\(\int \frac {(c+d x) (g+h x)}{(a+b x)^3 (e+f x)^{3/2}} \, dx\) [156]

Optimal result

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

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

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.25 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 (e+f x)^{3/2}} \, dx=-\frac {-a^3 d f h (e+f x)+a^2 b \left (c f (-8 f g+13 e h+5 f h x)+d \left (-14 e^2 h+e f (13 g-5 h x)+f^2 x (5 g+h x)\right )\right )+b^3 \left (4 d e x (3 f g x+e (g-2 h x))+c \left (-15 f^2 g x^2+2 e^2 (g+2 h x)+e f x (-5 g+12 h x)\right )\right )+a b^2 \left (d \left (3 f^2 g x^2+2 e^2 (g-12 h x)+e f x (21 g-8 h x)\right )+c \left (2 e^2 h+f^2 x (-25 g+3 h x)+e f (-9 g+21 h x)\right )\right )}{4 b (b e-a f)^3 (a+b x)^2 \sqrt {e+f x}}-\frac {\left (-a^2 d f^2 h+a b f (-3 d f g+8 d e h-3 c f h)+b^2 (3 c f (5 f g-4 e h)+4 d e (-3 f g+2 e h))\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{4 b^{3/2} (-b e+a f)^{7/2}} \] Input:

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

Output:

-1/4*(-(a^3*d*f*h*(e + f*x)) + a^2*b*(c*f*(-8*f*g + 13*e*h + 5*f*h*x) + d* 
(-14*e^2*h + e*f*(13*g - 5*h*x) + f^2*x*(5*g + h*x))) + b^3*(4*d*e*x*(3*f* 
g*x + e*(g - 2*h*x)) + c*(-15*f^2*g*x^2 + 2*e^2*(g + 2*h*x) + e*f*x*(-5*g 
+ 12*h*x))) + a*b^2*(d*(3*f^2*g*x^2 + 2*e^2*(g - 12*h*x) + e*f*x*(21*g - 8 
*h*x)) + c*(2*e^2*h + f^2*x*(-25*g + 3*h*x) + e*f*(-9*g + 21*h*x))))/(b*(b 
*e - a*f)^3*(a + b*x)^2*Sqrt[e + f*x]) - ((-(a^2*d*f^2*h) + a*b*f*(-3*d*f* 
g + 8*d*e*h - 3*c*f*h) + b^2*(3*c*f*(5*f*g - 4*e*h) + 4*d*e*(-3*f*g + 2*e* 
h)))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/(4*b^(3/2)*(-(b*e 
) + a*f)^(7/2))
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {161, 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.

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

\(\Big \downarrow \) 161

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

\(\Big \downarrow \) 52

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

-1/2*(b^2*c*e*f*g + a^2*d*e*f*h + 2*a*b*(2*c*f^2*g + 2*d*e^2*h - (5*e*f*(d 
*g + c*h))/2) + (a^2*d*f^2*h - a*b*f^2*(d*g + c*h) + b^2*(c*f*(5*f*g - 4*e 
*h) - 4*d*e*(f*g - e*h)))*x)/(b*f*(b*e - a*f)^2*(a + b*x)^2*Sqrt[e + f*x]) 
 + ((a^2*d*f^2*h + a*b*f*(3*d*f*g - 8*d*e*h + 3*c*f*h) - b^2*(3*c*f*(5*f*g 
 - 4*e*h) - 4*d*e*(3*f*g - 2*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))))/(4*b*f*(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 161

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_)) 
*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^2*c*d*e*g*(n + 1) + a^2*c*d*f*h*(n + 
 1) + a*b*(d^2*e*g*(m + 1) + c^2*f*h*(m + 1) - c*d*(f*g + e*h)*(m + n + 2)) 
 + (a^2*d^2*f*h*(n + 1) - a*b*d^2*(f*g + e*h)*(n + 1) + b^2*(c^2*f*h*(m + 1 
) - c*d*(f*g + e*h)*(m + 1) + d^2*e*g*(m + n + 2)))*x)/(b*d*(b*c - a*d)^2*( 
m + 1)*(n + 1)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f* 
h*(2 + 3*n + n^2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 
 3)) + b^2*(c^2*f*h*(2 + 3*m + m^2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + 
 d^2*e*g*(6 + m^2 + 5*n + n^2 + m*(2*n + 5))))/(b*d*(b*c - a*d)^2*(m + 1)*( 
n + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h}, x] && LtQ[m, -1] && LtQ[n, -1]
 

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 = 357, normalized size of antiderivative = 1.28

method	result	size
pseudoelliptic	\(-\frac {-\left (\left (-15 c g \,f^{2}+12 e \left (c h +d g \right ) f -8 d \,e^{2} h \right ) b^{2}+3 a \left (f \left (c h +d g \right )-\frac {8 d e h}{3}\right ) f b +a^{2} d \,f^{2} h \right ) \left (b x +a \right )^{2} \sqrt {f x +e}\, \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )+\sqrt {\left (a f -b e \right ) b}\, \left (\left (15 c \,f^{2} g \,x^{2}+5 x e \left (\frac {12 \left (-c h -d g \right ) x}{5}+c g \right ) f -2 \left (-4 d h \,x^{2}+2 \left (c h +d g \right ) x +c g \right ) e^{2}\right ) b^{3}-2 a \left (\frac {\left (3 \left (c h +d g \right ) x^{2}-25 c g x \right ) f^{2}}{2}-\frac {9 \left (\frac {8 d h \,x^{2}}{9}+\frac {7 \left (-c h -d g \right ) x}{3}+c g \right ) e f}{2}+e^{2} \left (-12 d h x +c h +d g \right )\right ) b^{2}-13 a^{2} \left (\frac {\left (d h \,x^{2}+5 \left (c h +d g \right ) x -8 c g \right ) f^{2}}{13}+e \left (-\frac {5}{13} d h x +c h +d g \right ) f -\frac {14 d \,e^{2} h}{13}\right ) b +a^{3} d f h \left (f x +e \right )\right )}{4 \sqrt {f x +e}\, \sqrt {\left (a f -b e \right ) b}\, \left (a f -b e \right )^{3} \left (b x +a \right )^{2} b}\)	\(357\)
derivativedivides	\(-\frac {2 \left (-c e f h +c g \,f^{2}+d \,e^{2} h -d e f g \right )}{\left (a f -b e \right )^{3} \sqrt {f x +e}}+\frac {\frac {2 \left (\left (\frac {1}{8} a^{2} d \,f^{2} h +\frac {3}{8} a b c \,f^{2} h -a b d e f h +\frac {3}{8} a b d \,f^{2} g +\frac {1}{2} c e f h \,b^{2}-\frac {7}{8} b^{2} c \,f^{2} g +\frac {1}{2} b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (a^{3} d \,f^{2} h -5 a^{2} b c \,f^{2} h +7 a^{2} b d e f h -5 a^{2} b d \,f^{2} g +a \,b^{2} c e f h +9 a \,b^{2} c \,f^{2} g -8 a \,b^{2} d \,e^{2} h +a \,b^{2} d e f g +4 b^{3} c \,e^{2} h -9 b^{3} c e f g +4 b^{3} d \,e^{2} g \right ) \sqrt {f x +e}}{8 b}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (a^{2} d \,f^{2} h +3 a b c \,f^{2} h -8 a b d e f h +3 a b d \,f^{2} g +12 c e f h \,b^{2}-15 b^{2} c \,f^{2} g -8 d \,e^{2} h \,b^{2}+12 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 b \sqrt {\left (a f -b e \right ) b}}}{\left (a f -b e \right )^{3}}\)	\(388\)
default	\(-\frac {2 \left (-c e f h +c g \,f^{2}+d \,e^{2} h -d e f g \right )}{\left (a f -b e \right )^{3} \sqrt {f x +e}}+\frac {\frac {2 \left (\left (\frac {1}{8} a^{2} d \,f^{2} h +\frac {3}{8} a b c \,f^{2} h -a b d e f h +\frac {3}{8} a b d \,f^{2} g +\frac {1}{2} c e f h \,b^{2}-\frac {7}{8} b^{2} c \,f^{2} g +\frac {1}{2} b^{2} d e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (a^{3} d \,f^{2} h -5 a^{2} b c \,f^{2} h +7 a^{2} b d e f h -5 a^{2} b d \,f^{2} g +a \,b^{2} c e f h +9 a \,b^{2} c \,f^{2} g -8 a \,b^{2} d \,e^{2} h +a \,b^{2} d e f g +4 b^{3} c \,e^{2} h -9 b^{3} c e f g +4 b^{3} d \,e^{2} g \right ) \sqrt {f x +e}}{8 b}\right )}{\left (\left (f x +e \right ) b +a f -b e \right )^{2}}+\frac {\left (a^{2} d \,f^{2} h +3 a b c \,f^{2} h -8 a b d e f h +3 a b d \,f^{2} g +12 c e f h \,b^{2}-15 b^{2} c \,f^{2} g -8 d \,e^{2} h \,b^{2}+12 b^{2} d e f g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{4 b \sqrt {\left (a f -b e \right ) b}}}{\left (a f -b e \right )^{3}}\)	\(388\)

Input:

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

Output:

-1/4/(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2)*(-((-15*c*g*f^2+12*e*(c*h+d*g)*f-8* 
d*e^2*h)*b^2+3*a*(f*(c*h+d*g)-8/3*d*e*h)*f*b+a^2*d*f^2*h)*(b*x+a)^2*(f*x+e 
)^(1/2)*arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))+((a*f-b*e)*b)^(1/2)*(( 
15*c*f^2*g*x^2+5*x*e*(12/5*(-c*h-d*g)*x+c*g)*f-2*(-4*d*h*x^2+2*(c*h+d*g)*x 
+c*g)*e^2)*b^3-2*a*(1/2*(3*(c*h+d*g)*x^2-25*c*g*x)*f^2-9/2*(8/9*d*h*x^2+7/ 
3*(-c*h-d*g)*x+c*g)*e*f+e^2*(-12*d*h*x+c*h+d*g))*b^2-13*a^2*(1/13*(d*h*x^2 
+5*(c*h+d*g)*x-8*c*g)*f^2+e*(-5/13*d*h*x+c*h+d*g)*f-14/13*d*e^2*h)*b+a^3*d 
*f*h*(f*x+e)))/(a*f-b*e)^3/(b*x+a)^2/b
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1246 vs. \(2 (255) = 510\).

Time = 0.25 (sec) , antiderivative size = 2506, normalized size of antiderivative = 9.01 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/8*(((3*(4*b^4*d*e*f^2 - (5*b^4*c - a*b^3*d)*f^3)*g - (8*b^4*d*e^2*f - 
4*(3*b^4*c - 2*a*b^3*d)*e*f^2 - (3*a*b^3*c + a^2*b^2*d)*f^3)*h)*x^3 + (3*( 
4*b^4*d*e^2*f - (5*b^4*c - 9*a*b^3*d)*e*f^2 - 2*(5*a*b^3*c - a^2*b^2*d)*f^ 
3)*g - (8*b^4*d*e^3 - 12*(b^4*c - 2*a*b^3*d)*e^2*f - 3*(9*a*b^3*c - 5*a^2* 
b^2*d)*e*f^2 - 2*(3*a^2*b^2*c + a^3*b*d)*f^3)*h)*x^2 + 3*(4*a^2*b^2*d*e^2* 
f - (5*a^2*b^2*c - a^3*b*d)*e*f^2)*g - (8*a^2*b^2*d*e^3 - 4*(3*a^2*b^2*c - 
 2*a^3*b*d)*e^2*f - (3*a^3*b*c + a^4*d)*e*f^2)*h + (3*(8*a*b^3*d*e^2*f - 2 
*(5*a*b^3*c - 3*a^2*b^2*d)*e*f^2 - (5*a^2*b^2*c - a^3*b*d)*f^3)*g - (16*a* 
b^3*d*e^3 - 24*(a*b^3*c - a^2*b^2*d)*e^2*f - 6*(3*a^2*b^2*c - a^3*b*d)*e*f 
^2 - (3*a^3*b*c + a^4*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*(4*b^5*d* 
e^2*f - (5*b^5*c + 3*a*b^4*d)*e*f^2 + (5*a*b^4*c - a^2*b^3*d)*f^3)*g - (8* 
b^5*d*e^3 - 12*b^5*c*e^2*f + 9*(a*b^4*c - a^2*b^3*d)*e*f^2 + (3*a^2*b^3*c 
+ a^3*b^2*d)*f^3)*h)*x^2 + (8*a^3*b^2*c*f^3 + 2*(b^5*c + a*b^4*d)*e^3 - 11 
*(a*b^4*c - a^2*b^3*d)*e^2*f + (a^2*b^3*c - 13*a^3*b^2*d)*e*f^2)*g + (2*(a 
*b^4*c - 7*a^2*b^3*d)*e^3 + (11*a^2*b^3*c + 13*a^3*b^2*d)*e^2*f - (13*a^3* 
b^2*c - a^4*b*d)*e*f^2)*h + ((4*b^5*d*e^3 - (5*b^5*c - 17*a*b^4*d)*e^2*f - 
 4*(5*a*b^4*c + 4*a^2*b^3*d)*e*f^2 + 5*(5*a^2*b^3*c - a^3*b^2*d)*f^3)*g + 
(4*(b^5*c - 6*a*b^4*d)*e^3 + (17*a*b^4*c + 19*a^2*b^3*d)*e^2*f - 4*(4*a^2* 
b^3*c - a^3*b^2*d)*e*f^2 - (5*a^3*b^2*c - a^4*b*d)*f^3)*h)*x)*sqrt(f*x ...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((d*x+c)*(h*x+g)/(b*x+a)^3/(f*x+e)^(3/2),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. 607 vs. \(2 (255) = 510\).

Time = 0.15 (sec) , antiderivative size = 607, normalized size of antiderivative = 2.18 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 (e+f x)^{3/2}} \, dx=-\frac {{\left (12 \, b^{2} d e f g - 15 \, b^{2} c f^{2} g + 3 \, a b d f^{2} g - 8 \, b^{2} d e^{2} h + 12 \, b^{2} c e f h - 8 \, a b d e f h + 3 \, a b c f^{2} h + a^{2} d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} b}{\sqrt {-b^{2} e + a b f}}\right )}{4 \, {\left (b^{4} e^{3} - 3 \, a b^{3} e^{2} f + 3 \, a^{2} b^{2} e f^{2} - a^{3} b f^{3}\right )} \sqrt {-b^{2} e + a b f}} - \frac {2 \, {\left (d e f g - c f^{2} g - d e^{2} h + c e f h\right )}}{{\left (b^{3} e^{3} - 3 \, a b^{2} e^{2} f + 3 \, a^{2} b e f^{2} - a^{3} f^{3}\right )} \sqrt {f x + e}} - \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} d e f g - 4 \, \sqrt {f x + e} b^{3} d e^{2} f g - 7 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c f^{2} g + 3 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d f^{2} g + 9 \, \sqrt {f x + e} b^{3} c e f^{2} g - \sqrt {f x + e} a b^{2} d e f^{2} g - 9 \, \sqrt {f x + e} a b^{2} c f^{3} g + 5 \, \sqrt {f x + e} a^{2} b d f^{3} g + 4 \, {\left (f x + e\right )}^{\frac {3}{2}} b^{3} c e f h - 8 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} d e f h - 4 \, \sqrt {f x + e} b^{3} c e^{2} f h + 8 \, \sqrt {f x + e} a b^{2} d e^{2} f h + 3 \, {\left (f x + e\right )}^{\frac {3}{2}} a b^{2} c f^{2} h + {\left (f x + e\right )}^{\frac {3}{2}} a^{2} b d f^{2} h - \sqrt {f x + e} a b^{2} c e f^{2} h - 7 \, \sqrt {f x + e} a^{2} b d e f^{2} h + 5 \, \sqrt {f x + e} a^{2} b c f^{3} h - \sqrt {f x + e} a^{3} d f^{3} h}{4 \, {\left (b^{4} e^{3} - 3 \, a b^{3} e^{2} f + 3 \, a^{2} b^{2} e f^{2} - a^{3} b f^{3}\right )} {\left ({\left (f x + e\right )} b - b e + a f\right )}^{2}} \] Input:

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

Output:

-1/4*(12*b^2*d*e*f*g - 15*b^2*c*f^2*g + 3*a*b*d*f^2*g - 8*b^2*d*e^2*h + 12 
*b^2*c*e*f*h - 8*a*b*d*e*f*h + 3*a*b*c*f^2*h + a^2*d*f^2*h)*arctan(sqrt(f* 
x + e)*b/sqrt(-b^2*e + a*b*f))/((b^4*e^3 - 3*a*b^3*e^2*f + 3*a^2*b^2*e*f^2 
 - a^3*b*f^3)*sqrt(-b^2*e + a*b*f)) - 2*(d*e*f*g - c*f^2*g - d*e^2*h + c*e 
*f*h)/((b^3*e^3 - 3*a*b^2*e^2*f + 3*a^2*b*e*f^2 - a^3*f^3)*sqrt(f*x + e)) 
- 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 - 7*( 
f*x + e)^(3/2)*b^3*c*f^2*g + 3*(f*x + e)^(3/2)*a*b^2*d*f^2*g + 9*sqrt(f*x 
+ e)*b^3*c*e*f^2*g - sqrt(f*x + e)*a*b^2*d*e*f^2*g - 9*sqrt(f*x + e)*a*b^2 
*c*f^3*g + 5*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 + 3*(f*x + e)^(3/2)*a*b^2*c*f^2*h + (f*x + e)^(3/ 
2)*a^2*b*d*f^2*h - sqrt(f*x + e)*a*b^2*c*e*f^2*h - 7*sqrt(f*x + e)*a^2*b*d 
*e*f^2*h + 5*sqrt(f*x + e)*a^2*b*c*f^3*h - sqrt(f*x + e)*a^3*d*f^3*h)/((b^ 
4*e^3 - 3*a*b^3*e^2*f + 3*a^2*b^2*e*f^2 - a^3*b*f^3)*((f*x + e)*b - b*e + 
a*f)^2)
 

Mupad [B] (verification not implemented)

Time = 2.80 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.59 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 (e+f x)^{3/2}} \, dx=-\frac {\frac {2\,\left (c\,f^2\,g+d\,e^2\,h-c\,e\,f\,h-d\,e\,f\,g\right )}{a\,f-b\,e}-\frac {{\left (e+f\,x\right )}^2\,\left (a^2\,d\,f^2\,h-15\,b^2\,c\,f^2\,g-8\,b^2\,d\,e^2\,h+3\,a\,b\,c\,f^2\,h+3\,a\,b\,d\,f^2\,g+12\,b^2\,c\,e\,f\,h+12\,b^2\,d\,e\,f\,g-8\,a\,b\,d\,e\,f\,h\right )}{4\,{\left (a\,f-b\,e\right )}^3}+\frac {\left (e+f\,x\right )\,\left (25\,b^2\,c\,f^2\,g+a^2\,d\,f^2\,h+16\,b^2\,d\,e^2\,h-5\,a\,b\,c\,f^2\,h-5\,a\,b\,d\,f^2\,g-20\,b^2\,c\,e\,f\,h-20\,b^2\,d\,e\,f\,g+8\,a\,b\,d\,e\,f\,h\right )}{4\,b\,{\left (a\,f-b\,e\right )}^2}}{b^2\,{\left (e+f\,x\right )}^{5/2}-{\left (e+f\,x\right )}^{3/2}\,\left (2\,b^2\,e-2\,a\,b\,f\right )+\sqrt {e+f\,x}\,\left (a^2\,f^2-2\,a\,b\,e\,f+b^2\,e^2\right )}-\frac {\mathrm {atan}\left (\frac {\sqrt {e+f\,x}\,\left (-a^3\,b\,f^3+3\,a^2\,b^2\,e\,f^2-3\,a\,b^3\,e^2\,f+b^4\,e^3\right )}{\sqrt {b}\,{\left (a\,f-b\,e\right )}^{7/2}}\right )\,\left (a^2\,d\,f^2\,h-15\,b^2\,c\,f^2\,g-8\,b^2\,d\,e^2\,h+3\,a\,b\,c\,f^2\,h+3\,a\,b\,d\,f^2\,g+12\,b^2\,c\,e\,f\,h+12\,b^2\,d\,e\,f\,g-8\,a\,b\,d\,e\,f\,h\right )}{4\,b^{3/2}\,{\left (a\,f-b\,e\right )}^{7/2}} \] Input:

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

Output:

- ((2*(c*f^2*g + d*e^2*h - c*e*f*h - d*e*f*g))/(a*f - b*e) - ((e + f*x)^2* 
(a^2*d*f^2*h - 15*b^2*c*f^2*g - 8*b^2*d*e^2*h + 3*a*b*c*f^2*h + 3*a*b*d*f^ 
2*g + 12*b^2*c*e*f*h + 12*b^2*d*e*f*g - 8*a*b*d*e*f*h))/(4*(a*f - b*e)^3) 
+ ((e + f*x)*(25*b^2*c*f^2*g + a^2*d*f^2*h + 16*b^2*d*e^2*h - 5*a*b*c*f^2* 
h - 5*a*b*d*f^2*g - 20*b^2*c*e*f*h - 20*b^2*d*e*f*g + 8*a*b*d*e*f*h))/(4*b 
*(a*f - b*e)^2))/(b^2*(e + f*x)^(5/2) - (e + f*x)^(3/2)*(2*b^2*e - 2*a*b*f 
) + (e + f*x)^(1/2)*(a^2*f^2 + b^2*e^2 - 2*a*b*e*f)) - (atan(((e + f*x)^(1 
/2)*(b^4*e^3 - a^3*b*f^3 + 3*a^2*b^2*e*f^2 - 3*a*b^3*e^2*f))/(b^(1/2)*(a*f 
 - b*e)^(7/2)))*(a^2*d*f^2*h - 15*b^2*c*f^2*g - 8*b^2*d*e^2*h + 3*a*b*c*f^ 
2*h + 3*a*b*d*f^2*g + 12*b^2*c*e*f*h + 12*b^2*d*e*f*g - 8*a*b*d*e*f*h))/(4 
*b^(3/2)*(a*f - b*e)^(7/2))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 2007, normalized size of antiderivative = 7.22 \[ \int \frac {(c+d x) (g+h x)}{(a+b x)^3 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqr 
t(a*f - b*e)))*a**4*d*f**2*h + 3*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*ata 
n((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*b*c*f**2*h - 8*sqrt(b) 
*sqrt(e + f*x)*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 + 3*sqrt(b)*sqrt(e + f*x)*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 + 2*sqrt(b)*sqrt(e 
 + f*x)*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 + 12*sqrt(b)*sqrt(e + f*x)*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 - 15*sqrt(b)*sqrt(e 
 + f*x)*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 + 6*sqrt(b)*sqrt(e + f*x)*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 - 8*sqrt(b)*sqrt 
(e + f*x)*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 + 12*sqrt(b)*sqrt(e + f*x)*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 - 16*sqrt(b)*sqr 
t(e + f*x)*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 + 6*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*atan((sqr 
t(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b**2*d*f**2*g*x + sqrt(b)*sq 
rt(e + f*x)*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 + 24*sqrt(b)*sqrt(e + f*x)*sqrt(a*f - b*e)*...