\(\int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx\) [112]

Optimal result

Integrand size = 27, antiderivative size = 329 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{3 b^2 (b e-a f) (a+b x)^3}-\frac {\left (7 a^2 d f h+b^2 (6 d e g-5 c f g+6 c e h)-a b (d f g+12 d e h+c f h)\right ) \sqrt {e+f x}}{12 b^2 (b e-a f)^2 (a+b x)^2}-\frac {\left (a^2 d f^2 h+a b f (d f g-4 d e h+c f h)+b^2 (c f (5 f g-6 e h)-2 d e (3 f g-4 e h))\right ) \sqrt {e+f x}}{8 b^2 (b e-a f)^3 (a+b x)}+\frac {f \left (a^2 d f^2 h+a b f (d f g-4 d e h+c f h)+b^2 (c f (5 f g-6 e h)-2 d e (3 f g-4 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{5/2} (b e-a f)^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\frac {\sqrt {e+f x} \left (-3 a^4 d f^2 h-a^3 b f (3 c f h+d (3 f g-10 e h+8 f h x))+a^2 b^2 \left (c f (33 f g-16 e h+8 f h x)+d \left (8 e^2 h-2 e f (8 g-7 h x)+f^2 x (8 g+3 h x)\right )\right )+b^4 \left (6 d e x (-3 f g x+2 e (g+2 h x))+c \left (15 f^2 g x^2+4 e^2 (2 g+3 h x)-2 e f x (5 g+9 h x)\right )\right )+a b^3 \left (d \left (3 f^2 g x^2+4 e^2 (g+6 h x)-2 e f x (25 g+6 h x)\right )+c \left (4 e^2 h+f^2 x (40 g+3 h x)-2 e f (13 g+25 h x)\right )\right )\right )}{24 b^2 (-b e+a f)^3 (a+b x)^3}+\frac {f \left (a^2 d f^2 h+a b f (d f g-4 d e h+c f h)+b^2 (c f (5 f g-6 e h)+2 d e (-3 f g+4 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{5/2} (-b e+a f)^{7/2}} \] Input:

Integrate[((c + d*x)*(g + h*x))/((a + b*x)^4*Sqrt[e + f*x]),x]
 

Output:

(Sqrt[e + f*x]*(-3*a^4*d*f^2*h - a^3*b*f*(3*c*f*h + d*(3*f*g - 10*e*h + 8* 
f*h*x)) + a^2*b^2*(c*f*(33*f*g - 16*e*h + 8*f*h*x) + d*(8*e^2*h - 2*e*f*(8 
*g - 7*h*x) + f^2*x*(8*g + 3*h*x))) + b^4*(6*d*e*x*(-3*f*g*x + 2*e*(g + 2* 
h*x)) + c*(15*f^2*g*x^2 + 4*e^2*(2*g + 3*h*x) - 2*e*f*x*(5*g + 9*h*x))) + 
a*b^3*(d*(3*f^2*g*x^2 + 4*e^2*(g + 6*h*x) - 2*e*f*x*(25*g + 6*h*x)) + c*(4 
*e^2*h + f^2*x*(40*g + 3*h*x) - 2*e*f*(13*g + 25*h*x)))))/(24*b^2*(-(b*e) 
+ a*f)^3*(a + b*x)^3) + (f*(a^2*d*f^2*h + a*b*f*(d*f*g - 4*d*e*h + c*f*h) 
+ b^2*(c*f*(5*f*g - 6*e*h) + 2*d*e*(-3*f*g + 4*e*h)))*ArcTan[(Sqrt[b]*Sqrt 
[e + f*x])/Sqrt[-(b*e) + a*f]])/(8*b^(5/2)*(-(b*e) + a*f)^(7/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 295, normalized size of antiderivative = 0.90, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {162, 52, 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.

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

\(\Big \downarrow \) 162

\(\displaystyle \frac {\left (a^2 d f^2 h+a b f (c f h-4 d e h+d f g)+b^2 (c f (5 f g-6 e h)-2 d e (3 f g-4 e h))\right ) \int \frac {1}{(a+b x)^2 \sqrt {e+f x}}dx}{8 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (7 a^2 d f h-a b (c f h+12 d e h+d f g)+b^2 (6 c e h-5 c f g+6 d e g)\right )+a^2 b (3 c f h-8 d e h+3 d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (a^2 d f^2 h+a b f (c f h-4 d e h+d f g)+b^2 (c f (5 f g-6 e h)-2 d e (3 f g-4 e h))\right ) \left (-\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 (b e-a f)}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{8 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (7 a^2 d f h-a b (c f h+12 d e h+d f g)+b^2 (6 c e h-5 c f g+6 d e g)\right )+a^2 b (3 c f h-8 d e h+3 d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (a^2 d f^2 h+a b f (c f h-4 d e h+d f g)+b^2 (c f (5 f g-6 e h)-2 d e (3 f g-4 e h))\right ) \left (-\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b e-a f}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{8 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (7 a^2 d f h-a b (c f h+12 d e h+d f g)+b^2 (6 c e h-5 c f g+6 d e g)\right )+a^2 b (3 c f h-8 d e h+3 d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right ) \left (a^2 d f^2 h+a b f (c f h-4 d e h+d f g)+b^2 (c f (5 f g-6 e h)-2 d e (3 f g-4 e h))\right )}{8 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (7 a^2 d f h-a b (c f h+12 d e h+d f g)+b^2 (6 c e h-5 c f g+6 d e g)\right )+a^2 b (3 c f h-8 d e h+3 d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+4 b^3 c e g\right )}{12 b^2 (a+b x)^3 (b e-a f)^2}\)

Input:

Int[((c + d*x)*(g + h*x))/((a + b*x)^4*Sqrt[e + f*x]),x]
 

Output:

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

Defintions of rubi rules used

rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( 
m + n + 2)/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
 

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 = 0.50 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.15

method	result	size
pseudoelliptic	\(-\frac {\sqrt {\left (a f -b e \right ) b}\, \left (\left (\left (a^{2} \left (-\frac {b x}{3}+a \right ) \left (3 b x +a \right ) f^{2}-\frac {10 a b \left (2 b x +a \right ) e \left (-\frac {3 b x}{5}+a \right ) f}{3}-\frac {8 b^{2} e^{2} \left (3 b^{2} x^{2}+3 a b x +a^{2}\right )}{3}\right ) h +\left (a \left (\frac {b x}{3}+a \right ) \left (-3 b x +a \right ) f^{2}+\frac {16 \left (\frac {9}{8} b^{2} x^{2}+\frac {25}{8} a b x +a^{2}\right ) b e f}{3}-\frac {4 b^{2} e^{2} \left (3 b x +a \right )}{3}\right ) b g \right ) d +c b \left (\left (a \left (\frac {b x}{3}+a \right ) \left (-3 b x +a \right ) f^{2}+\frac {16 \left (\frac {9}{8} b^{2} x^{2}+\frac {25}{8} a b x +a^{2}\right ) b e f}{3}-\frac {4 b^{2} e^{2} \left (3 b x +a \right )}{3}\right ) h -11 \left (\left (\frac {5}{11} b^{2} x^{2}+\frac {40}{33} a b x +a^{2}\right ) f^{2}-\frac {26 \left (\frac {5 b x}{13}+a \right ) b e f}{33}+\frac {8 b^{2} e^{2}}{33}\right ) b g \right )\right ) \sqrt {f x +e}-\arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right ) \left (\left (\left (a^{2} f^{2}-4 a b f e +8 b^{2} e^{2}\right ) h +b f g \left (a f -6 b e \right )\right ) d +c \left (\left (a f -6 b e \right ) h +5 b f g \right ) b f \right ) \left (b x +a \right )^{3} f}{8 \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{3} \left (a f -b e \right )^{3} b^{2}}\)	\(379\)
derivativedivides	\(2 f \left (-\frac {-\frac {\left (a^{2} d \,f^{2} h +a b c \,f^{2} h -4 a b d e f h +a b d \,f^{2} g -6 b^{2} c e f h +5 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (a^{2} d \,f^{2} h -a b c \,f^{2} h -a b d \,f^{2} g +6 b^{2} c e f h -5 b^{2} c \,f^{2} g -6 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{6 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (a^{2} d \,f^{2} h +a b c \,f^{2} h -4 a b d e f h +a b d \,f^{2} g +10 b^{2} c e f h -11 b^{2} c \,f^{2} g -8 b^{2} d \,e^{2} h +10 b^{2} d e f g \right ) \sqrt {f x +e}}{16 b^{2} \left (a f -b e \right )}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (a^{2} d \,f^{2} h +a b c \,f^{2} h -4 a b d e f h +a b d \,f^{2} g -6 b^{2} c e f h +5 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -6 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 b^{2} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(488\)
default	\(2 f \left (-\frac {-\frac {\left (a^{2} d \,f^{2} h +a b c \,f^{2} h -4 a b d e f h +a b d \,f^{2} g -6 b^{2} c e f h +5 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (a^{2} d \,f^{2} h -a b c \,f^{2} h -a b d \,f^{2} g +6 b^{2} c e f h -5 b^{2} c \,f^{2} g -6 b^{2} d \,e^{2} h +6 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{6 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (a^{2} d \,f^{2} h +a b c \,f^{2} h -4 a b d e f h +a b d \,f^{2} g +10 b^{2} c e f h -11 b^{2} c \,f^{2} g -8 b^{2} d \,e^{2} h +10 b^{2} d e f g \right ) \sqrt {f x +e}}{16 b^{2} \left (a f -b e \right )}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (a^{2} d \,f^{2} h +a b c \,f^{2} h -4 a b d e f h +a b d \,f^{2} g -6 b^{2} c e f h +5 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -6 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 b^{2} \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(488\)

Input:

int((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/8*(((a*f-b*e)*b)^(1/2)*(((a^2*(-1/3*b*x+a)*(3*b*x+a)*f^2-10/3*a*b*(2*b* 
x+a)*e*(-3/5*b*x+a)*f-8/3*b^2*e^2*(3*b^2*x^2+3*a*b*x+a^2))*h+(a*(1/3*b*x+a 
)*(-3*b*x+a)*f^2+16/3*(9/8*b^2*x^2+25/8*a*b*x+a^2)*b*e*f-4/3*b^2*e^2*(3*b* 
x+a))*b*g)*d+c*b*((a*(1/3*b*x+a)*(-3*b*x+a)*f^2+16/3*(9/8*b^2*x^2+25/8*a*b 
*x+a^2)*b*e*f-4/3*b^2*e^2*(3*b*x+a))*h-11*((5/11*b^2*x^2+40/33*a*b*x+a^2)* 
f^2-26/33*(5/13*b*x+a)*b*e*f+8/33*b^2*e^2)*b*g))*(f*x+e)^(1/2)-arctan(b*(f 
*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))*(((a^2*f^2-4*a*b*e*f+8*b^2*e^2)*h+b*f*g*( 
a*f-6*b*e))*d+c*((a*f-6*b*e)*h+5*b*f*g)*b*f)*(b*x+a)^3*f)/((a*f-b*e)*b)^(1 
/2)/(b*x+a)^3/(a*f-b*e)^3/b^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1183 vs. \(2 (309) = 618\).

Time = 0.28 (sec) , antiderivative size = 2380, normalized size of antiderivative = 7.23 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

[1/48*(3*(((6*b^5*d*e*f^2 - (5*b^5*c + a*b^4*d)*f^3)*g - (8*b^5*d*e^2*f - 
2*(3*b^5*c + 2*a*b^4*d)*e*f^2 + (a*b^4*c + a^2*b^3*d)*f^3)*h)*x^3 + 3*((6* 
a*b^4*d*e*f^2 - (5*a*b^4*c + a^2*b^3*d)*f^3)*g - (8*a*b^4*d*e^2*f - 2*(3*a 
*b^4*c + 2*a^2*b^3*d)*e*f^2 + (a^2*b^3*c + a^3*b^2*d)*f^3)*h)*x^2 + (6*a^3 
*b^2*d*e*f^2 - (5*a^3*b^2*c + a^4*b*d)*f^3)*g - (8*a^3*b^2*d*e^2*f - 2*(3* 
a^3*b^2*c + 2*a^4*b*d)*e*f^2 + (a^4*b*c + a^5*d)*f^3)*h + 3*((6*a^2*b^3*d* 
e*f^2 - (5*a^2*b^3*c + a^3*b^2*d)*f^3)*g - (8*a^2*b^3*d*e^2*f - 2*(3*a^2*b 
^3*c + 2*a^3*b^2*d)*e*f^2 + (a^3*b^2*c + a^4*b*d)*f^3)*h)*x)*sqrt(b^2*e - 
a*b*f)*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*(3*((6*b^6*d*e^2*f - (5*b^6*c + 7*a*b^5*d)*e*f^2 + (5*a*b^5*c 
+ a^2*b^4*d)*f^3)*g - (8*b^6*d*e^3 - 6*(b^6*c + 2*a*b^5*d)*e^2*f + (7*a*b^ 
5*c + 5*a^2*b^4*d)*e*f^2 - (a^2*b^4*c + a^3*b^3*d)*f^3)*h)*x^2 - (4*(2*b^6 
*c + a*b^5*d)*e^3 - 2*(17*a*b^5*c + 10*a^2*b^4*d)*e^2*f + (59*a^2*b^4*c + 
13*a^3*b^3*d)*e*f^2 - 3*(11*a^3*b^3*c - a^4*b^2*d)*f^3)*g - (4*(a*b^5*c + 
2*a^2*b^4*d)*e^3 - 2*(10*a^2*b^4*c - a^3*b^3*d)*e^2*f + 13*(a^3*b^3*c - a^ 
4*b^2*d)*e*f^2 + 3*(a^4*b^2*c + a^5*b*d)*f^3)*h - 2*((6*b^6*d*e^3 - (5*b^6 
*c + 31*a*b^5*d)*e^2*f + (25*a*b^5*c + 29*a^2*b^4*d)*e*f^2 - 4*(5*a^2*b^4* 
c + a^3*b^3*d)*f^3)*g + (6*(b^6*c + 2*a*b^5*d)*e^3 - (31*a*b^5*c + 5*a^2*b 
^4*d)*e^2*f + (29*a^2*b^4*c - 11*a^3*b^3*d)*e*f^2 - 4*(a^3*b^3*c - a^4*b^2 
*d)*f^3)*h)*x)*sqrt(f*x + e))/(a^3*b^7*e^4 - 4*a^4*b^6*e^3*f + 6*a^5*b^...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)**4/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^4/(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. 922 vs. \(2 (309) = 618\).

Time = 0.14 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.80 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^4/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

1/8*(6*b^2*d*e*f^2*g - 5*b^2*c*f^3*g - a*b*d*f^3*g - 8*b^2*d*e^2*f*h + 6*b 
^2*c*e*f^2*h + 4*a*b*d*e*f^2*h - a*b*c*f^3*h - a^2*d*f^3*h)*arctan(sqrt(f* 
x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*e^3 - 3*a*b^4*e^2*f + 3*a^2*b^3*e*f^2 
 - a^3*b^2*f^3)*sqrt(-b^2*e + a*b*f)) + 1/24*(18*(f*x + e)^(5/2)*b^4*d*e*f 
^2*g - 48*(f*x + e)^(3/2)*b^4*d*e^2*f^2*g + 30*sqrt(f*x + e)*b^4*d*e^3*f^2 
*g - 15*(f*x + e)^(5/2)*b^4*c*f^3*g - 3*(f*x + e)^(5/2)*a*b^3*d*f^3*g + 40 
*(f*x + e)^(3/2)*b^4*c*e*f^3*g + 56*(f*x + e)^(3/2)*a*b^3*d*e*f^3*g - 33*s 
qrt(f*x + e)*b^4*c*e^2*f^3*g - 57*sqrt(f*x + e)*a*b^3*d*e^2*f^3*g - 40*(f* 
x + e)^(3/2)*a*b^3*c*f^4*g - 8*(f*x + e)^(3/2)*a^2*b^2*d*f^4*g + 66*sqrt(f 
*x + e)*a*b^3*c*e*f^4*g + 24*sqrt(f*x + e)*a^2*b^2*d*e*f^4*g - 33*sqrt(f*x 
 + e)*a^2*b^2*c*f^5*g + 3*sqrt(f*x + e)*a^3*b*d*f^5*g - 24*(f*x + e)^(5/2) 
*b^4*d*e^2*f*h + 48*(f*x + e)^(3/2)*b^4*d*e^3*f*h - 24*sqrt(f*x + e)*b^4*d 
*e^4*f*h + 18*(f*x + e)^(5/2)*b^4*c*e*f^2*h + 12*(f*x + e)^(5/2)*a*b^3*d*e 
*f^2*h - 48*(f*x + e)^(3/2)*b^4*c*e^2*f^2*h - 48*(f*x + e)^(3/2)*a*b^3*d*e 
^2*f^2*h + 30*sqrt(f*x + e)*b^4*c*e^3*f^2*h + 36*sqrt(f*x + e)*a*b^3*d*e^3 
*f^2*h - 3*(f*x + e)^(5/2)*a*b^3*c*f^3*h - 3*(f*x + e)^(5/2)*a^2*b^2*d*f^3 
*h + 56*(f*x + e)^(3/2)*a*b^3*c*e*f^3*h - 8*(f*x + e)^(3/2)*a^2*b^2*d*e*f^ 
3*h - 57*sqrt(f*x + e)*a*b^3*c*e^2*f^3*h + 3*sqrt(f*x + e)*a^2*b^2*d*e^2*f 
^3*h - 8*(f*x + e)^(3/2)*a^2*b^2*c*f^4*h + 8*(f*x + e)^(3/2)*a^3*b*d*f^4*h 
 + 24*sqrt(f*x + e)*a^2*b^2*c*e*f^4*h - 18*sqrt(f*x + e)*a^3*b*d*e*f^4*...
 

Mupad [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 662, normalized size of antiderivative = 2.01 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx=\frac {\frac {{\left (e+f\,x\right )}^{5/2}\,\left (5\,b^2\,c\,f^3\,g+a^2\,d\,f^3\,h-6\,b^2\,c\,e\,f^2\,h-6\,b^2\,d\,e\,f^2\,g+8\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g-4\,a\,b\,d\,e\,f^2\,h\right )}{8\,{\left (a\,f-b\,e\right )}^3}-\frac {\sqrt {e+f\,x}\,\left (a^2\,d\,f^3\,h-11\,b^2\,c\,f^3\,g+10\,b^2\,c\,e\,f^2\,h+10\,b^2\,d\,e\,f^2\,g-8\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g-4\,a\,b\,d\,e\,f^2\,h\right )}{8\,b^2\,\left (a\,f-b\,e\right )}+\frac {{\left (e+f\,x\right )}^{3/2}\,\left (5\,b^2\,c\,f^3\,g-a^2\,d\,f^3\,h-6\,b^2\,c\,e\,f^2\,h-6\,b^2\,d\,e\,f^2\,g+6\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g\right )}{3\,b\,{\left (a\,f-b\,e\right )}^2}}{\left (e+f\,x\right )\,\left (3\,a^2\,b\,f^2-6\,a\,b^2\,e\,f+3\,b^3\,e^2\right )+b^3\,{\left (e+f\,x\right )}^3-{\left (e+f\,x\right )}^2\,\left (3\,b^3\,e-3\,a\,b^2\,f\right )+a^3\,f^3-b^3\,e^3+3\,a\,b^2\,e^2\,f-3\,a^2\,b\,e\,f^2}+\frac {f\,\mathrm {atan}\left (\frac {\sqrt {b}\,f\,\sqrt {e+f\,x}\,\left (5\,b^2\,c\,f^2\,g+a^2\,d\,f^2\,h+8\,b^2\,d\,e^2\,h+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g-4\,a\,b\,d\,e\,f\,h\right )}{\sqrt {a\,f-b\,e}\,\left (5\,b^2\,c\,f^3\,g+a^2\,d\,f^3\,h-6\,b^2\,c\,e\,f^2\,h-6\,b^2\,d\,e\,f^2\,g+8\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g-4\,a\,b\,d\,e\,f^2\,h\right )}\right )\,\left (5\,b^2\,c\,f^2\,g+a^2\,d\,f^2\,h+8\,b^2\,d\,e^2\,h+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g-6\,b^2\,c\,e\,f\,h-6\,b^2\,d\,e\,f\,g-4\,a\,b\,d\,e\,f\,h\right )}{8\,b^{5/2}\,{\left (a\,f-b\,e\right )}^{7/2}} \] Input:

int(((g + h*x)*(c + d*x))/((e + f*x)^(1/2)*(a + b*x)^4),x)
 

Output:

(((e + f*x)^(5/2)*(5*b^2*c*f^3*g + a^2*d*f^3*h - 6*b^2*c*e*f^2*h - 6*b^2*d 
*e*f^2*g + 8*b^2*d*e^2*f*h + a*b*c*f^3*h + a*b*d*f^3*g - 4*a*b*d*e*f^2*h)) 
/(8*(a*f - b*e)^3) - ((e + f*x)^(1/2)*(a^2*d*f^3*h - 11*b^2*c*f^3*g + 10*b 
^2*c*e*f^2*h + 10*b^2*d*e*f^2*g - 8*b^2*d*e^2*f*h + a*b*c*f^3*h + a*b*d*f^ 
3*g - 4*a*b*d*e*f^2*h))/(8*b^2*(a*f - b*e)) + ((e + f*x)^(3/2)*(5*b^2*c*f^ 
3*g - a^2*d*f^3*h - 6*b^2*c*e*f^2*h - 6*b^2*d*e*f^2*g + 6*b^2*d*e^2*f*h + 
a*b*c*f^3*h + a*b*d*f^3*g))/(3*b*(a*f - b*e)^2))/((e + f*x)*(3*b^3*e^2 + 3 
*a^2*b*f^2 - 6*a*b^2*e*f) + b^3*(e + f*x)^3 - (e + f*x)^2*(3*b^3*e - 3*a*b 
^2*f) + a^3*f^3 - b^3*e^3 + 3*a*b^2*e^2*f - 3*a^2*b*e*f^2) + (f*atan((b^(1 
/2)*f*(e + f*x)^(1/2)*(5*b^2*c*f^2*g + 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 - 6*b^2*c*e*f*h - 6*b^2*d*e*f*g - 4*a*b*d*e*f*h))/((a 
*f - b*e)^(1/2)*(5*b^2*c*f^3*g + a^2*d*f^3*h - 6*b^2*c*e*f^2*h - 6*b^2*d*e 
*f^2*g + 8*b^2*d*e^2*f*h + a*b*c*f^3*h + a*b*d*f^3*g - 4*a*b*d*e*f^2*h)))* 
(5*b^2*c*f^2*g + 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 - 
 6*b^2*c*e*f*h - 6*b^2*d*e*f*g - 4*a*b*d*e*f*h))/(8*b^(5/2)*(a*f - b*e)^(7 
/2))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 2680, normalized size of antiderivative = 8.15 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^4 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)*(h*x+g)/(b*x+a)^4/(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**5*d*f**3*h + 3*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt( 
b)*sqrt(a*f - b*e)))*a**4*b*c*f**3*h - 12*sqrt(b)*sqrt(a*f - b*e)*atan((sq 
rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d*e*f**2*h + 3*sqrt(b)*sq 
rt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d*f 
**3*g + 9*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
 - b*e)))*a**4*b*d*f**3*h*x - 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f* 
x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*e*f**2*h + 15*sqrt(b)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*c*f** 
3*g + 9*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a**3*b**2*c*f**3*h*x + 24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f 
*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d*e**2*f*h - 18*sqrt(b)*sqrt(a 
*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d*e* 
f**2*g - 36*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a 
*f - b*e)))*a**3*b**2*d*e*f**2*h*x + 9*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt( 
e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2*d*f**3*g*x + 9*sqrt(b)*sq 
rt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**2* 
d*f**3*h*x**2 - 54*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b) 
*sqrt(a*f - b*e)))*a**2*b**3*c*e*f**2*h*x + 45*sqrt(b)*sqrt(a*f - b*e)*ata 
n((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**3*c*f**3*g*x + 9...