\(\int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx\) [8]

Optimal result

Integrand size = 27, antiderivative size = 322 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx=-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{3 b^3 (a+b x)^3}-\frac {\left (13 a^2 d f h+b^2 (6 d e g+c f g+6 c e h)-a b (7 d f g+12 d e h+7 c f h)\right ) \sqrt {e+f x}}{12 b^3 (b e-a f) (a+b x)^2}-\frac {\left (11 a^2 d f^2 h-a b f (d f g+20 d e h+c f h)-b^2 (c f (f g-2 e h)-2 d e (f g+4 e h))\right ) \sqrt {e+f x}}{8 b^3 (b e-a f)^2 (a+b x)}-\frac {f \left (5 a^2 d f^2 h+a b f (d f g-12 d e h+c f h)-b^2 (2 d e (f g-4 e h)-c f (f g-2 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{8 b^{7/2} (b e-a f)^{5/2}} \] Output:

-1/3*(-a*d+b*c)*(-a*h+b*g)*(f*x+e)^(1/2)/b^3/(b*x+a)^3-1/12*(13*a^2*d*f*h+ 
b^2*(6*c*e*h+c*f*g+6*d*e*g)-a*b*(7*c*f*h+12*d*e*h+7*d*f*g))*(f*x+e)^(1/2)/ 
b^3/(-a*f+b*e)/(b*x+a)^2-1/8*(11*a^2*d*f^2*h-a*b*f*(c*f*h+20*d*e*h+d*f*g)- 
b^2*(c*f*(-2*e*h+f*g)-2*d*e*(4*e*h+f*g)))*(f*x+e)^(1/2)/b^3/(-a*f+b*e)^2/( 
b*x+a)-1/8*f*(5*a^2*d*f^2*h+a*b*f*(c*f*h-12*d*e*h+d*f*g)-b^2*(2*d*e*(-4*e* 
h+f*g)-c*f*(-2*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/ 
b^(7/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.16 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx=-\frac {\sqrt {e+f x} \left (15 a^4 d f^2 h+a^3 b f (3 c f h+d (3 f g-26 e h+40 f h x))+b^4 \left (6 d e x (f g x+2 e (g+2 h x))+c \left (-3 f^2 g x^2+2 e f x (g+3 h x)+4 e^2 (2 g+3 h x)\right )\right )+a b^3 \left (c \left (4 e^2 h-14 e f (g+h x)-f^2 x (8 g+3 h x)\right )+d \left (-3 f^2 g x^2+4 e^2 (g+6 h x)-2 e f x (7 g+30 h x)\right )\right )+a^2 b^2 \left (c f (3 f g-4 e h+8 f h x)+d \left (8 e^2 h+f^2 x (8 g+33 h x)-2 e f (2 g+35 h x)\right )\right )\right )}{24 b^3 (b e-a f)^2 (a+b x)^3}+\frac {f \left (5 a^2 d f^2 h+a b f (d f g-12 d e h+c f h)+b^2 (c f (f g-2 e h)+2 d e (-f g+4 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{8 b^{7/2} (-b e+a f)^{5/2}} \] Input:

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

Output:

-1/24*(Sqrt[e + f*x]*(15*a^4*d*f^2*h + a^3*b*f*(3*c*f*h + d*(3*f*g - 26*e* 
h + 40*f*h*x)) + b^4*(6*d*e*x*(f*g*x + 2*e*(g + 2*h*x)) + c*(-3*f^2*g*x^2 
+ 2*e*f*x*(g + 3*h*x) + 4*e^2*(2*g + 3*h*x))) + a*b^3*(c*(4*e^2*h - 14*e*f 
*(g + h*x) - f^2*x*(8*g + 3*h*x)) + d*(-3*f^2*g*x^2 + 4*e^2*(g + 6*h*x) - 
2*e*f*x*(7*g + 30*h*x))) + a^2*b^2*(c*f*(3*f*g - 4*e*h + 8*f*h*x) + d*(8*e 
^2*h + f^2*x*(8*g + 33*h*x) - 2*e*f*(2*g + 35*h*x)))))/(b^3*(b*e - a*f)^2* 
(a + b*x)^3) + (f*(5*a^2*d*f^2*h + a*b*f*(d*f*g - 12*d*e*h + c*f*h) + b^2* 
(c*f*(f*g - 2*e*h) + 2*d*e*(-(f*g) + 4*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x 
])/Sqrt[-(b*e) + a*f]])/(8*b^(7/2)*(-(b*e) + a*f)^(5/2))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 289, 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, 51, 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.

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

\(\Big \downarrow \) 162

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[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 = 0.53 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.17

method	result	size
pseudoelliptic	\(\frac {-\frac {5 \left (\left (\left (a^{2} \left (\frac {8}{3} a b x +\frac {11}{5} b^{2} x^{2}+a^{2}\right ) f^{2}-\frac {26 a \left (\frac {30}{13} b^{2} x^{2}+\frac {35}{13} a b x +a^{2}\right ) b e f}{15}+\frac {8 b^{2} e^{2} \left (3 b^{2} x^{2}+3 a b x +a^{2}\right )}{15}\right ) h +\frac {\left (a \left (3 b x +a \right ) \left (-\frac {b x}{3}+a \right ) f^{2}-\frac {4 \left (-\frac {3}{2} b^{2} x^{2}+\frac {7}{2} a b x +a^{2}\right ) b e f}{3}+\frac {4 b^{2} e^{2} \left (3 b x +a \right )}{3}\right ) b g}{5}\right ) d +\frac {c b \left (\left (a \left (3 b x +a \right ) \left (-\frac {b x}{3}+a \right ) f^{2}-\frac {4 \left (-\frac {3}{2} b^{2} x^{2}+\frac {7}{2} a b x +a^{2}\right ) b e f}{3}+\frac {4 b^{2} e^{2} \left (3 b x +a \right )}{3}\right ) h +\left (\left (\frac {b x}{3}+a \right ) f -\frac {2 b e}{3}\right ) b g \left (\left (-3 b x +a \right ) f -4 b e \right )\right )}{5}\right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}}{8}+\frac {5 \left (\left (\left (a^{2} f^{2}-\frac {12}{5} a b f e +\frac {8}{5} b^{2} e^{2}\right ) h +\frac {b f g \left (a f -2 b e \right )}{5}\right ) d +\frac {c \left (\left (a f -2 b e \right ) h +b f g \right ) b f}{5}\right ) \left (b x +a \right )^{3} f \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{8}}{\sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{3} \left (a f -b e \right )^{2} b^{3}}\)	\(376\)
derivativedivides	\(2 f \left (-\frac {\frac {\left (11 a^{2} d \,f^{2} h -a b c \,f^{2} h -20 a b d e f h -a b d \,f^{2} g +2 b^{2} c e f h -b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +2 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (5 a^{2} d \,f^{2} h +a b c \,f^{2} h -12 a b d e f h +a b d \,f^{2} g -b^{2} c \,f^{2} g +6 b^{2} d \,e^{2} h \right ) \left (f x +e \right )^{\frac {3}{2}}}{6 b^{2} \left (a f -b e \right )}+\frac {\left (5 a^{2} d \,f^{2} h +a b c \,f^{2} h -12 a b d e f h +a b d \,f^{2} g -2 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -2 b^{2} d e f g \right ) \sqrt {f x +e}}{16 b^{3}}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (5 a^{2} d \,f^{2} h +a b c \,f^{2} h -12 a b d e f h +a b d \,f^{2} g -2 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -2 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 b^{3} \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(430\)
default	\(2 f \left (-\frac {\frac {\left (11 a^{2} d \,f^{2} h -a b c \,f^{2} h -20 a b d e f h -a b d \,f^{2} g +2 b^{2} c e f h -b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +2 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{16 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (5 a^{2} d \,f^{2} h +a b c \,f^{2} h -12 a b d e f h +a b d \,f^{2} g -b^{2} c \,f^{2} g +6 b^{2} d \,e^{2} h \right ) \left (f x +e \right )^{\frac {3}{2}}}{6 b^{2} \left (a f -b e \right )}+\frac {\left (5 a^{2} d \,f^{2} h +a b c \,f^{2} h -12 a b d e f h +a b d \,f^{2} g -2 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -2 b^{2} d e f g \right ) \sqrt {f x +e}}{16 b^{3}}}{\left (\left (f x +e \right ) b +a f -b e \right )^{3}}+\frac {\left (5 a^{2} d \,f^{2} h +a b c \,f^{2} h -12 a b d e f h +a b d \,f^{2} g -2 b^{2} c e f h +b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h -2 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{16 b^{3} \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(430\)

Input:

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

Output:

5/8/((a*f-b*e)*b)^(1/2)*(-(((a^2*(8/3*a*b*x+11/5*b^2*x^2+a^2)*f^2-26/15*a* 
(30/13*b^2*x^2+35/13*a*b*x+a^2)*b*e*f+8/15*b^2*e^2*(3*b^2*x^2+3*a*b*x+a^2) 
)*h+1/5*(a*(3*b*x+a)*(-1/3*b*x+a)*f^2-4/3*(-3/2*b^2*x^2+7/2*a*b*x+a^2)*b*e 
*f+4/3*b^2*e^2*(3*b*x+a))*b*g)*d+1/5*c*b*((a*(3*b*x+a)*(-1/3*b*x+a)*f^2-4/ 
3*(-3/2*b^2*x^2+7/2*a*b*x+a^2)*b*e*f+4/3*b^2*e^2*(3*b*x+a))*h+((1/3*b*x+a) 
*f-2/3*b*e)*b*g*((-3*b*x+a)*f-4*b*e)))*((a*f-b*e)*b)^(1/2)*(f*x+e)^(1/2)+( 
((a^2*f^2-12/5*a*b*f*e+8/5*b^2*e^2)*h+1/5*b*f*g*(a*f-2*b*e))*d+1/5*c*((a*f 
-2*b*e)*h+b*f*g)*b*f)*(b*x+a)^3*f*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/ 
2)))/(b*x+a)^3/(a*f-b*e)^2/b^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1121 vs. \(2 (298) = 596\).

Time = 0.21 (sec) , antiderivative size = 2256, normalized size of antiderivative = 7.01 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/48*(3*(((2*b^5*d*e*f^2 - (b^5*c + a*b^4*d)*f^3)*g - (8*b^5*d*e^2*f - 2 
*(b^5*c + 6*a*b^4*d)*e*f^2 + (a*b^4*c + 5*a^2*b^3*d)*f^3)*h)*x^3 + 3*((2*a 
*b^4*d*e*f^2 - (a*b^4*c + a^2*b^3*d)*f^3)*g - (8*a*b^4*d*e^2*f - 2*(a*b^4* 
c + 6*a^2*b^3*d)*e*f^2 + (a^2*b^3*c + 5*a^3*b^2*d)*f^3)*h)*x^2 + (2*a^3*b^ 
2*d*e*f^2 - (a^3*b^2*c + a^4*b*d)*f^3)*g - (8*a^3*b^2*d*e^2*f - 2*(a^3*b^2 
*c + 6*a^4*b*d)*e*f^2 + (a^4*b*c + 5*a^5*d)*f^3)*h + 3*((2*a^2*b^3*d*e*f^2 
 - (a^2*b^3*c + a^3*b^2*d)*f^3)*g - (8*a^2*b^3*d*e^2*f - 2*(a^2*b^3*c + 6* 
a^3*b^2*d)*e*f^2 + (a^3*b^2*c + 5*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*((2*b^6*d*e^2*f - (b^6*c + 3*a*b^5*d)*e*f^2 + (a*b^5*c + a^2*b^4*d 
)*f^3)*g + (8*b^6*d*e^3 + 2*(b^6*c - 14*a*b^5*d)*e^2*f - (3*a*b^5*c - 31*a 
^2*b^4*d)*e*f^2 + (a^2*b^4*c - 11*a^3*b^3*d)*f^3)*h)*x^2 + (4*(2*b^6*c + a 
*b^5*d)*e^3 - 2*(11*a*b^5*c + 4*a^2*b^4*d)*e^2*f + (17*a^2*b^4*c + 7*a^3*b 
^3*d)*e*f^2 - 3*(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*(4*a^2*b^4*c + 17*a^3*b^3*d)*e^2*f + (7*a^3*b^3*c + 41*a^4*b^2*d 
)*e*f^2 - 3*(a^4*b^2*c + 5*a^5*b*d)*f^3)*h + 2*((6*b^6*d*e^3 + (b^6*c - 13 
*a*b^5*d)*e^2*f - (5*a*b^5*c - 11*a^2*b^4*d)*e*f^2 + 4*(a^2*b^4*c - a^3*b^ 
3*d)*f^3)*g + (6*(b^6*c + 2*a*b^5*d)*e^3 - (13*a*b^5*c + 47*a^2*b^4*d)*e^2 
*f + 11*(a^2*b^4*c + 5*a^3*b^3*d)*e*f^2 - 4*(a^3*b^3*c + 5*a^4*b^2*d)*f^3) 
*h)*x)*sqrt(f*x + e))/(a^3*b^7*e^3 - 3*a^4*b^6*e^2*f + 3*a^5*b^5*e*f^2 ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)*(f*x+e)^(1/2)*(h*x+g)/(b*x+a)^4,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. 852 vs. \(2 (298) = 596\).

Time = 0.14 (sec) , antiderivative size = 852, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

-1/8*(2*b^2*d*e*f^2*g - b^2*c*f^3*g - a*b*d*f^3*g - 8*b^2*d*e^2*f*h + 2*b^ 
2*c*e*f^2*h + 12*a*b*d*e*f^2*h - a*b*c*f^3*h - 5*a^2*d*f^3*h)*arctan(sqrt( 
f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^5*e^2 - 2*a*b^4*e*f + a^2*b^3*f^2)*sq 
rt(-b^2*e + a*b*f)) - 1/24*(6*(f*x + e)^(5/2)*b^4*d*e*f^2*g - 6*sqrt(f*x + 
 e)*b^4*d*e^3*f^2*g - 3*(f*x + e)^(5/2)*b^4*c*f^3*g - 3*(f*x + e)^(5/2)*a* 
b^3*d*f^3*g + 8*(f*x + e)^(3/2)*b^4*c*e*f^3*g - 8*(f*x + e)^(3/2)*a*b^3*d* 
e*f^3*g + 3*sqrt(f*x + e)*b^4*c*e^2*f^3*g + 15*sqrt(f*x + e)*a*b^3*d*e^2*f 
^3*g - 8*(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 
 - 6*sqrt(f*x + e)*a*b^3*c*e*f^4*g - 12*sqrt(f*x + e)*a^2*b^2*d*e*f^4*g + 
3*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 + 6*(f*x + e)^(5/2)*b^4*c*e*f^2*h - 60*(f*x + e)^(5/2)* 
a*b^3*d*e*f^2*h + 144*(f*x + e)^(3/2)*a*b^3*d*e^2*f^2*h - 6*sqrt(f*x + e)* 
b^4*c*e^3*f^2*h - 84*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 + 33*(f*x + e)^(5/2)*a^2*b^2*d*f^3*h - 8*(f*x + e)^(3/2)*a*b^ 
3*c*e*f^3*h - 136*(f*x + e)^(3/2)*a^2*b^2*d*e*f^3*h + 15*sqrt(f*x + e)*a*b 
^3*c*e^2*f^3*h + 111*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 + 40*(f*x + e)^(3/2)*a^3*b*d*f^4*h - 12*sqrt(f*x + e)*a^2 
*b^2*c*e*f^4*h - 66*sqrt(f*x + e)*a^3*b*d*e*f^4*h + 3*sqrt(f*x + e)*a^3*b* 
c*f^5*h + 15*sqrt(f*x + e)*a^4*d*f^5*h)/((b^5*e^2 - 2*a*b^4*e*f + a^2*b...
 

Mupad [B] (verification not implemented)

Time = 2.21 (sec) , antiderivative size = 644, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx=\frac {f\,\mathrm {atan}\left (\frac {\sqrt {b}\,f\,\sqrt {e+f\,x}\,\left (b^2\,c\,f^2\,g+5\,a^2\,d\,f^2\,h+8\,b^2\,d\,e^2\,h+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g-2\,b^2\,c\,e\,f\,h-2\,b^2\,d\,e\,f\,g-12\,a\,b\,d\,e\,f\,h\right )}{\sqrt {a\,f-b\,e}\,\left (b^2\,c\,f^3\,g+5\,a^2\,d\,f^3\,h-2\,b^2\,c\,e\,f^2\,h-2\,b^2\,d\,e\,f^2\,g+8\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g-12\,a\,b\,d\,e\,f^2\,h\right )}\right )\,\left (b^2\,c\,f^2\,g+5\,a^2\,d\,f^2\,h+8\,b^2\,d\,e^2\,h+a\,b\,c\,f^2\,h+a\,b\,d\,f^2\,g-2\,b^2\,c\,e\,f\,h-2\,b^2\,d\,e\,f\,g-12\,a\,b\,d\,e\,f\,h\right )}{8\,b^{7/2}\,{\left (a\,f-b\,e\right )}^{5/2}}-\frac {\frac {\sqrt {e+f\,x}\,\left (b^2\,c\,f^3\,g+5\,a^2\,d\,f^3\,h-2\,b^2\,c\,e\,f^2\,h-2\,b^2\,d\,e\,f^2\,g+8\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g-12\,a\,b\,d\,e\,f^2\,h\right )}{8\,b^3}-\frac {{\left (e+f\,x\right )}^{5/2}\,\left (b^2\,c\,f^3\,g-11\,a^2\,d\,f^3\,h-2\,b^2\,c\,e\,f^2\,h-2\,b^2\,d\,e\,f^2\,g-8\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g+20\,a\,b\,d\,e\,f^2\,h\right )}{8\,b\,{\left (a\,f-b\,e\right )}^2}+\frac {{\left (e+f\,x\right )}^{3/2}\,\left (5\,a^2\,d\,f^3\,h-b^2\,c\,f^3\,g+6\,b^2\,d\,e^2\,f\,h+a\,b\,c\,f^3\,h+a\,b\,d\,f^3\,g-12\,a\,b\,d\,e\,f^2\,h\right )}{3\,b^2\,\left (a\,f-b\,e\right )}}{\left (e+f\,x\right )\,\left (3\,a^2\,b\,f^2-6\,a\,b^2\,e\,f+3\,b^3\,e^2\right )+b^3\,{\left (e+f\,x\right )}^3-{\left (e+f\,x\right )}^2\,\left (3\,b^3\,e-3\,a\,b^2\,f\right )+a^3\,f^3-b^3\,e^3+3\,a\,b^2\,e^2\,f-3\,a^2\,b\,e\,f^2} \] Input:

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

Output:

(f*atan((b^(1/2)*f*(e + f*x)^(1/2)*(b^2*c*f^2*g + 5*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 - 2*b^2*c*e*f*h - 2*b^2*d*e*f*g - 12*a*b 
*d*e*f*h))/((a*f - b*e)^(1/2)*(b^2*c*f^3*g + 5*a^2*d*f^3*h - 2*b^2*c*e*f^2 
*h - 2*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 - 12*a* 
b*d*e*f^2*h)))*(b^2*c*f^2*g + 5*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 - 2*b^2*c*e*f*h - 2*b^2*d*e*f*g - 12*a*b*d*e*f*h))/(8*b^(7/2 
)*(a*f - b*e)^(5/2)) - (((e + f*x)^(1/2)*(b^2*c*f^3*g + 5*a^2*d*f^3*h - 2* 
b^2*c*e*f^2*h - 2*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 - 12*a*b*d*e*f^2*h))/(8*b^3) - ((e + f*x)^(5/2)*(b^2*c*f^3*g - 11*a^2* 
d*f^3*h - 2*b^2*c*e*f^2*h - 2*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 + 20*a*b*d*e*f^2*h))/(8*b*(a*f - b*e)^2) + ((e + f*x)^(3/2 
)*(5*a^2*d*f^3*h - b^2*c*f^3*g + 6*b^2*d*e^2*f*h + a*b*c*f^3*h + a*b*d*f^3 
*g - 12*a*b*d*e*f^2*h))/(3*b^2*(a*f - b*e)))/((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)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 2619, normalized size of antiderivative = 8.13 \[ \int \frac {(c+d x) \sqrt {e+f x} (g+h x)}{(a+b x)^4} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*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 - 36*sqrt(b)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d*e*f**2*h + 3*sqrt(b)*s 
qrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b*d* 
f**3*g + 45*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 - 6*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 + 3*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 - 6*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 - 108*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 + 45*sqrt(b)*s 
qrt(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 - 18*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 + 9*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...