x +c h +d g \right ) f -6 d e h \right ) b}{3}+a^{3} d f h \right ) \sqrt {f x +e}\right )}{4 \sqrt {\left (a f -b e \right ) b}\, \left (a f -b e \right )^{2} b^{2} \left (b x +a \right )^{2}}\)	\(248\)
derivativedivides	\(-\frac {2 \left (\frac {f \left (5 a^{2} d f h -a b c f h -8 a b d e h -a b d f g +4 b^{2} c e h -3 b^{2} c f g +4 b^{2} d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (3 a^{2} d f h +a b c f h -8 a b d e h +a b d f g +4 b^{2} c e h -5 b^{2} c f g +4 b^{2} d e g \right ) f \sqrt {f x +e}}{8 b^{2} \left (a f -b e \right )}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (3 a^{2} d \,f^{2} h +a b c \,f^{2} h -8 a b d e f h +a b d \,f^{2} g -4 b^{2} c e f h +3 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -4 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) b^{2} \sqrt {\left (a f -b e \right ) b}}\)	\(322\)
default	\(-\frac {2 \left (\frac {f \left (5 a^{2} d f h -a b c f h -8 a b d e h -a b d f g +4 b^{2} c e h -3 b^{2} c f g +4 b^{2} d e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (3 a^{2} d f h +a b c f h -8 a b d e h +a b d f g +4 b^{2} c e h -5 b^{2} c f g +4 b^{2} d e g \right ) f \sqrt {f x +e}}{8 b^{2} \left (a f -b e \right )}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (3 a^{2} d \,f^{2} h +a b c \,f^{2} h -8 a b d e f h +a b d \,f^{2} g -4 b^{2} c e f h +3 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -4 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) b^{2} \sqrt {\left (a f -b e \right ) b}}\)	\(322\)

Input:

int((d*x+c)*(h*x+g)/(b*x+a)^3/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 709 vs. \(2 (219) = 438\).

Time = 0.18 (sec) , antiderivative size = 1432, normalized size of antiderivative = 6.04 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^3/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

[-1/8*(sqrt(b^2*e - a*b*f)*(((4*b^4*d*e*f - (3*b^4*c + a*b^3*d)*f^2)*g - ( 
8*b^4*d*e^2 - 4*(b^4*c + 2*a*b^3*d)*e*f + (a*b^3*c + 3*a^2*b^2*d)*f^2)*h)* 
x^2 + (4*a^2*b^2*d*e*f - (3*a^2*b^2*c + a^3*b*d)*f^2)*g - (8*a^2*b^2*d*e^2 
 - 4*(a^2*b^2*c + 2*a^3*b*d)*e*f + (a^3*b*c + 3*a^4*d)*f^2)*h + 2*((4*a*b^ 
3*d*e*f - (3*a*b^3*c + a^2*b^2*d)*f^2)*g - (8*a*b^3*d*e^2 - 4*(a*b^3*c + 2 
*a^2*b^2*d)*e*f + (a^2*b^2*c + 3*a^3*b*d)*f^2)*h)*x)*log((b*f*x + 2*b*e - 
a*f - 2*sqrt(b^2*e - a*b*f)*sqrt(f*x + e))/(b*x + a)) + 2*((2*(b^5*c + a*b 
^4*d)*e^2 - (7*a*b^4*c + a^2*b^3*d)*e*f + (5*a^2*b^3*c - a^3*b^2*d)*f^2)*g 
 + (2*(a*b^4*c - 3*a^2*b^3*d)*e^2 - (a^2*b^3*c - 9*a^3*b^2*d)*e*f - (a^3*b 
^2*c + 3*a^4*b*d)*f^2)*h + ((4*b^5*d*e^2 - (3*b^5*c + 5*a*b^4*d)*e*f + (3* 
a*b^4*c + a^2*b^3*d)*f^2)*g + (4*(b^5*c - 2*a*b^4*d)*e^2 - (5*a*b^4*c - 13 
*a^2*b^3*d)*e*f + (a^2*b^3*c - 5*a^3*b^2*d)*f^2)*h)*x)*sqrt(f*x + e))/(a^2 
*b^6*e^3 - 3*a^3*b^5*e^2*f + 3*a^4*b^4*e*f^2 - a^5*b^3*f^3 + (b^8*e^3 - 3* 
a*b^7*e^2*f + 3*a^2*b^6*e*f^2 - a^3*b^5*f^3)*x^2 + 2*(a*b^7*e^3 - 3*a^2*b^ 
6*e^2*f + 3*a^3*b^5*e*f^2 - a^4*b^4*f^3)*x), -1/4*(sqrt(-b^2*e + a*b*f)*(( 
(4*b^4*d*e*f - (3*b^4*c + a*b^3*d)*f^2)*g - (8*b^4*d*e^2 - 4*(b^4*c + 2*a* 
b^3*d)*e*f + (a*b^3*c + 3*a^2*b^2*d)*f^2)*h)*x^2 + (4*a^2*b^2*d*e*f - (3*a 
^2*b^2*c + a^3*b*d)*f^2)*g - (8*a^2*b^2*d*e^2 - 4*(a^2*b^2*c + 2*a^3*b*d)* 
e*f + (a^3*b*c + 3*a^4*d)*f^2)*h + 2*((4*a*b^3*d*e*f - (3*a*b^3*c + a^2*b^ 
2*d)*f^2)*g - (8*a*b^3*d*e^2 - 4*(a*b^3*c + 2*a^2*b^2*d)*e*f + (a^2*b^2...
 

Sympy [F(-1)]

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

integrate((d*x+c)*(h*x+g)/(b*x+a)**3/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)*(h*x+g)/(b*x+a)^3/(f*x+e)^(1/2),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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 509 vs. \(2 (219) = 438\).

Time = 0.13 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.15 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx=-\frac {{\left (4 \, b^{2} d e f g - 3 \, b^{2} c f^{2} g - a b d f^{2} g - 8 \, b^{2} d e^{2} h + 4 \, b^{2} c e f h + 8 \, a b d e f h - a b c f^{2} h - 3 \, a^{2} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{4 \, {\left (b^{4} e^{2} - 2 \, a b^{3} e f + a^{2} b^{2} f^{2}\right )} \sqrt {-b^{2} e + a b f}} - \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d e f g - 4 \, \sqrt {f x + e} b^{3} d e^{2} f g - 3 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c f^{2} g - {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d f^{2} g + 5 \, \sqrt {f x + e} b^{3} c e f^{2} g + 3 \, \sqrt {f x + e} a b^{2} d e f^{2} g - 5 \, \sqrt {f x + e} a b^{2} c f^{3} g + \sqrt {f x + e} a^{2} b d f^{3} g + 4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c e f h - 8 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d e f h - 4 \, \sqrt {f x + e} b^{3} c e^{2} f h + 8 \, \sqrt {f x + e} a b^{2} d e^{2} f h - {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c f^{2} h + 5 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d f^{2} h + 3 \, \sqrt {f x + e} a b^{2} c e f^{2} h - 11 \, \sqrt {f x + e} a^{2} b d e f^{2} h + \sqrt {f x + e} a^{2} b c f^{3} h + 3 \, \sqrt {f x + e} a^{3} d f^{3} h}{4 \, {\left (b^{4} e^{2} - 2 \, a b^{3} e f + a^{2} b^{2} f^{2}\right )} {\left ({\left (f x + e\right )} b - b e + a f\right )}^{2}} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^3/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

-1/4*(4*b^2*d*e*f*g - 3*b^2*c*f^2*g - a*b*d*f^2*g - 8*b^2*d*e^2*h + 4*b^2* 
c*e*f*h + 8*a*b*d*e*f*h - a*b*c*f^2*h - 3*a^2*d*f^2*h)*arctan(sqrt(f*x + e 
)*b/sqrt(-b^2*e + a*b*f))/((b^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f^2)*sqrt(-b^2 
*e + a*b*f)) - 1/4*(4*(f*x + e)^(3/2)*b^3*d*e*f*g - 4*sqrt(f*x + e)*b^3*d* 
e^2*f*g - 3*(f*x + e)^(3/2)*b^3*c*f^2*g - (f*x + e)^(3/2)*a*b^2*d*f^2*g + 
5*sqrt(f*x + e)*b^3*c*e*f^2*g + 3*sqrt(f*x + e)*a*b^2*d*e*f^2*g - 5*sqrt(f 
*x + e)*a*b^2*c*f^3*g + sqrt(f*x + e)*a^2*b*d*f^3*g + 4*(f*x + e)^(3/2)*b^ 
3*c*e*f*h - 8*(f*x + e)^(3/2)*a*b^2*d*e*f*h - 4*sqrt(f*x + e)*b^3*c*e^2*f* 
h + 8*sqrt(f*x + e)*a*b^2*d*e^2*f*h - (f*x + e)^(3/2)*a*b^2*c*f^2*h + 5*(f 
*x + e)^(3/2)*a^2*b*d*f^2*h + 3*sqrt(f*x + e)*a*b^2*c*e*f^2*h - 11*sqrt(f* 
x + e)*a^2*b*d*e*f^2*h + sqrt(f*x + e)*a^2*b*c*f^3*h + 3*sqrt(f*x + e)*a^3 
*d*f^3*h)/((b^4*e^2 - 2*a*b^3*e*f + a^2*b^2*f^2)*((f*x + e)*b - b*e + a*f) 
^2)
 

Mupad [B] (verification not implemented)

Time = 2.74 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.42 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}}{\sqrt {a\,f-b\,e}}\right )\,\left (3\,b^2\,c\,f^2\,g+3\,a^2\,d\,f^2\,h+8\,b^2\,d\,e^2\,h+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g-4\,b^2\,c\,e\,f\,h-4\,b^2\,d\,e\,f\,g-8\,a\,b\,d\,e\,f\,h\right )}{4\,b^{5/2}\,{\left (a\,f-b\,e\right )}^{5/2}}-\frac {\frac {\sqrt {e+f\,x}\,\left (3\,a^2\,d\,f^2\,h-5\,b^2\,c\,f^2\,g+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g+4\,b^2\,c\,e\,f\,h+4\,b^2\,d\,e\,f\,g-8\,a\,b\,d\,e\,f\,h\right )}{4\,b^2\,\left (a\,f-b\,e\right )}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (3\,b^2\,c\,f^2\,g-5\,a^2\,d\,f^2\,h+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g-4\,b^2\,c\,e\,f\,h-4\,b^2\,d\,e\,f\,g+8\,a\,b\,d\,e\,f\,h\right )}{4\,b\,{\left (a\,f-b\,e\right )}^2}}{b^2\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,b^2\,e-2\,a\,b\,f\right )+a^2\,f^2+b^2\,e^2-2\,a\,b\,e\,f} \] Input:

int(((g + h*x)*(c + d*x))/((e + f*x)^(1/2)*(a + b*x)^3),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1691, normalized size of antiderivative = 7.14 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)*(h*x+g)/(b*x+a)^3/(f*x+e)^(1/2),x)
 

Output:

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