\(\int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx\) [113]

Optimal result

Integrand size = 27, antiderivative size = 435 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx=-\frac {(b c-a d) (b g-a h) \sqrt {e+f x}}{4 b^2 (b e-a f) (a+b x)^4}-\frac {\left (9 a^2 d f h+b^2 (8 d e g-7 c f g+8 c e h)-a b (d f g+16 d e h+c f h)\right ) \sqrt {e+f x}}{24 b^2 (b e-a f)^2 (a+b x)^3}-\frac {\left (3 a^2 d f^2 h+a b f (5 d f g-16 d e h+5 c f h)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \sqrt {e+f x}}{96 b^2 (b e-a f)^3 (a+b x)^2}+\frac {f \left (3 a^2 d f^2 h+a b f (5 d f g-16 d e h+5 c f h)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \sqrt {e+f x}}{64 b^2 (b e-a f)^4 (a+b x)}-\frac {f^2 \left (3 a^2 d f^2 h+a b f (5 d f g-16 d e h+5 c f h)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{64 b^{5/2} (b e-a f)^{9/2}} \] Output:

-1/4*(-a*d+b*c)*(-a*h+b*g)*(f*x+e)^(1/2)/b^2/(-a*f+b*e)/(b*x+a)^4-1/24*(9* 
a^2*d*f*h+b^2*(8*c*e*h-7*c*f*g+8*d*e*g)-a*b*(c*f*h+16*d*e*h+d*f*g))*(f*x+e 
)^(1/2)/b^2/(-a*f+b*e)^2/(b*x+a)^3-1/96*(3*a^2*d*f^2*h+a*b*f*(5*c*f*h-16*d 
*e*h+5*d*f*g)+b^2*(5*c*f*(-8*e*h+7*f*g)-8*d*e*(-6*e*h+5*f*g)))*(f*x+e)^(1/ 
2)/b^2/(-a*f+b*e)^3/(b*x+a)^2+1/64*f*(3*a^2*d*f^2*h+a*b*f*(5*c*f*h-16*d*e* 
h+5*d*f*g)+b^2*(5*c*f*(-8*e*h+7*f*g)-8*d*e*(-6*e*h+5*f*g)))*(f*x+e)^(1/2)/ 
b^2/(-a*f+b*e)^4/(b*x+a)-1/64*f^2*(3*a^2*d*f^2*h+a*b*f*(5*c*f*h-16*d*e*h+5 
*d*f*g)+b^2*(5*c*f*(-8*e*h+7*f*g)-8*d*e*(-6*e*h+5*f*g)))*arctanh(b^(1/2)*( 
f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(5/2)/(-a*f+b*e)^(9/2)
 

Mathematica [A] (verified)

Time = 2.57 (sec) , antiderivative size = 563, normalized size of antiderivative = 1.29 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx=\frac {\frac {\sqrt {b} \sqrt {e+f x} \left (-9 a^5 d f^3 h-3 a^4 b f^2 (5 c f h+d (5 f g-14 e h+11 f h x))-b^5 \left (8 d e x \left (15 f^2 g x^2+4 e^2 (2 g+3 h x)-2 e f x (5 g+9 h x)\right )+c \left (-105 f^3 g x^3+16 e^3 (3 g+4 h x)-8 e^2 f x (7 g+10 h x)+10 e f^2 x^2 (7 g+12 h x)\right )\right )+a^3 b^2 f \left (c f (279 f g-146 e h+73 f h x)+d \left (88 e^2 h+f^2 x (73 g+33 h x)+e f (-146 g+52 h x)\right )\right )+a b^4 \left (c \left (-16 e^3 h+5 f^3 x^2 (77 g+3 h x)-18 e f^2 x (14 g+25 h x)+8 e^2 f (25 g+37 h x)\right )+d \left (15 f^3 g x^3-16 e^3 (g+4 h x)-6 e f^2 x^2 (75 g+8 h x)+8 e^2 f x (37 g+70 h x)\right )\right )+a^2 b^3 \left (d \left (-16 e^3 h+72 e^2 f (g+5 h x)+f^3 x^2 (55 g+9 h x)-2 e f^2 x (310 g+91 h x)\right )+c f \left (72 e^2 h+f^2 x (511 g+55 h x)-2 e f (163 g+310 h x)\right )\right )\right )}{(b e-a f)^4 (a+b x)^4}+\frac {3 f^2 \left (3 a^2 d f^2 h+a b f (5 d f g-16 d e h+5 c f h)+b^2 (5 c f (7 f g-8 e h)+8 d e (-5 f g+6 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{(-b e+a f)^{9/2}}}{192 b^{5/2}} \] Input:

Integrate[((c + d*x)*(g + h*x))/((a + b*x)^5*Sqrt[e + f*x]),x]
 

Output:

((Sqrt[b]*Sqrt[e + f*x]*(-9*a^5*d*f^3*h - 3*a^4*b*f^2*(5*c*f*h + d*(5*f*g 
- 14*e*h + 11*f*h*x)) - b^5*(8*d*e*x*(15*f^2*g*x^2 + 4*e^2*(2*g + 3*h*x) - 
 2*e*f*x*(5*g + 9*h*x)) + c*(-105*f^3*g*x^3 + 16*e^3*(3*g + 4*h*x) - 8*e^2 
*f*x*(7*g + 10*h*x) + 10*e*f^2*x^2*(7*g + 12*h*x))) + a^3*b^2*f*(c*f*(279* 
f*g - 146*e*h + 73*f*h*x) + d*(88*e^2*h + f^2*x*(73*g + 33*h*x) + e*f*(-14 
6*g + 52*h*x))) + a*b^4*(c*(-16*e^3*h + 5*f^3*x^2*(77*g + 3*h*x) - 18*e*f^ 
2*x*(14*g + 25*h*x) + 8*e^2*f*(25*g + 37*h*x)) + d*(15*f^3*g*x^3 - 16*e^3* 
(g + 4*h*x) - 6*e*f^2*x^2*(75*g + 8*h*x) + 8*e^2*f*x*(37*g + 70*h*x))) + a 
^2*b^3*(d*(-16*e^3*h + 72*e^2*f*(g + 5*h*x) + f^3*x^2*(55*g + 9*h*x) - 2*e 
*f^2*x*(310*g + 91*h*x)) + c*f*(72*e^2*h + f^2*x*(511*g + 55*h*x) - 2*e*f* 
(163*g + 310*h*x)))))/((b*e - a*f)^4*(a + b*x)^4) + (3*f^2*(3*a^2*d*f^2*h 
+ a*b*f*(5*d*f*g - 16*d*e*h + 5*c*f*h) + b^2*(5*c*f*(7*f*g - 8*e*h) + 8*d* 
e*(-5*f*g + 6*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/( 
-(b*e) + a*f)^(9/2))/(192*b^(5/2))
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 344, normalized size of antiderivative = 0.79, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {162, 52, 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.

\(\displaystyle \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\left (3 a^2 d f^2 h+a b f (5 c f h-16 d e h+5 d f g)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \int \frac {1}{(a+b x)^3 \sqrt {e+f x}}dx}{48 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (9 a^2 d f h-a b (c f h+16 d e h+d f g)+b^2 (8 c e h-7 c f g+8 d e g)\right )+5 a^2 b (c f h-2 d e h+d f g)+2 a b^2 \left (c e h-\frac {13 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (3 a^2 d f^2 h+a b f (5 c f h-16 d e h+5 d f g)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \left (-\frac {3 f \int \frac {1}{(a+b x)^2 \sqrt {e+f x}}dx}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{48 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (9 a^2 d f h-a b (c f h+16 d e h+d f g)+b^2 (8 c e h-7 c f g+8 d e g)\right )+5 a^2 b (c f h-2 d e h+d f g)+2 a b^2 \left (c e h-\frac {13 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 52

\(\displaystyle \frac {\left (3 a^2 d f^2 h+a b f (5 c f h-16 d e h+5 d f g)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \left (-\frac {3 f \left (-\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 (b e-a f)}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{48 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (9 a^2 d f h-a b (c f h+16 d e h+d f g)+b^2 (8 c e h-7 c f g+8 d e g)\right )+5 a^2 b (c f h-2 d e h+d f g)+2 a b^2 \left (c e h-\frac {13 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (3 a^2 d f^2 h+a b f (5 c f h-16 d e h+5 d f g)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right ) \left (-\frac {3 f \left (-\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b e-a f}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right )}{48 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (9 a^2 d f h-a b (c f h+16 d e h+d f g)+b^2 (8 c e h-7 c f g+8 d e g)\right )+5 a^2 b (c f h-2 d e h+d f g)+2 a b^2 \left (c e h-\frac {13 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (-\frac {3 f \left (\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b e-a f)^{3/2}}-\frac {\sqrt {e+f x}}{(a+b x) (b e-a f)}\right )}{4 (b e-a f)}-\frac {\sqrt {e+f x}}{2 (a+b x)^2 (b e-a f)}\right ) \left (3 a^2 d f^2 h+a b f (5 c f h-16 d e h+5 d f g)+b^2 (5 c f (7 f g-8 e h)-8 d e (5 f g-6 e h))\right )}{48 b^2 (b e-a f)^2}-\frac {\sqrt {e+f x} \left (3 a^3 d f h+b x \left (9 a^2 d f h-a b (c f h+16 d e h+d f g)+b^2 (8 c e h-7 c f g+8 d e g)\right )+5 a^2 b (c f h-2 d e h+d f g)+2 a b^2 \left (c e h-\frac {13 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

Input:

Int[((c + d*x)*(g + h*x))/((a + b*x)^5*Sqrt[e + f*x]),x]
 

Output:

-1/24*(Sqrt[e + f*x]*(6*b^3*c*e*g + 3*a^3*d*f*h + 2*a*b^2*(d*e*g - (13*c*f 
*g)/2 + c*e*h) + 5*a^2*b*(d*f*g - 2*d*e*h + c*f*h) + b*(9*a^2*d*f*h + b^2* 
(8*d*e*g - 7*c*f*g + 8*c*e*h) - a*b*(d*f*g + 16*d*e*h + c*f*h))*x))/(b^2*( 
b*e - a*f)^2*(a + b*x)^4) + ((3*a^2*d*f^2*h + a*b*f*(5*d*f*g - 16*d*e*h + 
5*c*f*h) + b^2*(5*c*f*(7*f*g - 8*e*h) - 8*d*e*(5*f*g - 6*e*h)))*(-1/2*Sqrt 
[e + f*x]/((b*e - a*f)*(a + b*x)^2) - (3*f*(-(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))))/(4*(b*e - a*f))))/(48*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 = 0.59 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.22

method	result	size
pseudoelliptic	\(\frac {\frac {3 \left (b x +a \right )^{4} f^{2} \left (\left (\frac {35 c g \,f^{2}}{3}-\frac {40 e \left (c h +d g \right ) f}{3}+16 d \,e^{2} h \right ) b^{2}+\frac {5 a \left (f \left (c h +d g \right )-\frac {16 d e h}{5}\right ) f b}{3}+a^{2} d \,f^{2} h \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{64}-\frac {3 \left (\frac {\left (-35 c \,f^{3} g \,x^{3}+\frac {70 \left (\frac {12 \left (c h +d g \right ) x}{7}+c g \right ) x^{2} e \,f^{2}}{3}-\frac {56 x \left (\frac {18 d h \,x^{2}}{7}+\frac {10 \left (c h +d g \right ) x}{7}+c g \right ) e^{2} f}{3}+16 \left (2 d h \,x^{2}+\frac {4 \left (c h +d g \right ) x}{3}+c g \right ) e^{3}\right ) b^{5}}{3}+\frac {16 a \left (-\frac {385 x^{2} \left (\frac {3 \left (c h +d g \right ) x}{77}+c g \right ) f^{3}}{16}+\frac {63 x \left (\frac {4 d h \,x^{2}}{21}+\frac {25 \left (c h +d g \right ) x}{14}+c g \right ) e \,f^{2}}{4}-\frac {25 \left (\frac {14 d h \,x^{2}}{5}+\frac {37 \left (c h +d g \right ) x}{25}+c g \right ) e^{2} f}{2}+e^{3} \left (4 d h x +c h +d g \right )\right ) b^{4}}{9}-8 a^{2} \left (\frac {\left (d h \,x^{3}+\frac {55 \left (c h +d g \right ) x^{2}}{9}+\frac {511 c g x}{9}\right ) f^{3}}{8}-\frac {163 \left (\frac {91 d h \,x^{2}}{163}+\frac {310 \left (c h +d g \right ) x}{163}+c g \right ) e \,f^{2}}{36}+e^{2} \left (5 d h x +c h +d g \right ) f -\frac {2 d \,e^{3} h}{9}\right ) b^{3}+\frac {146 a^{3} \left (\frac {\left (-\frac {33 d h \,x^{2}}{73}+\left (-c h -d g \right ) x -\frac {279 c g}{73}\right ) f^{2}}{2}+e \left (-\frac {26}{73} d h x +c h +d g \right ) f -\frac {44 d \,e^{2} h}{73}\right ) f \,b^{2}}{9}+\frac {5 a^{4} f^{2} \left (\left (\frac {11}{5} d h x +c h +d g \right ) f -\frac {14 d e h}{5}\right ) b}{3}+a^{5} d \,f^{3} h \right ) \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}}{64}}{\sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{4} \left (a f -b e \right )^{4} b^{2}}\)	\(531\)
derivativedivides	\(2 f^{2} \left (-\frac {-\frac {\left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g -40 b^{2} c e f h +35 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) b \left (f x +e \right )^{\frac {7}{2}}}{128 \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right )}-\frac {11 \left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g -40 b^{2} c e f h +35 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{384 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (33 a^{2} d \,f^{2} h -73 a b c \,f^{2} h +80 a b d e f h -73 a b d \,f^{2} g +584 b^{2} c e f h -511 b^{2} c \,f^{2} g -624 b^{2} d \,e^{2} h +584 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{384 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g +88 b^{2} c e f h -93 b^{2} c \,f^{2} g -80 b^{2} d \,e^{2} h +88 b^{2} d e f g \right ) \sqrt {f x +e}}{128 b^{2} \left (a f -b e \right )}}{\left (\left (f x +e \right ) b +a f -b e \right )^{4}}+\frac {\left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g -40 b^{2} c e f h +35 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{128 b^{2} \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(657\)
default	\(2 f^{2} \left (-\frac {-\frac {\left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g -40 b^{2} c e f h +35 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) b \left (f x +e \right )^{\frac {7}{2}}}{128 \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right )}-\frac {11 \left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g -40 b^{2} c e f h +35 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{384 \left (a^{3} f^{3}-3 a^{2} b e \,f^{2}+3 a \,b^{2} e^{2} f -b^{3} e^{3}\right )}+\frac {\left (33 a^{2} d \,f^{2} h -73 a b c \,f^{2} h +80 a b d e f h -73 a b d \,f^{2} g +584 b^{2} c e f h -511 b^{2} c \,f^{2} g -624 b^{2} d \,e^{2} h +584 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{384 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g +88 b^{2} c e f h -93 b^{2} c \,f^{2} g -80 b^{2} d \,e^{2} h +88 b^{2} d e f g \right ) \sqrt {f x +e}}{128 b^{2} \left (a f -b e \right )}}{\left (\left (f x +e \right ) b +a f -b e \right )^{4}}+\frac {\left (3 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -16 a b d e f h +5 a b d \,f^{2} g -40 b^{2} c e f h +35 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{128 b^{2} \left (a^{4} f^{4}-4 a^{3} b e \,f^{3}+6 a^{2} b^{2} e^{2} f^{2}-4 a \,b^{3} e^{3} f +b^{4} e^{4}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(657\)

Input:

int((d*x+c)*(h*x+g)/(b*x+a)^5/(f*x+e)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

3/64*((b*x+a)^4*f^2*((35/3*c*g*f^2-40/3*e*(c*h+d*g)*f+16*d*e^2*h)*b^2+5/3* 
a*(f*(c*h+d*g)-16/5*d*e*h)*f*b+a^2*d*f^2*h)*arctan(b*(f*x+e)^(1/2)/((a*f-b 
*e)*b)^(1/2))-(1/3*(-35*c*f^3*g*x^3+70/3*(12/7*(c*h+d*g)*x+c*g)*x^2*e*f^2- 
56/3*x*(18/7*d*h*x^2+10/7*(c*h+d*g)*x+c*g)*e^2*f+16*(2*d*h*x^2+4/3*(c*h+d* 
g)*x+c*g)*e^3)*b^5+16/9*a*(-385/16*x^2*(3/77*(c*h+d*g)*x+c*g)*f^3+63/4*x*( 
4/21*d*h*x^2+25/14*(c*h+d*g)*x+c*g)*e*f^2-25/2*(14/5*d*h*x^2+37/25*(c*h+d* 
g)*x+c*g)*e^2*f+e^3*(4*d*h*x+c*h+d*g))*b^4-8*a^2*(1/8*(d*h*x^3+55/9*(c*h+d 
*g)*x^2+511/9*c*g*x)*f^3-163/36*(91/163*d*h*x^2+310/163*(c*h+d*g)*x+c*g)*e 
*f^2+e^2*(5*d*h*x+c*h+d*g)*f-2/9*d*e^3*h)*b^3+146/9*a^3*(1/2*(-33/73*d*h*x 
^2+(-c*h-d*g)*x-279/73*c*g)*f^2+e*(-26/73*d*h*x+c*h+d*g)*f-44/73*d*e^2*h)* 
f*b^2+5/3*a^4*f^2*((11/5*d*h*x+c*h+d*g)*f-14/5*d*e*h)*b+a^5*d*f^3*h)*(f*x+ 
e)^(1/2)*((a*f-b*e)*b)^(1/2))/((a*f-b*e)*b)^(1/2)/(b*x+a)^4/(a*f-b*e)^4/b^ 
2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1767 vs. \(2 (410) = 820\).

Time = 0.38 (sec) , antiderivative size = 3548, normalized size of antiderivative = 8.16 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^5/(f*x+e)^(1/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)**5/(f*x+e)**(1/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^5/(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. 1376 vs. \(2 (410) = 820\).

Time = 0.15 (sec) , antiderivative size = 1376, normalized size of antiderivative = 3.16 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(h*x+g)/(b*x+a)^5/(f*x+e)^(1/2),x, algorithm="giac")
 

Output:

-1/64*(40*b^2*d*e*f^3*g - 35*b^2*c*f^4*g - 5*a*b*d*f^4*g - 48*b^2*d*e^2*f^ 
2*h + 40*b^2*c*e*f^3*h + 16*a*b*d*e*f^3*h - 5*a*b*c*f^4*h - 3*a^2*d*f^4*h) 
*arctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6*e^4 - 4*a*b^5*e^3*f + 
6*a^2*b^4*e^2*f^2 - 4*a^3*b^3*e*f^3 + a^4*b^2*f^4)*sqrt(-b^2*e + a*b*f)) - 
 1/192*(120*(f*x + e)^(7/2)*b^5*d*e*f^3*g - 440*(f*x + e)^(5/2)*b^5*d*e^2* 
f^3*g + 584*(f*x + e)^(3/2)*b^5*d*e^3*f^3*g - 264*sqrt(f*x + e)*b^5*d*e^4* 
f^3*g - 105*(f*x + e)^(7/2)*b^5*c*f^4*g - 15*(f*x + e)^(7/2)*a*b^4*d*f^4*g 
 + 385*(f*x + e)^(5/2)*b^5*c*e*f^4*g + 495*(f*x + e)^(5/2)*a*b^4*d*e*f^4*g 
 - 511*(f*x + e)^(3/2)*b^5*c*e^2*f^4*g - 1241*(f*x + e)^(3/2)*a*b^4*d*e^2* 
f^4*g + 279*sqrt(f*x + e)*b^5*c*e^3*f^4*g + 777*sqrt(f*x + e)*a*b^4*d*e^3* 
f^4*g - 385*(f*x + e)^(5/2)*a*b^4*c*f^5*g - 55*(f*x + e)^(5/2)*a^2*b^3*d*f 
^5*g + 1022*(f*x + e)^(3/2)*a*b^4*c*e*f^5*g + 730*(f*x + e)^(3/2)*a^2*b^3* 
d*e*f^5*g - 837*sqrt(f*x + e)*a*b^4*c*e^2*f^5*g - 747*sqrt(f*x + e)*a^2*b^ 
3*d*e^2*f^5*g - 511*(f*x + e)^(3/2)*a^2*b^3*c*f^6*g - 73*(f*x + e)^(3/2)*a 
^3*b^2*d*f^6*g + 837*sqrt(f*x + e)*a^2*b^3*c*e*f^6*g + 219*sqrt(f*x + e)*a 
^3*b^2*d*e*f^6*g - 279*sqrt(f*x + e)*a^3*b^2*c*f^7*g + 15*sqrt(f*x + e)*a^ 
4*b*d*f^7*g - 144*(f*x + e)^(7/2)*b^5*d*e^2*f^2*h + 528*(f*x + e)^(5/2)*b^ 
5*d*e^3*f^2*h - 624*(f*x + e)^(3/2)*b^5*d*e^4*f^2*h + 240*sqrt(f*x + e)*b^ 
5*d*e^5*f^2*h + 120*(f*x + e)^(7/2)*b^5*c*e*f^3*h + 48*(f*x + e)^(7/2)*a*b 
^4*d*e*f^3*h - 440*(f*x + e)^(5/2)*b^5*c*e^2*f^3*h - 704*(f*x + e)^(5/2...
 

Mupad [B] (verification not implemented)

Time = 3.02 (sec) , antiderivative size = 869, normalized size of antiderivative = 2.00 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int(((g + h*x)*(c + d*x))/((e + f*x)^(1/2)*(a + b*x)^5),x)
 

Output:

((11*(e + f*x)^(5/2)*(35*b^2*c*f^4*g + 3*a^2*d*f^4*h - 40*b^2*c*e*f^3*h - 
40*b^2*d*e*f^3*g + 48*b^2*d*e^2*f^2*h + 5*a*b*c*f^4*h + 5*a*b*d*f^4*g - 16 
*a*b*d*e*f^3*h))/(192*(a*f - b*e)^3) + (b*(e + f*x)^(7/2)*(35*b^2*c*f^4*g 
+ 3*a^2*d*f^4*h - 40*b^2*c*e*f^3*h - 40*b^2*d*e*f^3*g + 48*b^2*d*e^2*f^2*h 
 + 5*a*b*c*f^4*h + 5*a*b*d*f^4*g - 16*a*b*d*e*f^3*h))/(64*(a*f - b*e)^4) - 
 ((e + f*x)^(1/2)*(3*a^2*d*f^4*h - 93*b^2*c*f^4*g + 88*b^2*c*e*f^3*h + 88* 
b^2*d*e*f^3*g - 80*b^2*d*e^2*f^2*h + 5*a*b*c*f^4*h + 5*a*b*d*f^4*g - 16*a* 
b*d*e*f^3*h))/(64*b^2*(a*f - b*e)) + ((e + f*x)^(3/2)*(511*b^2*c*f^4*g - 3 
3*a^2*d*f^4*h - 584*b^2*c*e*f^3*h - 584*b^2*d*e*f^3*g + 624*b^2*d*e^2*f^2* 
h + 73*a*b*c*f^4*h + 73*a*b*d*f^4*g - 80*a*b*d*e*f^3*h))/(192*b*(a*f - b*e 
)^2))/(b^4*(e + f*x)^4 - (e + f*x)^3*(4*b^4*e - 4*a*b^3*f) - (e + f*x)*(4* 
b^4*e^3 - 4*a^3*b*f^3 + 12*a^2*b^2*e*f^2 - 12*a*b^3*e^2*f) + a^4*f^4 + b^4 
*e^4 + (e + f*x)^2*(6*b^4*e^2 + 6*a^2*b^2*f^2 - 12*a*b^3*e*f) + 6*a^2*b^2* 
e^2*f^2 - 4*a*b^3*e^3*f - 4*a^3*b*e*f^3) + (f^2*atan((b^(1/2)*f^2*(e + f*x 
)^(1/2)*(35*b^2*c*f^2*g + 3*a^2*d*f^2*h + 48*b^2*d*e^2*h + 5*a*b*c*f^2*h + 
 5*a*b*d*f^2*g - 40*b^2*c*e*f*h - 40*b^2*d*e*f*g - 16*a*b*d*e*f*h))/((a*f 
- b*e)^(1/2)*(35*b^2*c*f^4*g + 3*a^2*d*f^4*h - 40*b^2*c*e*f^3*h - 40*b^2*d 
*e*f^3*g + 48*b^2*d*e^2*f^2*h + 5*a*b*c*f^4*h + 5*a*b*d*f^4*g - 16*a*b*d*e 
*f^3*h)))*(35*b^2*c*f^2*g + 3*a^2*d*f^2*h + 48*b^2*d*e^2*h + 5*a*b*c*f^2*h 
 + 5*a*b*d*f^2*g - 40*b^2*c*e*f*h - 40*b^2*d*e*f*g - 16*a*b*d*e*f*h))/(...
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 3797, normalized size of antiderivative = 8.73 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^5 \sqrt {e+f x}} \, dx =\text {Too large to display} \] Input:

int((d*x+c)*(h*x+g)/(b*x+a)^5/(f*x+e)^(1/2),x)
 

Output:

(9*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a**6*d*f**4*h + 15*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt 
(b)*sqrt(a*f - b*e)))*a**5*b*c*f**4*h - 48*sqrt(b)*sqrt(a*f - b*e)*atan((s 
qrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d*e*f**3*h + 15*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**5*b*d 
*f**4*g + 36*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt( 
a*f - b*e)))*a**5*b*d*f**4*h*x - 120*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
+ f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*c*e*f**3*h + 105*sqrt(b)*sq 
rt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2* 
c*f**4*g + 60*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt 
(a*f - b*e)))*a**4*b**2*c*f**4*h*x + 144*sqrt(b)*sqrt(a*f - b*e)*atan((sqr 
t(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d*e**2*f**2*h - 120*sqr 
t(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a** 
4*b**2*d*e*f**3*g - 192*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**4*b**2*d*e*f**3*h*x + 60*sqrt(b)*sqrt(a*f - b*e 
)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d*f**4*g*x + 
 54*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e 
)))*a**4*b**2*d*f**4*h*x**2 - 480*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f 
*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**3*c*e*f**3*h*x + 420*sqrt(b)*sqr 
t(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**...