\(\int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx\) [68]

Optimal result

Integrand size = 27, antiderivative size = 414 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=-\frac {\left (19 a^2 d f h+b^2 (8 d e g+3 c f g+8 c e h)-a b (11 d f g+16 d e h+11 c f h)\right ) \sqrt {e+f x}}{24 b^4 (a+b x)^3}-\frac {\left (163 a^2 d f^2 h-a b f (59 d f g+208 d e h+59 c f h)+b^2 (8 d e (7 f g+6 e h)+c f (3 f g+56 e h))\right ) \sqrt {e+f x}}{96 b^4 (b e-a f) (a+b x)^2}-\frac {f \left (93 a^2 d f^2 h-a b f (5 d f g+176 d e h+5 c f h)-b^2 (c f (3 f g-8 e h)-8 d e (f g+10 e h))\right ) \sqrt {e+f x}}{64 b^4 (b e-a f)^2 (a+b x)}-\frac {(b c-a d) (b g-a h) (e+f x)^{3/2}}{4 b^3 (a+b x)^4}-\frac {f^2 \left (35 a^2 d f^2 h+5 a b f (d f g-16 d e h+c f h)+b^2 (c f (3 f g-8 e h)-8 d e (f g-6 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{64 b^{9/2} (b e-a f)^{5/2}} \] Output:

-1/24*(19*a^2*d*f*h+b^2*(8*c*e*h+3*c*f*g+8*d*e*g)-a*b*(11*c*f*h+16*d*e*h+1 
1*d*f*g))*(f*x+e)^(1/2)/b^4/(b*x+a)^3-1/96*(163*a^2*d*f^2*h-a*b*f*(59*c*f* 
h+208*d*e*h+59*d*f*g)+b^2*(8*d*e*(6*e*h+7*f*g)+c*f*(56*e*h+3*f*g)))*(f*x+e 
)^(1/2)/b^4/(-a*f+b*e)/(b*x+a)^2-1/64*f*(93*a^2*d*f^2*h-a*b*f*(5*c*f*h+176 
*d*e*h+5*d*f*g)-b^2*(c*f*(-8*e*h+3*f*g)-8*d*e*(10*e*h+f*g)))*(f*x+e)^(1/2) 
/b^4/(-a*f+b*e)^2/(b*x+a)-1/4*(-a*d+b*c)*(-a*h+b*g)*(f*x+e)^(3/2)/b^3/(b*x 
+a)^4-1/64*f^2*(35*a^2*d*f^2*h+5*a*b*f*(c*f*h-16*d*e*h+d*f*g)+b^2*(c*f*(-8 
*e*h+3*f*g)-8*d*e*(-6*e*h+f*g)))*arctanh(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^ 
(1/2))/b^(9/2)/(-a*f+b*e)^(5/2)
 

Mathematica [A] (verified)

Time = 3.27 (sec) , antiderivative size = 557, normalized size of antiderivative = 1.35 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=-\frac {\sqrt {e+f x} \left (105 a^5 d f^3 h+5 a^4 b f^2 (3 c f h+d (3 f g-34 e h+77 f h x))+b^5 \left (c \left (-9 f^3 g x^3+6 e f^2 x^2 (g+4 h x)+16 e^3 (3 g+4 h x)+8 e^2 f x (9 g+14 h x)\right )+8 d e x \left (3 f^2 g x^2+4 e^2 (2 g+3 h x)+2 e f x (7 g+15 h x)\right )\right )+a b^4 \left (c \left (16 e^3 h-8 e^2 f (9 g+5 h x)-3 f^3 x^2 (11 g+5 h x)-2 e f^2 x (66 g+79 h x)\right )+d \left (-15 f^3 g x^3+16 e^3 (g+4 h x)+8 e^2 f x (-5 g+26 h x)-2 e f^2 x^2 (79 g+264 h x)\right )\right )+a^2 b^3 \left (c f \left (-8 e^2 h+e f (6 g-52 h x)+f^2 x (33 g+73 h x)\right )+d \left (16 e^3 h-8 e^2 f (g-19 h x)+f^3 x^2 (73 g+279 h x)-2 e f^2 x (26 g+421 h x)\right )\right )+a^3 b^2 f \left (c f (9 f g-14 e h+55 f h x)+d \left (40 e^2 h-2 e f (7 g+314 h x)+f^2 x (55 g+511 h x)\right )\right )\right )}{192 b^4 (b e-a f)^2 (a+b x)^4}+\frac {f^2 \left (35 a^2 d f^2 h+5 a b f (d f g-16 d e h+c f h)+b^2 (c f (3 f g-8 e h)+8 d e (-f g+6 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{64 b^{9/2} (-b e+a f)^{5/2}} \] Input:

Integrate[((c + d*x)*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^5,x]
 

Output:

-1/192*(Sqrt[e + f*x]*(105*a^5*d*f^3*h + 5*a^4*b*f^2*(3*c*f*h + d*(3*f*g - 
 34*e*h + 77*f*h*x)) + b^5*(c*(-9*f^3*g*x^3 + 6*e*f^2*x^2*(g + 4*h*x) + 16 
*e^3*(3*g + 4*h*x) + 8*e^2*f*x*(9*g + 14*h*x)) + 8*d*e*x*(3*f^2*g*x^2 + 4* 
e^2*(2*g + 3*h*x) + 2*e*f*x*(7*g + 15*h*x))) + a*b^4*(c*(16*e^3*h - 8*e^2* 
f*(9*g + 5*h*x) - 3*f^3*x^2*(11*g + 5*h*x) - 2*e*f^2*x*(66*g + 79*h*x)) + 
d*(-15*f^3*g*x^3 + 16*e^3*(g + 4*h*x) + 8*e^2*f*x*(-5*g + 26*h*x) - 2*e*f^ 
2*x^2*(79*g + 264*h*x))) + a^2*b^3*(c*f*(-8*e^2*h + e*f*(6*g - 52*h*x) + f 
^2*x*(33*g + 73*h*x)) + d*(16*e^3*h - 8*e^2*f*(g - 19*h*x) + f^3*x^2*(73*g 
 + 279*h*x) - 2*e*f^2*x*(26*g + 421*h*x))) + a^3*b^2*f*(c*f*(9*f*g - 14*e* 
h + 55*f*h*x) + d*(40*e^2*h - 2*e*f*(7*g + 314*h*x) + f^2*x*(55*g + 511*h* 
x)))))/(b^4*(b*e - a*f)^2*(a + b*x)^4) + (f^2*(35*a^2*d*f^2*h + 5*a*b*f*(d 
*f*g - 16*d*e*h + c*f*h) + b^2*(c*f*(3*f*g - 8*e*h) + 8*d*e*(-(f*g) + 6*e* 
h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(64*b^(9/2)*(-(b* 
e) + a*f)^(5/2))
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {162, 51, 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.

\(\displaystyle \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx\)

\(\Big \downarrow \) 162

\(\displaystyle \frac {\left (35 a^2 d f^2 h+5 a b f (c f h-16 d e h+d f g)+b^2 (c f (3 f g-8 e h)-8 d e (f g-6 e h))\right ) \int \frac {(e+f x)^{3/2}}{(a+b x)^3}dx}{48 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (13 a^2 d f h-a b (5 c f h+16 d e h+5 d f g)+b^2 (8 c e h-3 c f g+8 d e g)\right )+a^2 b (c f h-10 d e h+d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\left (35 a^2 d f^2 h+5 a b f (c f h-16 d e h+d f g)+b^2 (c f (3 f g-8 e h)-8 d e (f g-6 e h))\right ) \left (\frac {3 f \int \frac {\sqrt {e+f x}}{(a+b x)^2}dx}{4 b}-\frac {(e+f x)^{3/2}}{2 b (a+b x)^2}\right )}{48 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (13 a^2 d f h-a b (5 c f h+16 d e h+5 d f g)+b^2 (8 c e h-3 c f g+8 d e g)\right )+a^2 b (c f h-10 d e h+d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {\left (35 a^2 d f^2 h+5 a b f (c f h-16 d e h+d f g)+b^2 (c f (3 f g-8 e h)-8 d e (f g-6 e h))\right ) \left (\frac {3 f \left (\frac {f \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{2 b}-\frac {\sqrt {e+f x}}{b (a+b x)}\right )}{4 b}-\frac {(e+f x)^{3/2}}{2 b (a+b x)^2}\right )}{48 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (13 a^2 d f h-a b (5 c f h+16 d e h+5 d f g)+b^2 (8 c e h-3 c f g+8 d e g)\right )+a^2 b (c f h-10 d e h+d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (35 a^2 d f^2 h+5 a b f (c f h-16 d e h+d f g)+b^2 (c f (3 f g-8 e h)-8 d e (f g-6 e h))\right ) \left (\frac {3 f \left (\frac {\int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{b}-\frac {\sqrt {e+f x}}{b (a+b x)}\right )}{4 b}-\frac {(e+f x)^{3/2}}{2 b (a+b x)^2}\right )}{48 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (13 a^2 d f h-a b (5 c f h+16 d e h+5 d f g)+b^2 (8 c e h-3 c f g+8 d e g)\right )+a^2 b (c f h-10 d e h+d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (\frac {3 f \left (-\frac {f \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{b^{3/2} \sqrt {b e-a f}}-\frac {\sqrt {e+f x}}{b (a+b x)}\right )}{4 b}-\frac {(e+f x)^{3/2}}{2 b (a+b x)^2}\right ) \left (35 a^2 d f^2 h+5 a b f (c f h-16 d e h+d f g)+b^2 (c f (3 f g-8 e h)-8 d e (f g-6 e h))\right )}{48 b^2 (b e-a f)^2}-\frac {(e+f x)^{5/2} \left (7 a^3 d f h+b x \left (13 a^2 d f h-a b (5 c f h+16 d e h+5 d f g)+b^2 (8 c e h-3 c f g+8 d e g)\right )+a^2 b (c f h-10 d e h+d f g)+2 a b^2 \left (c e h-\frac {9 c f g}{2}+d e g\right )+6 b^3 c e g\right )}{24 b^2 (a+b x)^4 (b e-a f)^2}\)

Input:

Int[((c + d*x)*(e + f*x)^(3/2)*(g + h*x))/(a + b*x)^5,x]
 

Output:

-1/24*((e + f*x)^(5/2)*(6*b^3*c*e*g + 7*a^3*d*f*h + 2*a*b^2*(d*e*g - (9*c* 
f*g)/2 + c*e*h) + a^2*b*(d*f*g - 10*d*e*h + c*f*h) + b*(13*a^2*d*f*h + b^2 
*(8*d*e*g - 3*c*f*g + 8*c*e*h) - a*b*(5*d*f*g + 16*d*e*h + 5*c*f*h))*x))/( 
b^2*(b*e - a*f)^2*(a + b*x)^4) + ((35*a^2*d*f^2*h + 5*a*b*f*(d*f*g - 16*d* 
e*h + c*f*h) + b^2*(c*f*(3*f*g - 8*e*h) - 8*d*e*(f*g - 6*e*h)))*(-1/2*(e + 
 f*x)^(3/2)/(b*(a + b*x)^2) + (3*f*(-(Sqrt[e + f*x]/(b*(a + b*x))) - (f*Ar 
cTanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/(b^(3/2)*Sqrt[b*e - a*f])) 
)/(4*b)))/(48*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 = 0.54 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(\frac {\frac {35 \left (b x +a \right )^{4} f^{2} \left (\frac {\left (3 c g \,f^{2}-8 e \left (c h +d g \right ) f +48 d \,e^{2} h \right ) b^{2}}{35}+\frac {a \left (f \left (c h +d g \right )-16 d e h \right ) f b}{7}+a^{2} d \,f^{2} h \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{64}-\frac {35 \left (\frac {\left (-3 c \,f^{3} g \,x^{3}+2 \left (4 \left (c h +d g \right ) x +c g \right ) x^{2} e \,f^{2}+24 x \left (\frac {10 d h \,x^{2}}{3}+\frac {14 \left (c h +d g \right ) x}{9}+c g \right ) e^{2} f +16 \left (2 d h \,x^{2}+\frac {4 \left (c h +d g \right ) x}{3}+c g \right ) e^{3}\right ) b^{5}}{35}+\frac {16 a \left (-\frac {33 x^{2} \left (\frac {5 \left (c h +d g \right ) x}{11}+c g \right ) f^{3}}{16}-\frac {33 x \left (4 d h \,x^{2}+\frac {79 \left (c h +d g \right ) x}{66}+c g \right ) e \,f^{2}}{4}-\frac {9 \left (-\frac {26 d h \,x^{2}}{9}+\frac {5 \left (c h +d g \right ) x}{9}+c g \right ) e^{2} f}{2}+e^{3} \left (4 d h x +c h +d g \right )\right ) b^{4}}{105}-\frac {8 a^{2} \left (\frac {\left (-279 d h \,x^{3}+73 \left (-c h -d g \right ) x^{2}-33 c g x \right ) f^{3}}{8}-\frac {3 \left (-\frac {421 d h \,x^{2}}{3}+\frac {26 \left (-c h -d g \right ) x}{3}+c g \right ) e \,f^{2}}{4}+e^{2} \left (-19 d h x +c h +d g \right ) f -2 d \,e^{3} h \right ) b^{3}}{105}-\frac {2 a^{3} f \left (\frac {\left (-73 d h \,x^{2}+\frac {55 \left (-c h -d g \right ) x}{7}-\frac {9 c g}{7}\right ) f^{2}}{2}+e \left (\frac {314}{7} d h x +c h +d g \right ) f -\frac {20 d \,e^{2} h}{7}\right ) b^{2}}{15}+\frac {a^{4} \left (\left (\frac {77}{3} d h x +c h +d g \right ) f -\frac {34 d e h}{3}\right ) f^{2} b}{7}+a^{5} d \,f^{3} h \right ) \sqrt {\left (a f -b e \right ) b}\, \sqrt {f x +e}}{64}}{\sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{4} \left (a f -b e \right )^{2} b^{4}}\)	\(538\)
derivativedivides	\(2 f^{2} \left (-\frac {\frac {\left (93 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -176 a b d e f h -5 a b d \,f^{2} g +8 b^{2} c e f h -3 b^{2} c \,f^{2} g +80 b^{2} d \,e^{2} h +8 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {7}{2}}}{128 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (511 a^{2} d \,f^{2} h +73 a b c \,f^{2} h -1168 a b d e f h +73 a b d \,f^{2} g -40 b^{2} c e f h -33 b^{2} c \,f^{2} g +624 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{384 b^{2} \left (a f -b e \right )}+\frac {11 \left (35 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -80 a b d e f h +5 a b d \,f^{2} g -8 b^{2} c e f h +3 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{384 b^{3}}+\frac {\left (35 a^{3} d \,f^{3} h +5 a^{2} b c \,f^{3} h -115 a^{2} b d e \,f^{2} h +5 a^{2} b d \,f^{3} g -13 a \,b^{2} c e \,f^{2} h +3 a \,b^{2} c \,f^{3} g +128 a \,b^{2} d \,e^{2} f h -13 a \,b^{2} d e \,f^{2} g +8 b^{3} c \,e^{2} f h -3 b^{3} c e \,f^{2} g -48 b^{3} d \,e^{3} h +8 b^{3} d \,e^{2} f g \right ) \sqrt {f x +e}}{128 b^{4}}}{\left (\left (f x +e \right ) b +a f -b e \right )^{4}}+\frac {\left (35 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -80 a b d e f h +5 a b d \,f^{2} g -8 b^{2} c e f h +3 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{128 b^{4} \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(605\)
default	\(2 f^{2} \left (-\frac {\frac {\left (93 a^{2} d \,f^{2} h -5 a b c \,f^{2} h -176 a b d e f h -5 a b d \,f^{2} g +8 b^{2} c e f h -3 b^{2} c \,f^{2} g +80 b^{2} d \,e^{2} h +8 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {7}{2}}}{128 b \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right )}+\frac {\left (511 a^{2} d \,f^{2} h +73 a b c \,f^{2} h -1168 a b d e f h +73 a b d \,f^{2} g -40 b^{2} c e f h -33 b^{2} c \,f^{2} g +624 b^{2} d \,e^{2} h -40 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {5}{2}}}{384 b^{2} \left (a f -b e \right )}+\frac {11 \left (35 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -80 a b d e f h +5 a b d \,f^{2} g -8 b^{2} c e f h +3 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}}{384 b^{3}}+\frac {\left (35 a^{3} d \,f^{3} h +5 a^{2} b c \,f^{3} h -115 a^{2} b d e \,f^{2} h +5 a^{2} b d \,f^{3} g -13 a \,b^{2} c e \,f^{2} h +3 a \,b^{2} c \,f^{3} g +128 a \,b^{2} d \,e^{2} f h -13 a \,b^{2} d e \,f^{2} g +8 b^{3} c \,e^{2} f h -3 b^{3} c e \,f^{2} g -48 b^{3} d \,e^{3} h +8 b^{3} d \,e^{2} f g \right ) \sqrt {f x +e}}{128 b^{4}}}{\left (\left (f x +e \right ) b +a f -b e \right )^{4}}+\frac {\left (35 a^{2} d \,f^{2} h +5 a b c \,f^{2} h -80 a b d e f h +5 a b d \,f^{2} g -8 b^{2} c e f h +3 b^{2} c \,f^{2} g +48 b^{2} d \,e^{2} h -8 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{128 b^{4} \left (a^{2} f^{2}-2 a b f e +b^{2} e^{2}\right ) \sqrt {\left (a f -b e \right ) b}}\right )\)	\(605\)

Input:

int((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5,x,method=_RETURNVERBOSE)
 

Output:

35/64/((a*f-b*e)*b)^(1/2)*((b*x+a)^4*f^2*(1/35*(3*c*g*f^2-8*e*(c*h+d*g)*f+ 
48*d*e^2*h)*b^2+1/7*a*(f*(c*h+d*g)-16*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))-(1/35*(-3*c*f^3*g*x^3+2*(4*(c*h+d*g)*x+c*g 
)*x^2*e*f^2+24*x*(10/3*d*h*x^2+14/9*(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+16/105*a*(-33/16*x^2*(5/11*(c*h+d*g)*x+c*g)*f 
^3-33/4*x*(4*d*h*x^2+79/66*(c*h+d*g)*x+c*g)*e*f^2-9/2*(-26/9*d*h*x^2+5/9*( 
c*h+d*g)*x+c*g)*e^2*f+e^3*(4*d*h*x+c*h+d*g))*b^4-8/105*a^2*(1/8*(-279*d*h* 
x^3+73*(-c*h-d*g)*x^2-33*c*g*x)*f^3-3/4*(-421/3*d*h*x^2+26/3*(-c*h-d*g)*x+ 
c*g)*e*f^2+e^2*(-19*d*h*x+c*h+d*g)*f-2*d*e^3*h)*b^3-2/15*a^3*f*(1/2*(-73*d 
*h*x^2+55/7*(-c*h-d*g)*x-9/7*c*g)*f^2+e*(314/7*d*h*x+c*h+d*g)*f-20/7*d*e^2 
*h)*b^2+1/7*a^4*((77/3*d*h*x+c*h+d*g)*f-34/3*d*e*h)*f^2*b+a^5*d*f^3*h)*((a 
*f-b*e)*b)^(1/2)*(f*x+e)^(1/2))/(b*x+a)^4/(a*f-b*e)^2/b^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1605 vs. \(2 (388) = 776\).

Time = 0.32 (sec) , antiderivative size = 3224, normalized size of antiderivative = 7.79 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)*(f*x+e)**(3/2)*(h*x+g)/(b*x+a)**5,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5,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. 1320 vs. \(2 (388) = 776\).

Time = 0.17 (sec) , antiderivative size = 1320, normalized size of antiderivative = 3.19 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5,x, algorithm="giac")
 

Output:

-1/64*(8*b^2*d*e*f^3*g - 3*b^2*c*f^4*g - 5*a*b*d*f^4*g - 48*b^2*d*e^2*f^2* 
h + 8*b^2*c*e*f^3*h + 80*a*b*d*e*f^3*h - 5*a*b*c*f^4*h - 35*a^2*d*f^4*h)*a 
rctan(sqrt(f*x + e)*b/sqrt(-b^2*e + a*b*f))/((b^6*e^2 - 2*a*b^5*e*f + a^2* 
b^4*f^2)*sqrt(-b^2*e + a*b*f)) - 1/192*(24*(f*x + e)^(7/2)*b^5*d*e*f^3*g + 
 40*(f*x + e)^(5/2)*b^5*d*e^2*f^3*g - 88*(f*x + e)^(3/2)*b^5*d*e^3*f^3*g + 
 24*sqrt(f*x + e)*b^5*d*e^4*f^3*g - 9*(f*x + e)^(7/2)*b^5*c*f^4*g - 15*(f* 
x + e)^(7/2)*a*b^4*d*f^4*g + 33*(f*x + e)^(5/2)*b^5*c*e*f^4*g - 113*(f*x + 
 e)^(5/2)*a*b^4*d*e*f^4*g + 33*(f*x + e)^(3/2)*b^5*c*e^2*f^4*g + 231*(f*x 
+ e)^(3/2)*a*b^4*d*e^2*f^4*g - 9*sqrt(f*x + e)*b^5*c*e^3*f^4*g - 87*sqrt(f 
*x + e)*a*b^4*d*e^3*f^4*g - 33*(f*x + e)^(5/2)*a*b^4*c*f^5*g + 73*(f*x + e 
)^(5/2)*a^2*b^3*d*f^5*g - 66*(f*x + e)^(3/2)*a*b^4*c*e*f^5*g - 198*(f*x + 
e)^(3/2)*a^2*b^3*d*e*f^5*g + 27*sqrt(f*x + e)*a*b^4*c*e^2*f^5*g + 117*sqrt 
(f*x + e)*a^2*b^3*d*e^2*f^5*g + 33*(f*x + e)^(3/2)*a^2*b^3*c*f^6*g + 55*(f 
*x + e)^(3/2)*a^3*b^2*d*f^6*g - 27*sqrt(f*x + e)*a^2*b^3*c*e*f^6*g - 69*sq 
rt(f*x + e)*a^3*b^2*d*e*f^6*g + 9*sqrt(f*x + e)*a^3*b^2*c*f^7*g + 15*sqrt( 
f*x + e)*a^4*b*d*f^7*g + 240*(f*x + e)^(7/2)*b^5*d*e^2*f^2*h - 624*(f*x + 
e)^(5/2)*b^5*d*e^3*f^2*h + 528*(f*x + e)^(3/2)*b^5*d*e^4*f^2*h - 144*sqrt( 
f*x + e)*b^5*d*e^5*f^2*h + 24*(f*x + e)^(7/2)*b^5*c*e*f^3*h - 528*(f*x + e 
)^(7/2)*a*b^4*d*e*f^3*h + 40*(f*x + e)^(5/2)*b^5*c*e^2*f^3*h + 1792*(f*x + 
 e)^(5/2)*a*b^4*d*e^2*f^3*h - 88*(f*x + e)^(3/2)*b^5*c*e^3*f^3*h - 1936...
 

Mupad [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 863, normalized size of antiderivative = 2.08 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx=\frac {f^2\,\mathrm {atan}\left (\frac {\sqrt {b}\,f^2\,\sqrt {e+f\,x}\,\left (3\,b^2\,c\,f^2\,g+35\,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-8\,b^2\,c\,e\,f\,h-8\,b^2\,d\,e\,f\,g-80\,a\,b\,d\,e\,f\,h\right )}{\sqrt {a\,f-b\,e}\,\left (3\,b^2\,c\,f^4\,g+35\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,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-80\,a\,b\,d\,e\,f^3\,h\right )}\right )\,\left (3\,b^2\,c\,f^2\,g+35\,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-8\,b^2\,c\,e\,f\,h-8\,b^2\,d\,e\,f\,g-80\,a\,b\,d\,e\,f\,h\right )}{64\,b^{9/2}\,{\left (a\,f-b\,e\right )}^{5/2}}-\frac {\frac {11\,{\left (e+f\,x\right )}^{3/2}\,\left (3\,b^2\,c\,f^4\,g+35\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,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-80\,a\,b\,d\,e\,f^3\,h\right )}{192\,b^3}+\frac {\sqrt {e+f\,x}\,\left (a\,f-b\,e\right )\,\left (3\,b^2\,c\,f^4\,g+35\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,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-80\,a\,b\,d\,e\,f^3\,h\right )}{64\,b^4}-\frac {{\left (e+f\,x\right )}^{7/2}\,\left (3\,b^2\,c\,f^4\,g-93\,a^2\,d\,f^4\,h-8\,b^2\,c\,e\,f^3\,h-8\,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+176\,a\,b\,d\,e\,f^3\,h\right )}{64\,b\,{\left (a\,f-b\,e\right )}^2}-\frac {{\left (e+f\,x\right )}^{5/2}\,\left (33\,b^2\,c\,f^4\,g-511\,a^2\,d\,f^4\,h+40\,b^2\,c\,e\,f^3\,h+40\,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+1168\,a\,b\,d\,e\,f^3\,h\right )}{192\,b^2\,\left (a\,f-b\,e\right )}}{b^4\,{\left (e+f\,x\right )}^4-{\left (e+f\,x\right )}^3\,\left (4\,b^4\,e-4\,a\,b^3\,f\right )-\left (e+f\,x\right )\,\left (-4\,a^3\,b\,f^3+12\,a^2\,b^2\,e\,f^2-12\,a\,b^3\,e^2\,f+4\,b^4\,e^3\right )+a^4\,f^4+b^4\,e^4+{\left (e+f\,x\right )}^2\,\left (6\,a^2\,b^2\,f^2-12\,a\,b^3\,e\,f+6\,b^4\,e^2\right )+6\,a^2\,b^2\,e^2\,f^2-4\,a\,b^3\,e^3\,f-4\,a^3\,b\,e\,f^3} \] Input:

int(((e + f*x)^(3/2)*(g + h*x)*(c + d*x))/(a + b*x)^5,x)
 

Output:

(f^2*atan((b^(1/2)*f^2*(e + f*x)^(1/2)*(3*b^2*c*f^2*g + 35*a^2*d*f^2*h + 4 
8*b^2*d*e^2*h + 5*a*b*c*f^2*h + 5*a*b*d*f^2*g - 8*b^2*c*e*f*h - 8*b^2*d*e* 
f*g - 80*a*b*d*e*f*h))/((a*f - b*e)^(1/2)*(3*b^2*c*f^4*g + 35*a^2*d*f^4*h 
- 8*b^2*c*e*f^3*h - 8*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 - 80*a*b*d*e*f^3*h)))*(3*b^2*c*f^2*g + 35*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 - 8*b^2*c*e*f*h - 8*b^2*d*e*f* 
g - 80*a*b*d*e*f*h))/(64*b^(9/2)*(a*f - b*e)^(5/2)) - ((11*(e + f*x)^(3/2) 
*(3*b^2*c*f^4*g + 35*a^2*d*f^4*h - 8*b^2*c*e*f^3*h - 8*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 - 80*a*b*d*e*f^3*h))/(192* 
b^3) + ((e + f*x)^(1/2)*(a*f - b*e)*(3*b^2*c*f^4*g + 35*a^2*d*f^4*h - 8*b^ 
2*c*e*f^3*h - 8*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 - 80*a*b*d*e*f^3*h))/(64*b^4) - ((e + f*x)^(7/2)*(3*b^2*c*f^4*g - 
 93*a^2*d*f^4*h - 8*b^2*c*e*f^3*h - 8*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 + 176*a*b*d*e*f^3*h))/(64*b*(a*f - b*e)^2) 
- ((e + f*x)^(5/2)*(33*b^2*c*f^4*g - 511*a^2*d*f^4*h + 40*b^2*c*e*f^3*h + 
40*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 + 
 1168*a*b*d*e*f^3*h))/(192*b^2*(a*f - b*e)))/(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*...
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 3637, normalized size of antiderivative = 8.79 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^5} \, dx =\text {Too large to display} \] Input:

int((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^5,x)
 

Output:

(105*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)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**5*b*c*f**4*h - 240*sqrt(b)*sqrt(a*f - b*e)*atan 
((sqrt(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 + 420*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*s 
qrt(a*f - b*e)))*a**5*b*d*f**4*h*x - 24*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 + 9*sqrt(b)*s 
qrt(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)*sqr 
t(a*f - b*e)))*a**4*b**2*c*f**4*h*x + 144*sqrt(b)*sqrt(a*f - b*e)*atan((sq 
rt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**4*b**2*d*e**2*f**2*h - 24*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 - 960*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 + 
 630*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 - 96*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 + 36*sqrt(b)*sqrt 
(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b**3...