\(\int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx\) [101]

Optimal result

Integrand size = 29, antiderivative size = 491 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx=-\frac {(d e-c f) (3 b d g-b c h-2 a d h) \sqrt {e+f x}}{2 b d (b c-a d)^2 (c+d x)^2}-\frac {\left (4 a^2 d^2 f h-a b d (9 d f g+8 d e h-9 c f h)+b^2 \left (12 d^2 e g-c^2 f h-c d (3 f g+4 e h)\right )\right ) \sqrt {e+f x}}{4 b d (b c-a d)^3 (c+d x)}-\frac {(b g-a h) (e+f x)^{3/2}}{b (b c-a d) (a+b x) (c+d x)^2}+\frac {\sqrt {b e-a f} \left (a^2 d f h+b^2 (6 d e g-3 c f g-2 c e h)-a b (3 d f g+4 d e h-5 c f h)\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right )}{\sqrt {b} (b c-a d)^4}-\frac {\left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (24 d^3 e^2 g+c^3 f^2 h-8 c d^2 e (3 f g+e h)+c^2 d f (3 f g+4 e h)\right )-2 a b d \left (5 c^2 f^2 h+4 d^2 e (3 f g+2 e h)-c d f (9 f g+16 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{3/2} (b c-a d)^4 \sqrt {d e-c f}} \] Output:

-1/2*(-c*f+d*e)*(-2*a*d*h-b*c*h+3*b*d*g)*(f*x+e)^(1/2)/b/d/(-a*d+b*c)^2/(d 
*x+c)^2-1/4*(4*a^2*d^2*f*h-a*b*d*(-9*c*f*h+8*d*e*h+9*d*f*g)+b^2*(12*d^2*e* 
g-c^2*f*h-c*d*(4*e*h+3*f*g)))*(f*x+e)^(1/2)/b/d/(-a*d+b*c)^3/(d*x+c)-(-a*h 
+b*g)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(b*x+a)/(d*x+c)^2+(-a*f+b*e)^(1/2)*(a^2*d 
*f*h+b^2*(-2*c*e*h-3*c*f*g+6*d*e*g)-a*b*(-5*c*f*h+4*d*e*h+3*d*f*g))*arctan 
h(b^(1/2)*(f*x+e)^(1/2)/(-a*f+b*e)^(1/2))/b^(1/2)/(-a*d+b*c)^4-1/4*(3*a^2* 
d^2*f*(-5*c*f*h+4*d*e*h+d*f*g)+b^2*(24*d^3*e^2*g+c^3*f^2*h-8*c*d^2*e*(e*h+ 
3*f*g)+c^2*d*f*(4*e*h+3*f*g))-2*a*b*d*(5*c^2*f^2*h+4*d^2*e*(2*e*h+3*f*g)-c 
*d*f*(16*e*h+9*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2)/(-c*f+d*e)^(1/2))/d^(3 
/2)/(-a*d+b*c)^4/(-c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 3.83 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx=\frac {1}{4} \left (-\frac {\sqrt {e+f x} \left (a b \left (-c^3 f h+d^3 x (-6 e g+9 f g x+8 e h x)+c^2 d (9 f g+10 e h-6 f h x)+c d^2 \left (-10 e g+14 f g x+14 e h x-9 f h x^2\right )\right )+b^2 \left (-c^3 f h x-12 d^3 e g x^2+c d^2 x (-18 e g+3 f g x+4 e h x)+c^2 d \left (-4 e g+5 f g x+6 e h x+f h x^2\right )\right )+a^2 d \left (-11 c^2 f h+c d (3 f g+2 e h-17 f h x)+d^2 (f x (5 g-4 h x)+2 e (g+2 h x))\right )\right )}{d (-b c+a d)^3 (a+b x) (c+d x)^2}+\frac {4 \sqrt {-b e+a f} \left (a^2 d f h+b^2 (6 d e g-3 c f g-2 c e h)+a b (-3 d f g-4 d e h+5 c f h)\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {-b e+a f}}\right )}{\sqrt {b} (b c-a d)^4}+\frac {\left (3 a^2 d^2 f (d f g+4 d e h-5 c f h)+b^2 \left (24 d^3 e^2 g+c^3 f^2 h-8 c d^2 e (3 f g+e h)+c^2 d f (3 f g+4 e h)\right )-2 a b d \left (5 c^2 f^2 h+4 d^2 e (3 f g+2 e h)-c d f (9 f g+16 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (b c-a d)^4 \sqrt {-d e+c f}}\right ) \] Input:

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

Output:

(-((Sqrt[e + f*x]*(a*b*(-(c^3*f*h) + d^3*x*(-6*e*g + 9*f*g*x + 8*e*h*x) + 
c^2*d*(9*f*g + 10*e*h - 6*f*h*x) + c*d^2*(-10*e*g + 14*f*g*x + 14*e*h*x - 
9*f*h*x^2)) + b^2*(-(c^3*f*h*x) - 12*d^3*e*g*x^2 + c*d^2*x*(-18*e*g + 3*f* 
g*x + 4*e*h*x) + c^2*d*(-4*e*g + 5*f*g*x + 6*e*h*x + f*h*x^2)) + a^2*d*(-1 
1*c^2*f*h + c*d*(3*f*g + 2*e*h - 17*f*h*x) + d^2*(f*x*(5*g - 4*h*x) + 2*e* 
(g + 2*h*x)))))/(d*(-(b*c) + a*d)^3*(a + b*x)*(c + d*x)^2)) + (4*Sqrt[-(b* 
e) + a*f]*(a^2*d*f*h + b^2*(6*d*e*g - 3*c*f*g - 2*c*e*h) + a*b*(-3*d*f*g - 
 4*d*e*h + 5*c*f*h))*ArcTan[(Sqrt[b]*Sqrt[e + f*x])/Sqrt[-(b*e) + a*f]])/( 
Sqrt[b]*(b*c - a*d)^4) + ((3*a^2*d^2*f*(d*f*g + 4*d*e*h - 5*c*f*h) + b^2*( 
24*d^3*e^2*g + c^3*f^2*h - 8*c*d^2*e*(3*f*g + e*h) + c^2*d*f*(3*f*g + 4*e* 
h)) - 2*a*b*d*(5*c^2*f^2*h + 4*d^2*e*(3*f*g + 2*e*h) - c*d*f*(9*f*g + 16*e 
*h)))*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[-(d*e) + c*f]])/(d^(3/2)*(b*c - 
a*d)^4*Sqrt[-(d*e) + c*f]))/4
 

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {166, 27, 25, 166, 168, 27, 174, 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.

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

\(\Big \downarrow \) 166

\(\displaystyle \frac {\int -\frac {\sqrt {e+f x} (6 b d e g-3 b c f g-2 b c e h-4 a d e h+3 a c f h+f (3 b d g-2 b c h-a d h) x)}{2 (a+b x) (c+d x)^3}dx}{b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {\sqrt {e+f x} (a (4 d e-3 c f) h-b (6 d e g-3 c f g-2 c e h)-f (3 b d g-2 b c h-a d h) x)}{(a+b x) (c+d x)^3}dx}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sqrt {e+f x} (a (4 d e-3 c f) h-b (6 d e g-3 c f g-2 c e h)-f (3 b d g-2 b c h-a d h) x)}{(a+b x) (c+d x)^3}dx}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {-\frac {\int \frac {2 d f (2 d e-c f) h a^2-b \left (e (9 f g+8 e h) d^2-c f (3 f g+11 e h) d+c^2 f^2 h\right ) a+2 b^2 d e (6 d e g-3 c f g-2 c e h)+f \left (\left (-f h c^2-3 d (f g+e h) c+9 d^2 e g\right ) b^2+2 a d (4 c f h-3 d (f g+e h)) b+2 a^2 d^2 f h\right ) x}{(a+b x) (c+d x)^2 \sqrt {e+f x}}dx}{2 d (b c-a d)}-\frac {\sqrt {e+f x} (d e-c f) (-2 a d h-b c h+3 b d g)}{d (c+d x)^2 (b c-a d)}}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {-\frac {\frac {\int \frac {b (d e-c f) \left (-d f (11 c f h-3 d (f g+4 e h)) a^2-b \left (8 e (3 f g+2 e h) d^2-3 c f (3 f g+8 e h) d+c^2 f^2 h\right ) a+4 b^2 d e (6 d e g-3 c f g-2 c e h)+f \left (\left (-f h c^2-d (3 f g+4 e h) c+12 d^2 e g\right ) b^2-a d (9 d f g+8 d e h-9 c f h) b+4 a^2 d^2 f h\right ) x\right )}{2 (a+b x) (c+d x) \sqrt {e+f x}}dx}{(b c-a d) (d e-c f)}+\frac {\sqrt {e+f x} \left (4 a^2 d^2 f h-a b d (-9 c f h+8 d e h+9 d f g)+b^2 \left (c^2 (-f) h-c d (4 e h+3 f g)+12 d^2 e g\right )\right )}{(c+d x) (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} (d e-c f) (-2 a d h-b c h+3 b d g)}{d (c+d x)^2 (b c-a d)}}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {b \int \frac {-d f (11 c f h-3 d (f g+4 e h)) a^2-b \left (8 e (3 f g+2 e h) d^2-3 c f (3 f g+8 e h) d+c^2 f^2 h\right ) a+4 b^2 d e (6 d e g-3 c f g-2 c e h)+f \left (\left (-f h c^2-d (3 f g+4 e h) c+12 d^2 e g\right ) b^2-a d (9 d f g+8 d e h-9 c f h) b+4 a^2 d^2 f h\right ) x}{(a+b x) (c+d x) \sqrt {e+f x}}dx}{2 (b c-a d)}+\frac {\sqrt {e+f x} \left (4 a^2 d^2 f h-a b d (-9 c f h+8 d e h+9 d f g)+b^2 \left (c^2 (-f) h-c d (4 e h+3 f g)+12 d^2 e g\right )\right )}{(c+d x) (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} (d e-c f) (-2 a d h-b c h+3 b d g)}{d (c+d x)^2 (b c-a d)}}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 174

\(\displaystyle \frac {-\frac {\frac {b \left (\frac {4 d (b e-a f) \left (a^2 d f h-a b (-5 c f h+4 d e h+3 d f g)+b^2 (-2 c e h-3 c f g+6 d e g)\right ) \int \frac {1}{(a+b x) \sqrt {e+f x}}dx}{b c-a d}-\frac {\left (3 a^2 d^2 f (-5 c f h+4 d e h+d f g)-2 a b d \left (5 c^2 f^2 h-c d f (16 e h+9 f g)+4 d^2 e (2 e h+3 f g)\right )+b^2 \left (c^3 f^2 h+c^2 d f (4 e h+3 f g)-8 c d^2 e (e h+3 f g)+24 d^3 e^2 g\right )\right ) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{b c-a d}\right )}{2 (b c-a d)}+\frac {\sqrt {e+f x} \left (4 a^2 d^2 f h-a b d (-9 c f h+8 d e h+9 d f g)+b^2 \left (c^2 (-f) h-c d (4 e h+3 f g)+12 d^2 e g\right )\right )}{(c+d x) (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} (d e-c f) (-2 a d h-b c h+3 b d g)}{d (c+d x)^2 (b c-a d)}}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {-\frac {\frac {b \left (\frac {8 d (b e-a f) \left (a^2 d f h-a b (-5 c f h+4 d e h+3 d f g)+b^2 (-2 c e h-3 c f g+6 d e g)\right ) \int \frac {1}{a+\frac {b (e+f x)}{f}-\frac {b e}{f}}d\sqrt {e+f x}}{f (b c-a d)}-\frac {2 \left (3 a^2 d^2 f (-5 c f h+4 d e h+d f g)-2 a b d \left (5 c^2 f^2 h-c d f (16 e h+9 f g)+4 d^2 e (2 e h+3 f g)\right )+b^2 \left (c^3 f^2 h+c^2 d f (4 e h+3 f g)-8 c d^2 e (e h+3 f g)+24 d^3 e^2 g\right )\right ) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{f (b c-a d)}\right )}{2 (b c-a d)}+\frac {\sqrt {e+f x} \left (4 a^2 d^2 f h-a b d (-9 c f h+8 d e h+9 d f g)+b^2 \left (c^2 (-f) h-c d (4 e h+3 f g)+12 d^2 e g\right )\right )}{(c+d x) (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} (d e-c f) (-2 a d h-b c h+3 b d g)}{d (c+d x)^2 (b c-a d)}}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {\frac {b \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right ) \left (3 a^2 d^2 f (-5 c f h+4 d e h+d f g)-2 a b d \left (5 c^2 f^2 h-c d f (16 e h+9 f g)+4 d^2 e (2 e h+3 f g)\right )+b^2 \left (c^3 f^2 h+c^2 d f (4 e h+3 f g)-8 c d^2 e (e h+3 f g)+24 d^3 e^2 g\right )\right )}{\sqrt {d} (b c-a d) \sqrt {d e-c f}}-\frac {8 d \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e+f x}}{\sqrt {b e-a f}}\right ) \left (a^2 d f h-a b (-5 c f h+4 d e h+3 d f g)+b^2 (-2 c e h-3 c f g+6 d e g)\right )}{\sqrt {b} (b c-a d)}\right )}{2 (b c-a d)}+\frac {\sqrt {e+f x} \left (4 a^2 d^2 f h-a b d (-9 c f h+8 d e h+9 d f g)+b^2 \left (c^2 (-f) h-c d (4 e h+3 f g)+12 d^2 e g\right )\right )}{(c+d x) (b c-a d)}}{2 d (b c-a d)}-\frac {\sqrt {e+f x} (d e-c f) (-2 a d h-b c h+3 b d g)}{d (c+d x)^2 (b c-a d)}}{2 b (b c-a d)}-\frac {(e+f x)^{3/2} (b g-a h)}{b (a+b x) (c+d x)^2 (b c-a d)}\)

Input:

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

Output:

-(((b*g - a*h)*(e + f*x)^(3/2))/(b*(b*c - a*d)*(a + b*x)*(c + d*x)^2)) + ( 
-(((d*e - c*f)*(3*b*d*g - b*c*h - 2*a*d*h)*Sqrt[e + f*x])/(d*(b*c - a*d)*( 
c + d*x)^2)) - (((4*a^2*d^2*f*h - a*b*d*(9*d*f*g + 8*d*e*h - 9*c*f*h) + b^ 
2*(12*d^2*e*g - c^2*f*h - c*d*(3*f*g + 4*e*h)))*Sqrt[e + f*x])/((b*c - a*d 
)*(c + d*x)) + (b*((-8*d*Sqrt[b*e - a*f]*(a^2*d*f*h + b^2*(6*d*e*g - 3*c*f 
*g - 2*c*e*h) - a*b*(3*d*f*g + 4*d*e*h - 5*c*f*h))*ArcTanh[(Sqrt[b]*Sqrt[e 
 + f*x])/Sqrt[b*e - a*f]])/(Sqrt[b]*(b*c - a*d)) + (2*(3*a^2*d^2*f*(d*f*g 
+ 4*d*e*h - 5*c*f*h) + b^2*(24*d^3*e^2*g + c^3*f^2*h - 8*c*d^2*e*(3*f*g + 
e*h) + c^2*d*f*(3*f*g + 4*e*h)) - 2*a*b*d*(5*c^2*f^2*h + 4*d^2*e*(3*f*g + 
2*e*h) - c*d*f*(9*f*g + 16*e*h)))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e 
 - c*f]])/(Sqrt[d]*(b*c - a*d)*Sqrt[d*e - c*f])))/(2*(b*c - a*d)))/(2*d*(b 
*c - a*d)))/(2*b*(b*c - a*d))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - 
a*f)*(m + 1))   Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* 
c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h 
)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 168

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

rule 174

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* 
((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d)   Int[(e + f*x)^ 
p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d)   Int[(e + f*x)^p/(c + d 
*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
 

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 = 1.17 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.14

method	result	size
pseudoelliptic	\(\frac {\frac {\left (11 \left (a d -b c \right ) \sqrt {\left (c f -d e \right ) d}\, \left (\left (\frac {\left (c \left (-x d +c \right ) h -5 \left (\frac {3 x d}{5}+c \right ) d g \right ) x c \,b^{2}}{11}+\frac {a \left (c \left (3 x d +c \right )^{2} h -9 d g \left (d^{2} x^{2}+\frac {14}{9} c d x +c^{2}\right )\right ) b}{11}+a^{2} d \left (\left (\frac {17}{11} c d x +\frac {4}{11} d^{2} x^{2}+c^{2}\right ) h -\frac {3 \left (\frac {5 x d}{3}+c \right ) d g}{11}\right )\right ) f -\frac {2 d e \left (\left (\left (2 c d \,x^{2}+3 c^{2} x \right ) h -2 \left (3 d^{2} x^{2}+\frac {9}{2} c d x +c^{2}\right ) g \right ) b^{2}+5 a \left (\left (\frac {4}{5} d^{2} x^{2}+\frac {7}{5} c d x +c^{2}\right ) h -\left (\frac {3 x d}{5}+c \right ) d g \right ) b +a^{2} d \left (\left (2 x d +c \right ) h +d g \right )\right )}{11}\right ) \sqrt {f x +e}-15 \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right ) \left (\left (-\frac {c^{2} \left (c h +3 d g \right ) b^{2}}{15}+\frac {2 a \left (c h -\frac {9 d g}{5}\right ) c d b}{3}+a^{2} d^{2} \left (c h -\frac {d g}{5}\right )\right ) f^{2}-\frac {4 \left (\left (\frac {1}{3} h \,c^{2}-2 c d g \right ) b^{2}+2 a \left (\frac {4}{3} c d h -d^{2} g \right ) b +a^{2} d^{2} h \right ) d e f}{5}+\frac {16 d^{2} \left (\frac {\left (c h -3 d g \right ) b}{2}+a d h \right ) b \,e^{2}}{15}\right ) \left (x d +c \right )^{2} \left (b x +a \right )\right ) \sqrt {\left (a f -b e \right ) b}}{4}+\arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right ) \left (\left (-3 b^{2} c g +a \left (5 c h -3 d g \right ) b +a^{2} d h \right ) f -4 \left (\frac {\left (c h -3 d g \right ) b}{2}+a d h \right ) b e \right ) d \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right )^{2} \left (b x +a \right ) \left (a f -b e \right )}{\sqrt {\left (a f -b e \right ) b}\, \sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right )^{2} \left (a d -b c \right )^{4} \left (b x +a \right ) d}\)	\(560\)
derivativedivides	\(2 f^{3} \left (-\frac {\frac {\left (-\frac {9}{8} a^{2} c \,d^{2} f^{2} h +\frac {1}{2} a^{2} d^{3} e f h +\frac {5}{8} a^{2} d^{3} f^{2} g +\frac {5}{4} a b \,c^{2} d \,f^{2} h -\frac {1}{4} a b c \,d^{2} f^{2} g -a b \,d^{3} e f g -\frac {1}{8} b^{2} c^{3} f^{2} h -\frac {1}{2} b^{2} c^{2} d e f h -\frac {3}{8} b^{2} c^{2} d \,f^{2} g +b^{2} c \,d^{2} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (7 a^{2} c^{2} d^{2} f^{2} h -11 a^{2} c \,d^{3} e f h -3 a^{2} c \,d^{3} f^{2} g +4 a^{2} d^{4} e^{2} h +3 a^{2} d^{4} e f g -6 a b \,c^{3} d \,f^{2} h +6 a b \,c^{2} d^{2} e f h -2 a b \,c^{2} d^{2} f^{2} g +10 a b c \,d^{3} e f g -8 a b \,d^{4} e^{2} g -b^{2} c^{4} f^{2} h +5 b^{2} c^{3} d e f h +5 b^{2} c^{3} d \,f^{2} g -4 b^{2} c^{2} d^{2} e^{2} h -13 b^{2} c^{2} d^{2} e f g +8 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8 d}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a^{2} c \,d^{2} f^{2} h -12 a^{2} d^{3} e f h -3 a^{2} d^{3} f^{2} g +10 a b \,c^{2} d \,f^{2} h -32 a b c \,d^{2} e f h -18 a b c \,d^{2} f^{2} g +16 e^{2} h b a \,d^{3}+24 a b \,d^{3} e f g -b^{2} c^{3} f^{2} h -4 b^{2} c^{2} d e f h -3 b^{2} c^{2} d \,f^{2} g +8 b^{2} c \,d^{2} e^{2} h +24 b^{2} c \,d^{2} e f g -24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 d \sqrt {\left (c f -d e \right ) d}}}{f^{3} \left (a d -b c \right )^{4}}+\frac {\left (a f -b e \right ) \left (\frac {\left (\frac {1}{2} a^{2} d f h -\frac {1}{2} a b c f h -\frac {1}{2} a b d f g +\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (a^{2} d f h +5 a b c f h -4 a b d e h -3 a b d f g -2 b^{2} c e h -3 b^{2} c f g +6 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{4} f^{3}}\right )\)	\(760\)
default	\(2 f^{3} \left (-\frac {\frac {\left (-\frac {9}{8} a^{2} c \,d^{2} f^{2} h +\frac {1}{2} a^{2} d^{3} e f h +\frac {5}{8} a^{2} d^{3} f^{2} g +\frac {5}{4} a b \,c^{2} d \,f^{2} h -\frac {1}{4} a b c \,d^{2} f^{2} g -a b \,d^{3} e f g -\frac {1}{8} b^{2} c^{3} f^{2} h -\frac {1}{2} b^{2} c^{2} d e f h -\frac {3}{8} b^{2} c^{2} d \,f^{2} g +b^{2} c \,d^{2} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (7 a^{2} c^{2} d^{2} f^{2} h -11 a^{2} c \,d^{3} e f h -3 a^{2} c \,d^{3} f^{2} g +4 a^{2} d^{4} e^{2} h +3 a^{2} d^{4} e f g -6 a b \,c^{3} d \,f^{2} h +6 a b \,c^{2} d^{2} e f h -2 a b \,c^{2} d^{2} f^{2} g +10 a b c \,d^{3} e f g -8 a b \,d^{4} e^{2} g -b^{2} c^{4} f^{2} h +5 b^{2} c^{3} d e f h +5 b^{2} c^{3} d \,f^{2} g -4 b^{2} c^{2} d^{2} e^{2} h -13 b^{2} c^{2} d^{2} e f g +8 b^{2} c \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8 d}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a^{2} c \,d^{2} f^{2} h -12 a^{2} d^{3} e f h -3 a^{2} d^{3} f^{2} g +10 a b \,c^{2} d \,f^{2} h -32 a b c \,d^{2} e f h -18 a b c \,d^{2} f^{2} g +16 e^{2} h b a \,d^{3}+24 a b \,d^{3} e f g -b^{2} c^{3} f^{2} h -4 b^{2} c^{2} d e f h -3 b^{2} c^{2} d \,f^{2} g +8 b^{2} c \,d^{2} e^{2} h +24 b^{2} c \,d^{2} e f g -24 b^{2} d^{3} e^{2} g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 d \sqrt {\left (c f -d e \right ) d}}}{f^{3} \left (a d -b c \right )^{4}}+\frac {\left (a f -b e \right ) \left (\frac {\left (\frac {1}{2} a^{2} d f h -\frac {1}{2} a b c f h -\frac {1}{2} a b d f g +\frac {1}{2} b^{2} c f g \right ) \sqrt {f x +e}}{\left (f x +e \right ) b +a f -b e}+\frac {\left (a^{2} d f h +5 a b c f h -4 a b d e h -3 a b d f g -2 b^{2} c e h -3 b^{2} c f g +6 b^{2} d e g \right ) \arctan \left (\frac {b \sqrt {f x +e}}{\sqrt {\left (a f -b e \right ) b}}\right )}{2 \sqrt {\left (a f -b e \right ) b}}\right )}{\left (a d -b c \right )^{4} f^{3}}\right )\)	\(760\)

Input:

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

Output:

1/((a*f-b*e)*b)^(1/2)/((c*f-d*e)*d)^(1/2)*(1/4*(11*(a*d-b*c)*((c*f-d*e)*d) 
^(1/2)*((1/11*(c*(-d*x+c)*h-5*(3/5*x*d+c)*d*g)*x*c*b^2+1/11*a*(c*(3*d*x+c) 
^2*h-9*d*g*(d^2*x^2+14/9*c*d*x+c^2))*b+a^2*d*((17/11*c*d*x+4/11*d^2*x^2+c^ 
2)*h-3/11*(5/3*x*d+c)*d*g))*f-2/11*d*e*(((2*c*d*x^2+3*c^2*x)*h-2*(3*d^2*x^ 
2+9/2*c*d*x+c^2)*g)*b^2+5*a*((4/5*d^2*x^2+7/5*c*d*x+c^2)*h-(3/5*x*d+c)*d*g 
)*b+a^2*d*((2*d*x+c)*h+d*g)))*(f*x+e)^(1/2)-15*arctan(d*(f*x+e)^(1/2)/((c* 
f-d*e)*d)^(1/2))*((-1/15*c^2*(c*h+3*d*g)*b^2+2/3*a*(c*h-9/5*d*g)*c*d*b+a^2 
*d^2*(c*h-1/5*d*g))*f^2-4/5*((1/3*h*c^2-2*c*d*g)*b^2+2*a*(4/3*c*d*h-d^2*g) 
*b+a^2*d^2*h)*d*e*f+16/15*d^2*(1/2*(c*h-3*d*g)*b+a*d*h)*b*e^2)*(d*x+c)^2*( 
b*x+a))*((a*f-b*e)*b)^(1/2)+arctan(b*(f*x+e)^(1/2)/((a*f-b*e)*b)^(1/2))*(( 
-3*b^2*c*g+a*(5*c*h-3*d*g)*b+a^2*d*h)*f-4*(1/2*(c*h-3*d*g)*b+a*d*h)*b*e)*d 
*((c*f-d*e)*d)^(1/2)*(d*x+c)^2*(b*x+a)*(a*f-b*e))/(d*x+c)^2/(a*d-b*c)^4/(b 
*x+a)/d
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3105 vs. \(2 (459) = 918\).

Time = 23.30 (sec) , antiderivative size = 12465, normalized size of antiderivative = 25.39 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2/(d*x+c)^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. 959 vs. \(2 (459) = 918\).

Time = 0.19 (sec) , antiderivative size = 959, normalized size of antiderivative = 1.95 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

-(6*b^3*d*e^2*g - 3*b^3*c*e*f*g - 9*a*b^2*d*e*f*g + 3*a*b^2*c*f^2*g + 3*a^ 
2*b*d*f^2*g - 2*b^3*c*e^2*h - 4*a*b^2*d*e^2*h + 7*a*b^2*c*e*f*h + 5*a^2*b* 
d*e*f*h - 5*a^2*b*c*f^2*h - a^3*d*f^2*h)*arctan(sqrt(f*x + e)*b/sqrt(-b^2* 
e + a*b*f))/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 
+ a^4*d^4)*sqrt(-b^2*e + a*b*f)) + 1/4*(24*b^2*d^3*e^2*g - 24*b^2*c*d^2*e* 
f*g - 24*a*b*d^3*e*f*g + 3*b^2*c^2*d*f^2*g + 18*a*b*c*d^2*f^2*g + 3*a^2*d^ 
3*f^2*g - 8*b^2*c*d^2*e^2*h - 16*a*b*d^3*e^2*h + 4*b^2*c^2*d*e*f*h + 32*a* 
b*c*d^2*e*f*h + 12*a^2*d^3*e*f*h + b^2*c^3*f^2*h - 10*a*b*c^2*d*f^2*h - 15 
*a^2*c*d^2*f^2*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((b^4*c^4*d 
 - 4*a*b^3*c^3*d^2 + 6*a^2*b^2*c^2*d^3 - 4*a^3*b*c*d^4 + a^4*d^5)*sqrt(-d^ 
2*e + c*d*f)) - (sqrt(f*x + e)*b^2*e*f*g - sqrt(f*x + e)*a*b*f^2*g - sqrt( 
f*x + e)*a*b*e*f*h + sqrt(f*x + e)*a^2*f^2*h)/((b^3*c^3 - 3*a*b^2*c^2*d + 
3*a^2*b*c*d^2 - a^3*d^3)*((f*x + e)*b - b*e + a*f)) - 1/4*(8*(f*x + e)^(3/ 
2)*b*d^3*e*f*g - 8*sqrt(f*x + e)*b*d^3*e^2*f*g - 3*(f*x + e)^(3/2)*b*c*d^2 
*f^2*g - 5*(f*x + e)^(3/2)*a*d^3*f^2*g + 13*sqrt(f*x + e)*b*c*d^2*e*f^2*g 
+ 3*sqrt(f*x + e)*a*d^3*e*f^2*g - 5*sqrt(f*x + e)*b*c^2*d*f^3*g - 3*sqrt(f 
*x + e)*a*c*d^2*f^3*g - 4*(f*x + e)^(3/2)*b*c*d^2*e*f*h - 4*(f*x + e)^(3/2 
)*a*d^3*e*f*h + 4*sqrt(f*x + e)*b*c*d^2*e^2*f*h + 4*sqrt(f*x + e)*a*d^3*e^ 
2*f*h - (f*x + e)^(3/2)*b*c^2*d*f^2*h + 9*(f*x + e)^(3/2)*a*c*d^2*f^2*h - 
5*sqrt(f*x + e)*b*c^2*d*e*f^2*h - 11*sqrt(f*x + e)*a*c*d^2*e*f^2*h + sq...
 

Mupad [B] (verification not implemented)

Time = 116.80 (sec) , antiderivative size = 97984, normalized size of antiderivative = 199.56 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(log((((((b^2*d*f^3*(a*f - b*e)*(8*a*d^2*e*h + 3*a*d^2*f*g - 12*b*d^2*e*g 
- b*c^2*f*h - 11*a*c*d*f*h + 4*b*c*d*e*h + 9*b*c*d*f*g))/(a*d - b*c) - (b^ 
2*d^2*f^2*(e + f*x)^(1/2)*(a*d - b*c)^2*(a*d*f + b*c*f - 2*b*d*e)*(-(2*(f^ 
6*(a*d - b*c)^24*(16*a*d^2*e*h^2 + b*c^2*f*h^2 + 9*b*d^2*f*g^2 - 16*a*c*d* 
f*h^2 + 8*b*c*d*e*h^2 - 24*b*d^2*e*g*h + 6*b*c*d*f*g*h)^2)^(1/2) + 2*b^13* 
c^14*f^4*h^2 + 2304*a^8*b^5*d^14*e^4*g^2 + 1024*a^10*b^3*d^14*e^4*h^2 + 23 
04*b^13*c^8*d^6*e^4*g^2 + 256*b^13*c^10*d^4*e^4*h^2 + 18*b^13*c^12*d^2*f^4 
*g^2 + 18*a^12*b*d^14*f^4*g^2 + 32*a^13*c*d^13*f^4*h^2 - 32*a^13*d^14*e*f^ 
3*h^2 - 56*a*b^12*c^13*d*f^4*h^2 - 3072*a^9*b^4*d^14*e^4*g*h + 16*b^13*c^1 
3*d*e*f^3*h^2 - 1536*b^13*c^9*d^5*e^4*g*h - 18432*a*b^12*c^7*d^7*e^4*g^2 - 
 18432*a^7*b^6*c*d^13*e^4*g^2 - 1024*a*b^12*c^9*d^5*e^4*h^2 + 360*a*b^12*c 
^11*d^3*f^4*g^2 - 7168*a^9*b^4*c*d^13*e^4*h^2 + 360*a^11*b^2*c*d^13*f^4*g^ 
2 + 514*a^12*b*c^2*d^12*f^4*h^2 - 4608*a^9*b^4*d^14*e^3*f*g^2 - 576*a^11*b 
^2*d^14*e*f^3*g^2 - 1536*a^11*b^2*d^14*e^3*f*h^2 + 576*a^12*b*d^14*e^2*f^2 
*h^2 - 4608*b^13*c^9*d^5*e^3*f*g^2 - 576*b^13*c^11*d^3*e*f^3*g^2 - 256*b^1 
3*c^11*d^3*e^3*f*h^2 + 12*b^13*c^13*d*f^4*g*h + 64512*a^2*b^11*c^6*d^8*e^4 
*g^2 - 129024*a^3*b^10*c^5*d^9*e^4*g^2 + 161280*a^4*b^9*c^4*d^10*e^4*g^2 - 
 129024*a^5*b^8*c^3*d^11*e^4*g^2 + 64512*a^6*b^7*c^2*d^12*e^4*g^2 - 2268*a 
^2*b^11*c^10*d^4*f^4*g^2 + 6144*a^3*b^10*c^7*d^7*e^4*h^2 + 3528*a^3*b^10*c 
^9*d^5*f^4*g^2 - 10752*a^4*b^9*c^6*d^8*e^4*h^2 + 4302*a^4*b^9*c^8*d^6*f...
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 8595, normalized size of antiderivative = 17.51 \[ \int \frac {(e+f x)^{3/2} (g+h x)}{(a+b x)^2 (c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e) 
))*a**3*c**3*d**3*f**2*h - 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b 
)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*c**2*d**4*e*f*h + 8*sqrt(b)*sqrt(a*f - b 
*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*c**2*d**4*f**2* 
h*x - 8*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - 
 b*e)))*a**3*c*d**5*e*f*h*x + 4*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x 
)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*c*d**5*f**2*h*x**2 - 4*sqrt(b)*sqrt(a 
*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**3*d**6*e*f* 
h*x**2 + 20*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a 
*f - b*e)))*a**2*b*c**4*d**2*f**2*h - 36*sqrt(b)*sqrt(a*f - b*e)*atan((sqr 
t(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c**3*d**3*e*f*h - 12*sqrt( 
b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2* 
b*c**3*d**3*f**2*g + 44*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sq 
rt(b)*sqrt(a*f - b*e)))*a**2*b*c**3*d**3*f**2*h*x + 16*sqrt(b)*sqrt(a*f - 
b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c**2*d**4*e* 
*2*h + 12*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)/(sqrt(b)*sqrt(a*f 
 - b*e)))*a**2*b*c**2*d**4*e*f*g - 76*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e 
 + f*x)*b)/(sqrt(b)*sqrt(a*f - b*e)))*a**2*b*c**2*d**4*e*f*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 
*c**2*d**4*f**2*g*x + 28*sqrt(b)*sqrt(a*f - b*e)*atan((sqrt(e + f*x)*b)...