\(\int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx\) [143]

Optimal result

Integrand size = 27, antiderivative size = 239 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=\frac {(b c-a d) (d g-c h) \sqrt {e+f x}}{2 d^2 (d e-c f) (c+d x)^2}+\frac {\left (a d (3 d f g-4 d e h+c f h)-b \left (4 d^2 e g+5 c^2 f h-c d (f g+8 e h)\right )\right ) \sqrt {e+f x}}{4 d^2 (d e-c f)^2 (c+d x)}-\frac {\left (a d f (3 d f g-4 d e h+c f h)+b \left (3 c^2 f^2 h+c d f (f g-8 e h)-4 d^2 e (f g-2 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{5/2} (d e-c f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.74 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=-\frac {\sqrt {e+f x} \left (a d \left (c^2 f h+d^2 (2 e g-3 f g x+4 e h x)-c d (5 f g-2 e h+f h x)\right )+b \left (3 c^3 f h+4 d^3 e g x+c d^2 (2 e g-f g x-8 e h x)+c^2 d (f g-6 e h+5 f h x)\right )\right )}{4 d^2 (d e-c f)^2 (c+d x)^2}+\frac {\left (a d f (3 d f g-4 d e h+c f h)+b \left (3 c^2 f^2 h+c d f (f g-8 e h)+4 d^2 e (-f g+2 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{5/2} (-d e+c f)^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.06, 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.

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

\(\Big \downarrow \) 162

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/4*(Sqrt[e + f*x]*(a*d*(2*d^2*e*g - 5*c*d*f*g + 2*c*d*e*h + c^2*f*h) + b 
*c*(2*d^2*e*g + 3*c^2*f*h + c*d*(f*g - 6*e*h)) - d*(a*d*(3*d*f*g - 4*d*e*h 
 + c*f*h) - b*(4*d^2*e*g + 5*c^2*f*h - c*d*(f*g + 8*e*h)))*x))/(d^2*(d*e - 
 c*f)^2*(c + d*x)^2) - ((a*d*f*(3*d*f*g - 4*d*e*h + c*f*h) + b*(3*c^2*f^2* 
h + c*d*f*(f*g - 8*e*h) - 4*d^2*e*(f*g - 2*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e 
+ f*x])/Sqrt[d*e - c*f]])/(4*d^(5/2)*(d*e - c*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 = 249, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {-\left (x d +c \right )^{2} \left (\left (3 g \,f^{2} a -4 e \left (a h +b g \right ) f +8 b \,e^{2} h \right ) d^{2}+c \left (\left (a h +b g \right ) f -8 e h b \right ) f d +3 b \,c^{2} f^{2} h \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+\sqrt {\left (c f -d e \right ) d}\, \sqrt {f x +e}\, \left (\left (-3 x a f g +2 e \left (2 \left (a h +b g \right ) x +g a \right )\right ) d^{3}+2 c \left (\frac {\left (\left (-a h -b g \right ) x -5 g a \right ) f}{2}+e \left (-4 b h x +a h +b g \right )\right ) d^{2}+c^{2} \left (\left (5 b h x +a h +b g \right ) f -6 e h b \right ) d +3 c^{3} h b f \right )}{4 \sqrt {\left (c f -d e \right ) d}\, \left (c f -d e \right )^{2} d^{2} \left (x d +c \right )^{2}}\)	\(249\)
derivativedivides	\(-\frac {2 \left (-\frac {f \left (a c d f h -4 a \,d^{2} e h +3 a \,d^{2} f g -5 b \,c^{2} f h +8 b c d e h +b c d f g -4 b \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 d \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a c d f h +4 a \,d^{2} e h -5 a \,d^{2} f g +3 b \,c^{2} f h -8 b c d e h +b c d f g +4 b \,d^{2} e g \right ) f \sqrt {f x +e}}{8 d^{2} \left (c f -d e \right )}\right )}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (a c d \,f^{2} h -4 a \,d^{2} e f h +3 a \,d^{2} f^{2} g +3 b \,c^{2} f^{2} h -8 b c d e f h +b c d \,f^{2} g +8 b \,d^{2} e^{2} h -4 b \,d^{2} e f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) d^{2} \sqrt {\left (c f -d e \right ) d}}\)	\(320\)
default	\(-\frac {2 \left (-\frac {f \left (a c d f h -4 a \,d^{2} e h +3 a \,d^{2} f g -5 b \,c^{2} f h +8 b c d e h +b c d f g -4 b \,d^{2} e g \right ) \left (f x +e \right )^{\frac {3}{2}}}{8 d \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right )}+\frac {\left (a c d f h +4 a \,d^{2} e h -5 a \,d^{2} f g +3 b \,c^{2} f h -8 b c d e h +b c d f g +4 b \,d^{2} e g \right ) f \sqrt {f x +e}}{8 d^{2} \left (c f -d e \right )}\right )}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (a c d \,f^{2} h -4 a \,d^{2} e f h +3 a \,d^{2} f^{2} g +3 b \,c^{2} f^{2} h -8 b c d e f h +b c d \,f^{2} g +8 b \,d^{2} e^{2} h -4 b \,d^{2} e f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4 \left (c^{2} f^{2}-2 c d e f +d^{2} e^{2}\right ) d^{2} \sqrt {\left (c f -d e \right ) d}}\)	\(320\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 714 vs. \(2 (219) = 438\).

Time = 0.15 (sec) , antiderivative size = 1442, normalized size of antiderivative = 6.03 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)*(h*x+g)/(d*x+c)^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(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. 509 vs. \(2 (219) = 438\).

Time = 0.13 (sec) , antiderivative size = 509, normalized size of antiderivative = 2.13 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=-\frac {{\left (4 \, b d^{2} e f g - b c d f^{2} g - 3 \, a d^{2} f^{2} g - 8 \, b d^{2} e^{2} h + 8 \, b c d e f h + 4 \, a d^{2} e f h - 3 \, b c^{2} f^{2} h - a c d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{4 \, {\left (d^{4} e^{2} - 2 \, c d^{3} e f + c^{2} d^{2} f^{2}\right )} \sqrt {-d^{2} e + c d f}} - \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b d^{3} e f g - 4 \, \sqrt {f x + e} b d^{3} e^{2} f g - {\left (f x + e\right )}^{\frac {3}{2}} b c d^{2} f^{2} g - 3 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} f^{2} g + 3 \, \sqrt {f x + e} b c d^{2} e f^{2} g + 5 \, \sqrt {f x + e} a d^{3} e f^{2} g + \sqrt {f x + e} b c^{2} d f^{3} g - 5 \, \sqrt {f x + e} a c d^{2} f^{3} g - 8 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d^{2} e f h + 4 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} e f h + 8 \, \sqrt {f x + e} b c d^{2} e^{2} f h - 4 \, \sqrt {f x + e} a d^{3} e^{2} f h + 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b c^{2} d f^{2} h - {\left (f x + e\right )}^{\frac {3}{2}} a c d^{2} f^{2} h - 11 \, \sqrt {f x + e} b c^{2} d e f^{2} h + 3 \, \sqrt {f x + e} a c d^{2} e f^{2} h + 3 \, \sqrt {f x + e} b c^{3} f^{3} h + \sqrt {f x + e} a c^{2} d f^{3} h}{4 \, {\left (d^{4} e^{2} - 2 \, c d^{3} e f + c^{2} d^{2} f^{2}\right )} {\left ({\left (f x + e\right )} d - d e + c f\right )}^{2}} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {c\,f-d\,e}}\right )\,\left (3\,a\,d^2\,f^2\,g+3\,b\,c^2\,f^2\,h+8\,b\,d^2\,e^2\,h+a\,c\,d\,f^2\,h+b\,c\,d\,f^2\,g-4\,a\,d^2\,e\,f\,h-4\,b\,d^2\,e\,f\,g-8\,b\,c\,d\,e\,f\,h\right )}{4\,d^{5/2}\,{\left (c\,f-d\,e\right )}^{5/2}}-\frac {\frac {\sqrt {e+f\,x}\,\left (3\,b\,c^2\,f^2\,h-5\,a\,d^2\,f^2\,g+a\,c\,d\,f^2\,h+b\,c\,d\,f^2\,g+4\,a\,d^2\,e\,f\,h+4\,b\,d^2\,e\,f\,g-8\,b\,c\,d\,e\,f\,h\right )}{4\,d^2\,\left (c\,f-d\,e\right )}-\frac {{\left (e+f\,x\right )}^{3/2}\,\left (3\,a\,d^2\,f^2\,g-5\,b\,c^2\,f^2\,h+a\,c\,d\,f^2\,h+b\,c\,d\,f^2\,g-4\,a\,d^2\,e\,f\,h-4\,b\,d^2\,e\,f\,g+8\,b\,c\,d\,e\,f\,h\right )}{4\,d\,{\left (c\,f-d\,e\right )}^2}}{d^2\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,d^2\,e-2\,c\,d\,f\right )+c^2\,f^2+d^2\,e^2-2\,c\,d\,e\,f} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1691, normalized size of antiderivative = 7.08 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^3 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e))) 
*a*c**3*d*f**2*h - 4*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt( 
d)*sqrt(c*f - d*e)))*a*c**2*d**2*e*f*h + 3*sqrt(d)*sqrt(c*f - d*e)*atan((s 
qrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c**2*d**2*f**2*g + 2*sqrt(d)* 
sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c**2*d 
**2*f**2*h*x - 8*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*s 
qrt(c*f - d*e)))*a*c*d**3*e*f*h*x + 6*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e 
 + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**3*f**2*g*x + sqrt(d)*sqrt(c*f 
 - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**3*f**2*h* 
x**2 - 4*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f 
- d*e)))*a*d**4*e*f*h*x**2 + 3*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x) 
*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*d**4*f**2*g*x**2 + 3*sqrt(d)*sqrt(c*f - d 
*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**4*f**2*h - 8*sq 
rt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b* 
c**3*d*e*f*h + sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqr 
t(c*f - d*e)))*b*c**3*d*f**2*g + 6*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + 
f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**3*d*f**2*h*x + 8*sqrt(d)*sqrt(c*f 
- d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**2*d**2*e**2* 
h - 4*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d 
*e)))*b*c**2*d**2*e*f*g - 16*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x...