\(\int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx\) [66]

Optimal result

Integrand size = 27, antiderivative size = 278 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=-\frac {2 (3 a d f h-b (d f g+d e h+c f h)) \sqrt {e+f x}}{b^4}-\frac {\left (11 a^2 d f h+b^2 (4 d e g+3 c f g+4 c e h)-a b (7 d f g+8 d e h+7 c f h)\right ) \sqrt {e+f x}}{4 b^4 (a+b x)}+\frac {2 d h (e+f x)^{3/2}}{3 b^3}-\frac {(b c-a d) (b g-a h) (e+f x)^{3/2}}{2 b^3 (a+b x)^2}-\frac {\left (35 a^2 d f^2 h-5 a b f (3 d f g+8 d e h+3 c f h)+b^2 (4 d e (3 f g+2 e h)+3 c f (f g+4 e h))\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{4 b^{9/2} \sqrt {b e-a f}} \] Output:

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

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.98 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\frac {\sqrt {e+f x} \left (-105 a^3 d f h+5 a^2 b (9 c f h+d (9 f g+10 e h-35 f h x))+a b^2 (d f x (75 g-56 h x)+d e (-6 g+88 h x)-3 c (3 f g+2 e h-25 f h x))+b^3 (-3 c (f x (5 g-8 h x)+2 e (g+2 h x))+4 d x (2 f x (3 g+h x)+e (-3 g+8 h x)))\right )}{12 b^4 (a+b x)^2}+\frac {\left (35 a^2 d f^2 h-5 a b f (3 d f g+8 d e h+3 c f h)+b^2 (4 d e (3 f g+2 e h)+3 c f (f g+4 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{4 b^{9/2} \sqrt {-b e+a f}} \] Input:

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

Output:

(Sqrt[e + f*x]*(-105*a^3*d*f*h + 5*a^2*b*(9*c*f*h + d*(9*f*g + 10*e*h - 35 
*f*h*x)) + a*b^2*(d*f*x*(75*g - 56*h*x) + d*e*(-6*g + 88*h*x) - 3*c*(3*f*g 
 + 2*e*h - 25*f*h*x)) + b^3*(-3*c*(f*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x)) + 
4*d*x*(2*f*x*(3*g + h*x) + e*(-3*g + 8*h*x)))))/(12*b^4*(a + b*x)^2) + ((3 
5*a^2*d*f^2*h - 5*a*b*f*(3*d*f*g + 8*d*e*h + 3*c*f*h) + b^2*(4*d*e*(3*f*g 
+ 2*e*h) + 3*c*f*(f*g + 4*e*h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e 
) + a*f]])/(4*b^(9/2)*Sqrt[-(b*e) + a*f])
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {162, 60, 60, 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.

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

\(\Big \downarrow \) 162

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/4*((e + f*x)^(5/2)*(2*b^3*c*e*g + 7*a^3*d*f*h + a*b^2*(2*d*e*g - c*f*g 
+ 2*c*e*h) - 3*a^2*b*(d*f*g + 2*d*e*h + c*f*h) + b*(9*a^2*d*f*h + b^2*(4*d 
*e*g + c*f*g + 4*c*e*h) - a*b*(5*d*f*g + 8*d*e*h + 5*c*f*h))*x))/(b^2*(b*e 
 - a*f)^2*(a + b*x)^2) + ((35*a^2*d*f^2*h - 5*a*b*f*(3*d*f*g + 8*d*e*h + 3 
*c*f*h) + b^2*(4*d*e*(3*f*g + 2*e*h) + 3*c*f*(f*g + 4*e*h)))*((2*(e + f*x) 
^(3/2))/(3*b) + ((b*e - a*f)*((2*Sqrt[e + f*x])/b - (2*Sqrt[b*e - a*f]*Arc 
Tanh[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[b*e - a*f]])/b^(3/2)))/b))/(8*b^2*(b*e - 
 a*f)^2)
 

Defintions of rubi rules used

rule 60

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

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 = 0.51 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(-\frac {35 \left (-\left (\frac {\left (3 c g \,f^{2}+12 e \left (c h +d g \right ) f +8 d \,e^{2} h \right ) b^{2}}{35}-\frac {3 a \left (f \left (c h +d g \right )+\frac {8 d e h}{3}\right ) f b}{7}+a^{2} d \,f^{2} h \right ) \left (b x +a \right )^{2} \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (\frac {\left (x \left (-\frac {8 d h \,x^{2}}{15}+\frac {8 \left (-c h -d g \right ) x}{5}+c g \right ) f +\frac {2 \left (-\frac {16 d h \,x^{2}}{3}+2 \left (c h +d g \right ) x +c g \right ) e}{5}\right ) b^{3}}{7}+\frac {2 a \left (\left (\frac {28 d h \,x^{2}}{3}+\frac {25 \left (-c h -d g \right ) x}{2}+\frac {3 c g}{2}\right ) f +e \left (-\frac {44}{3} d h x +c h +d g \right )\right ) b^{2}}{35}-\frac {3 a^{2} \left (\left (-\frac {35}{9} d h x +c h +d g \right ) f +\frac {10 d e h}{9}\right ) b}{7}+a^{3} d f h \right ) \sqrt {f x +e}\right )}{4 \sqrt {\left (a f -b e \right ) b}\, \left (b x +a \right )^{2} b^{4}}\)	\(275\)
risch	\(-\frac {2 \left (-d h x b f +9 a d f h -3 b c f h -4 b d e h -3 b d f g \right ) \sqrt {f x +e}}{3 b^{4}}+\frac {\frac {2 \left (-\frac {13}{8} a^{2} b d \,f^{2} h +\frac {9}{8} a \,b^{2} c \,f^{2} h +a \,b^{2} d e f h +\frac {9}{8} a \,b^{2} d \,f^{2} g -\frac {1}{2} b^{3} c e f h -\frac {5}{8} b^{3} c \,f^{2} g -\frac {1}{2} b^{3} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (11 a^{3} d \,f^{2} h -7 a^{2} b c \,f^{2} h -19 a^{2} b d e f h -7 a^{2} b d \,f^{2} g +11 a \,b^{2} c e f h +3 a \,b^{2} c \,f^{2} g +8 a \,b^{2} d \,e^{2} h +11 a \,b^{2} d e f g -4 b^{3} c \,e^{2} h -3 b^{3} c e f g -4 b^{3} d \,e^{2} g \right ) \sqrt {f x +e}}{4}}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{2} d \,f^{2} h -15 a b c \,f^{2} h -40 a b d e f h -15 a b d \,f^{2} g +12 b^{2} c e f h +3 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +12 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \sqrt {\left (a f -b e \right ) b}}}{b^{4}}\)	\(385\)
derivativedivides	\(-\frac {2 \left (-\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-b c f h \sqrt {f x +e}-b d e h \sqrt {f x +e}-b d f g \sqrt {f x +e}\right )}{b^{4}}+\frac {\frac {2 \left (\left (-\frac {13}{8} a^{2} b d \,f^{2} h +\frac {9}{8} a \,b^{2} c \,f^{2} h +a \,b^{2} d e f h +\frac {9}{8} a \,b^{2} d \,f^{2} g -\frac {1}{2} b^{3} c e f h -\frac {5}{8} b^{3} c \,f^{2} g -\frac {1}{2} b^{3} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (11 a^{3} d \,f^{2} h -7 a^{2} b c \,f^{2} h -19 a^{2} b d e f h -7 a^{2} b d \,f^{2} g +11 a \,b^{2} c e f h +3 a \,b^{2} c \,f^{2} g +8 a \,b^{2} d \,e^{2} h +11 a \,b^{2} d e f g -4 b^{3} c \,e^{2} h -3 b^{3} c e f g -4 b^{3} d \,e^{2} g \right ) \sqrt {f x +e}}{8}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{2} d \,f^{2} h -15 a b c \,f^{2} h -40 a b d e f h -15 a b d \,f^{2} g +12 b^{2} c e f h +3 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +12 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \sqrt {\left (a f -b e \right ) b}}}{b^{4}}\)	\(411\)
default	\(-\frac {2 \left (-\frac {d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+3 a d f h \sqrt {f x +e}-b c f h \sqrt {f x +e}-b d e h \sqrt {f x +e}-b d f g \sqrt {f x +e}\right )}{b^{4}}+\frac {\frac {2 \left (\left (-\frac {13}{8} a^{2} b d \,f^{2} h +\frac {9}{8} a \,b^{2} c \,f^{2} h +a \,b^{2} d e f h +\frac {9}{8} a \,b^{2} d \,f^{2} g -\frac {1}{2} b^{3} c e f h -\frac {5}{8} b^{3} c \,f^{2} g -\frac {1}{2} b^{3} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (11 a^{3} d \,f^{2} h -7 a^{2} b c \,f^{2} h -19 a^{2} b d e f h -7 a^{2} b d \,f^{2} g +11 a \,b^{2} c e f h +3 a \,b^{2} c \,f^{2} g +8 a \,b^{2} d \,e^{2} h +11 a \,b^{2} d e f g -4 b^{3} c \,e^{2} h -3 b^{3} c e f g -4 b^{3} d \,e^{2} g \right ) \sqrt {f x +e}}{8}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (35 a^{2} d \,f^{2} h -15 a b c \,f^{2} h -40 a b d e f h -15 a b d \,f^{2} g +12 b^{2} c e f h +3 b^{2} c \,f^{2} g +8 b^{2} d \,e^{2} h +12 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 \sqrt {\left (a f -b e \right ) b}}}{b^{4}}\)	\(411\)

Input:

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

Output:

-35/4/((a*f-b*e)*b)^(1/2)*(-(1/35*(3*c*g*f^2+12*e*(c*h+d*g)*f+8*d*e^2*h)*b 
^2-3/7*a*(f*(c*h+d*g)+8/3*d*e*h)*f*b+a^2*d*f^2*h)*(b*x+a)^2*arctan(b*(f*x+ 
e)^(1/2)/((a*f-b*e)*b)^(1/2))+((a*f-b*e)*b)^(1/2)*(1/7*(x*(-8/15*d*h*x^2+8 
/5*(-c*h-d*g)*x+c*g)*f+2/5*(-16/3*d*h*x^2+2*(c*h+d*g)*x+c*g)*e)*b^3+2/35*a 
*((28/3*d*h*x^2+25/2*(-c*h-d*g)*x+3/2*c*g)*f+e*(-44/3*d*h*x+c*h+d*g))*b^2- 
3/7*a^2*((-35/9*d*h*x+c*h+d*g)*f+10/9*d*e*h)*b+a^3*d*f*h)*(f*x+e)^(1/2))/( 
b*x+a)^2/b^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (252) = 504\).

Time = 0.17 (sec) , antiderivative size = 1500, normalized size of antiderivative = 5.40 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/24*(3*sqrt(b^2*e - a*b*f)*((3*(4*b^4*d*e*f + (b^4*c - 5*a*b^3*d)*f^2)*g 
 + (8*b^4*d*e^2 + 4*(3*b^4*c - 10*a*b^3*d)*e*f - 5*(3*a*b^3*c - 7*a^2*b^2* 
d)*f^2)*h)*x^2 + 3*(4*a^2*b^2*d*e*f + (a^2*b^2*c - 5*a^3*b*d)*f^2)*g + (8* 
a^2*b^2*d*e^2 + 4*(3*a^2*b^2*c - 10*a^3*b*d)*e*f - 5*(3*a^3*b*c - 7*a^4*d) 
*f^2)*h + 2*(3*(4*a*b^3*d*e*f + (a*b^3*c - 5*a^2*b^2*d)*f^2)*g + (8*a*b^3* 
d*e^2 + 4*(3*a*b^3*c - 10*a^2*b^2*d)*e*f - 5*(3*a^2*b^2*c - 7*a^3*b*d)*f^2 
)*h)*x)*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*(8*(b^5*d*e*f - a*b^4*d*f^2)*h*x^3 + 8*(3*(b^5*d*e*f - a*b^4* 
d*f^2)*g + (4*b^5*d*e^2 + (3*b^5*c - 11*a*b^4*d)*e*f - (3*a*b^4*c - 7*a^2* 
b^3*d)*f^2)*h)*x^2 - 3*(2*(b^5*c + a*b^4*d)*e^2 + (a*b^4*c - 17*a^2*b^3*d) 
*e*f - 3*(a^2*b^3*c - 5*a^3*b^2*d)*f^2)*g - (2*(3*a*b^4*c - 25*a^2*b^3*d)* 
e^2 - (51*a^2*b^3*c - 155*a^3*b^2*d)*e*f + 15*(3*a^3*b^2*c - 7*a^4*b*d)*f^ 
2)*h - (3*(4*b^5*d*e^2 + (5*b^5*c - 29*a*b^4*d)*e*f - 5*(a*b^4*c - 5*a^2*b 
^3*d)*f^2)*g + (4*(3*b^5*c - 22*a*b^4*d)*e^2 - (87*a*b^4*c - 263*a^2*b^3*d 
)*e*f + 25*(3*a^2*b^3*c - 7*a^3*b^2*d)*f^2)*h)*x)*sqrt(f*x + e))/(a^2*b^6* 
e - a^3*b^5*f + (b^8*e - a*b^7*f)*x^2 + 2*(a*b^7*e - a^2*b^6*f)*x), 1/12*( 
3*sqrt(-b^2*e + a*b*f)*((3*(4*b^4*d*e*f + (b^4*c - 5*a*b^3*d)*f^2)*g + (8* 
b^4*d*e^2 + 4*(3*b^4*c - 10*a*b^3*d)*e*f - 5*(3*a*b^3*c - 7*a^2*b^2*d)*f^2 
)*h)*x^2 + 3*(4*a^2*b^2*d*e*f + (a^2*b^2*c - 5*a^3*b*d)*f^2)*g + (8*a^2*b^ 
2*d*e^2 + 4*(3*a^2*b^2*c - 10*a^3*b*d)*e*f - 5*(3*a^3*b*c - 7*a^4*d)*f^...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^3,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. 541 vs. \(2 (252) = 504\).

Time = 0.14 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.95 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\frac {{\left (12 \, b^{2} d e f g + 3 \, b^{2} c f^{2} g - 15 \, a b d f^{2} g + 8 \, b^{2} d e^{2} h + 12 \, b^{2} c e f h - 40 \, a b d e f h - 15 \, a b c f^{2} h + 35 \, a^{2} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{4 \, \sqrt {-b^{2} e + a b f} b^{4}} - \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d e f g - 4 \, \sqrt {f x + e} b^{3} d e^{2} f g + 5 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c f^{2} g - 9 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d f^{2} g - 3 \, \sqrt {f x + e} b^{3} c e f^{2} g + 11 \, \sqrt {f x + e} a b^{2} d e f^{2} g + 3 \, \sqrt {f x + e} a b^{2} c f^{3} g - 7 \, \sqrt {f x + e} a^{2} b d f^{3} g + 4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c e f h - 8 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d e f h - 4 \, \sqrt {f x + e} b^{3} c e^{2} f h + 8 \, \sqrt {f x + e} a b^{2} d e^{2} f h - 9 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c f^{2} h + 13 \, {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d f^{2} h + 11 \, \sqrt {f x + e} a b^{2} c e f^{2} h - 19 \, \sqrt {f x + e} a^{2} b d e f^{2} h - 7 \, \sqrt {f x + e} a^{2} b c f^{3} h + 11 \, \sqrt {f x + e} a^{3} d f^{3} h}{4 \, {\left ({\left (f x + e\right )} b - b e + a f\right )}^{2} b^{4}} + \frac {2 \, {\left (3 \, \sqrt {f x + e} b^{6} d f g + {\left (f x + e\right )}^{\frac {3}{2}} b^{6} d h + 3 \, \sqrt {f x + e} b^{6} d e h + 3 \, \sqrt {f x + e} b^{6} c f h - 9 \, \sqrt {f x + e} a b^{5} d f h\right )}}{3 \, b^{9}} \] Input:

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

Output:

1/4*(12*b^2*d*e*f*g + 3*b^2*c*f^2*g - 15*a*b*d*f^2*g + 8*b^2*d*e^2*h + 12* 
b^2*c*e*f*h - 40*a*b*d*e*f*h - 15*a*b*c*f^2*h + 35*a^2*d*f^2*h)*arctan(sqr 
t(f*x + e)*b/sqrt(-b^2*e + a*b*f))/(sqrt(-b^2*e + a*b*f)*b^4) - 1/4*(4*(f* 
x + e)^(3/2)*b^3*d*e*f*g - 4*sqrt(f*x + e)*b^3*d*e^2*f*g + 5*(f*x + e)^(3/ 
2)*b^3*c*f^2*g - 9*(f*x + e)^(3/2)*a*b^2*d*f^2*g - 3*sqrt(f*x + e)*b^3*c*e 
*f^2*g + 11*sqrt(f*x + e)*a*b^2*d*e*f^2*g + 3*sqrt(f*x + e)*a*b^2*c*f^3*g 
- 7*sqrt(f*x + e)*a^2*b*d*f^3*g + 4*(f*x + e)^(3/2)*b^3*c*e*f*h - 8*(f*x + 
 e)^(3/2)*a*b^2*d*e*f*h - 4*sqrt(f*x + e)*b^3*c*e^2*f*h + 8*sqrt(f*x + e)* 
a*b^2*d*e^2*f*h - 9*(f*x + e)^(3/2)*a*b^2*c*f^2*h + 13*(f*x + e)^(3/2)*a^2 
*b*d*f^2*h + 11*sqrt(f*x + e)*a*b^2*c*e*f^2*h - 19*sqrt(f*x + e)*a^2*b*d*e 
*f^2*h - 7*sqrt(f*x + e)*a^2*b*c*f^3*h + 11*sqrt(f*x + e)*a^3*d*f^3*h)/((( 
f*x + e)*b - b*e + a*f)^2*b^4) + 2/3*(3*sqrt(f*x + e)*b^6*d*f*g + (f*x + e 
)^(3/2)*b^6*d*h + 3*sqrt(f*x + e)*b^6*d*e*h + 3*sqrt(f*x + e)*b^6*c*f*h - 
9*sqrt(f*x + e)*a*b^5*d*f*h)/b^9
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.62 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx=\sqrt {e+f\,x}\,\left (\frac {2\,c\,f\,h-4\,d\,e\,h+2\,d\,f\,g}{b^3}+\frac {2\,d\,h\,\left (3\,b^3\,e-3\,a\,b^2\,f\right )}{b^6}\right )+\frac {\sqrt {e+f\,x}\,\left (\frac {7\,a^2\,b\,c\,f^3\,h}{4}-\frac {3\,a\,b^2\,c\,f^3\,g}{4}-\frac {11\,a^3\,d\,f^3\,h}{4}+\frac {7\,a^2\,b\,d\,f^3\,g}{4}+\frac {3\,b^3\,c\,e\,f^2\,g}{4}+b^3\,c\,e^2\,f\,h+b^3\,d\,e^2\,f\,g-\frac {11\,a\,b^2\,c\,e\,f^2\,h}{4}-\frac {11\,a\,b^2\,d\,e\,f^2\,g}{4}-2\,a\,b^2\,d\,e^2\,f\,h+\frac {19\,a^2\,b\,d\,e\,f^2\,h}{4}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {5\,b^3\,c\,f^2\,g}{4}-\frac {9\,a\,b^2\,c\,f^2\,h}{4}-\frac {9\,a\,b^2\,d\,f^2\,g}{4}+\frac {13\,a^2\,b\,d\,f^2\,h}{4}+b^3\,c\,e\,f\,h+b^3\,d\,e\,f\,g-2\,a\,b^2\,d\,e\,f\,h\right )}{b^6\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,b^6\,e-2\,a\,b^5\,f\right )+b^6\,e^2+a^2\,b^4\,f^2-2\,a\,b^5\,e\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {e+f\,x}}{\sqrt {a\,f-b\,e}}\right )\,\left (3\,b^2\,c\,f^2\,g+35\,a^2\,d\,f^2\,h+8\,b^2\,d\,e^2\,h-15\,a\,b\,c\,f^2\,h-15\,a\,b\,d\,f^2\,g+12\,b^2\,c\,e\,f\,h+12\,b^2\,d\,e\,f\,g-40\,a\,b\,d\,e\,f\,h\right )}{4\,b^{9/2}\,\sqrt {a\,f-b\,e}}+\frac {2\,d\,h\,{\left (e+f\,x\right )}^{3/2}}{3\,b^3} \] Input:

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

Output:

(e + f*x)^(1/2)*((2*c*f*h - 4*d*e*h + 2*d*f*g)/b^3 + (2*d*h*(3*b^3*e - 3*a 
*b^2*f))/b^6) + ((e + f*x)^(1/2)*((7*a^2*b*c*f^3*h)/4 - (3*a*b^2*c*f^3*g)/ 
4 - (11*a^3*d*f^3*h)/4 + (7*a^2*b*d*f^3*g)/4 + (3*b^3*c*e*f^2*g)/4 + b^3*c 
*e^2*f*h + b^3*d*e^2*f*g - (11*a*b^2*c*e*f^2*h)/4 - (11*a*b^2*d*e*f^2*g)/4 
 - 2*a*b^2*d*e^2*f*h + (19*a^2*b*d*e*f^2*h)/4) - (e + f*x)^(3/2)*((5*b^3*c 
*f^2*g)/4 - (9*a*b^2*c*f^2*h)/4 - (9*a*b^2*d*f^2*g)/4 + (13*a^2*b*d*f^2*h) 
/4 + b^3*c*e*f*h + b^3*d*e*f*g - 2*a*b^2*d*e*f*h))/(b^6*(e + f*x)^2 - (e + 
 f*x)*(2*b^6*e - 2*a*b^5*f) + b^6*e^2 + a^2*b^4*f^2 - 2*a*b^5*e*f) + (atan 
((b^(1/2)*(e + f*x)^(1/2))/(a*f - b*e)^(1/2))*(3*b^2*c*f^2*g + 35*a^2*d*f^ 
2*h + 8*b^2*d*e^2*h - 15*a*b*c*f^2*h - 15*a*b*d*f^2*g + 12*b^2*c*e*f*h + 1 
2*b^2*d*e*f*g - 40*a*b*d*e*f*h))/(4*b^(9/2)*(a*f - b*e)^(1/2)) + (2*d*h*(e 
 + f*x)^(3/2))/(3*b^3)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1779, normalized size of antiderivative = 6.40 \[ \int \frac {(c+d x) (e+f x)^{3/2} (g+h x)}{(a+b x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(105*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b* 
e)))*a**4*d*f**2*h - 45*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**3*b*c*f**2*h - 120*sqrt(b)*sqrt(a*f - b*e)*atan 
((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*d*e*f*h - 45*sqrt(b)* 
sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*d 
*f**2*g + 210*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt 
(a*f - b*e)))*a**3*b*d*f**2*h*x + 36*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
+ f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c*e*f*h + 9*sqrt(b)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*c*f** 
2*g - 90*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
- b*e)))*a**2*b**2*c*f**2*h*x + 24*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + 
f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d*e**2*h + 36*sqrt(b)*sqrt(a* 
f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d*e*f 
*g - 240*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
- b*e)))*a**2*b**2*d*e*f*h*x - 90*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f 
*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d*f**2*g*x + 105*sqrt(b)*sqrt( 
a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d*f 
**2*h*x**2 + 72*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sq 
rt(a*f - b*e)))*a*b**3*c*e*f*h*x + 18*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
 + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a*b**3*c*f**2*g*x - 45*sqrt(b)*sq...