\(\int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx\) [179]

Optimal result

Integrand size = 27, antiderivative size = 181 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {2 (b e-a f) (f g-e h)}{f (d e-c f)^2 \sqrt {e+f x}}+\frac {(b c-a d) (d g-c h) \sqrt {e+f x}}{d (d e-c f)^2 (c+d x)}+\frac {\left (a d (3 d f g-2 d e h-c f h)-b \left (2 d^2 e g+c^2 f h+c d (f g-4 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2} (d e-c f)^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.65 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {-b \left (2 d^2 e (-f g+e h) x+c^2 f h (e+f x)+c d \left (-3 e f g+2 e^2 h-f^2 g x\right )\right )+a d f (-d (e g+3 f g x-2 e h x)+c (-2 f g+3 e h+f h x))}{d f (d e-c f)^2 (c+d x) \sqrt {e+f x}}+\frac {\left (a d (-3 d f g+2 d e h+c f h)+b \left (2 d^2 e g+c^2 f h+c d (f g-4 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{d^{3/2} (-d e+c f)^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.25, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {161, 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.

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

\(\Big \downarrow \) 161

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 = 209, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-a f h e +g \,f^{2} a +b \,e^{2} h -b f g e \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {2 f \left (\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 d \left (\left (f x +e \right ) d +c f -d e \right )}+\frac {\left (a c d f h +2 a \,d^{2} e h -3 a \,d^{2} f g +b \,c^{2} f h -4 b c d e h +b c d f g +2 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 d \sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2}}}{f}\)	\(209\)
default	\(\frac {-\frac {2 \left (-a f h e +g \,f^{2} a +b \,e^{2} h -b f g e \right )}{\left (c f -d e \right )^{2} \sqrt {f x +e}}+\frac {2 f \left (\frac {f \left (a c d h -a \,d^{2} g -b \,c^{2} h +b c d g \right ) \sqrt {f x +e}}{2 d \left (\left (f x +e \right ) d +c f -d e \right )}+\frac {\left (a c d f h +2 a \,d^{2} e h -3 a \,d^{2} f g +b \,c^{2} f h -4 b c d e h +b c d f g +2 b \,d^{2} e g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{2 d \sqrt {\left (c f -d e \right ) d}}\right )}{\left (c f -d e \right )^{2}}}{f}\)	\(209\)
pseudoelliptic	\(\frac {\left (\left (-3 a f g +2 \left (a h +b g \right ) e \right ) d^{2}+c \left (\left (a h +b g \right ) f -4 e h b \right ) d +b \,c^{2} f h \right ) \left (x d +c \right ) \sqrt {f x +e}\, f \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )+3 \left (\left (-a \,f^{2} g x -\frac {\left (-2 x g b +a \left (-2 h x +g \right )\right ) e f}{3}-\frac {2 b \,e^{2} h x}{3}\right ) d^{2}+c \left (\frac {\left (x g b -2 a \left (-\frac {h x}{2}+g \right )\right ) f^{2}}{3}+e \left (a h +b g \right ) f -\frac {2 b \,e^{2} h}{3}\right ) d -\frac {b \,c^{2} f h \left (f x +e \right )}{3}\right ) \sqrt {\left (c f -d e \right ) d}}{\sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}\, f \left (c f -d e \right )^{2} \left (x d +c \right ) d}\)	\(234\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 715 vs. \(2 (167) = 334\).

Time = 0.14 (sec) , antiderivative size = 1445, normalized size of antiderivative = 7.98 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/2*(sqrt(d^2*e - c*d*f)*(((2*b*d^3*e*f^2 + (b*c*d^2 - 3*a*d^3)*f^3)*g - 
(2*(2*b*c*d^2 - a*d^3)*e*f^2 - (b*c^2*d + a*c*d^2)*f^3)*h)*x^2 + (2*b*c*d^ 
2*e^2*f + (b*c^2*d - 3*a*c*d^2)*e*f^2)*g - (2*(2*b*c^2*d - a*c*d^2)*e^2*f 
- (b*c^3 + a*c^2*d)*e*f^2)*h + ((2*b*d^3*e^2*f + 3*(b*c*d^2 - a*d^3)*e*f^2 
 + (b*c^2*d - 3*a*c*d^2)*f^3)*g - (2*(2*b*c*d^2 - a*d^3)*e^2*f + 3*(b*c^2* 
d - a*c*d^2)*e*f^2 - (b*c^3 + a*c^2*d)*f^3)*h)*x)*log((d*f*x + 2*d*e - c*f 
 - 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/(d*x + c)) + 2*((2*a*c^2*d^2*f^3 + 
 (3*b*c*d^3 - a*d^4)*e^2*f - (3*b*c^2*d^2 + a*c*d^3)*e*f^2)*g - (2*b*c*d^3 
*e^3 - (b*c^2*d^2 + 3*a*c*d^3)*e^2*f - (b*c^3*d - 3*a*c^2*d^2)*e*f^2)*h + 
((2*b*d^4*e^2*f - (b*c*d^3 + 3*a*d^4)*e*f^2 - (b*c^2*d^2 - 3*a*c*d^3)*f^3) 
*g - (2*b*d^4*e^3 - 2*(b*c*d^3 + a*d^4)*e^2*f + (b*c^2*d^2 + a*c*d^3)*e*f^ 
2 - (b*c^3*d - a*c^2*d^2)*f^3)*h)*x)*sqrt(f*x + e))/(c*d^5*e^4*f - 3*c^2*d 
^4*e^3*f^2 + 3*c^3*d^3*e^2*f^3 - c^4*d^2*e*f^4 + (d^6*e^3*f^2 - 3*c*d^5*e^ 
2*f^3 + 3*c^2*d^4*e*f^4 - c^3*d^3*f^5)*x^2 + (d^6*e^4*f - 2*c*d^5*e^3*f^2 
+ 2*c^3*d^3*e*f^4 - c^4*d^2*f^5)*x), (sqrt(-d^2*e + c*d*f)*(((2*b*d^3*e*f^ 
2 + (b*c*d^2 - 3*a*d^3)*f^3)*g - (2*(2*b*c*d^2 - a*d^3)*e*f^2 - (b*c^2*d + 
 a*c*d^2)*f^3)*h)*x^2 + (2*b*c*d^2*e^2*f + (b*c^2*d - 3*a*c*d^2)*e*f^2)*g 
- (2*(2*b*c^2*d - a*c*d^2)*e^2*f - (b*c^3 + a*c^2*d)*e*f^2)*h + ((2*b*d^3* 
e^2*f + 3*(b*c*d^2 - a*d^3)*e*f^2 + (b*c^2*d - 3*a*c*d^2)*f^3)*g - (2*(2*b 
*c*d^2 - a*d^3)*e^2*f + 3*(b*c^2*d - a*c*d^2)*e*f^2 - (b*c^3 + a*c^2*d)...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)*(h*x+g)/(d*x+c)^2/(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(c*f-d*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. 360 vs. \(2 (167) = 334\).

Time = 0.13 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {{\left (2 \, b d^{2} e g + b c d f g - 3 \, a d^{2} f g - 4 \, b c d e h + 2 \, a d^{2} e h + b c^{2} f h + a c d f h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{{\left (d^{3} e^{2} - 2 \, c d^{2} e f + c^{2} d f^{2}\right )} \sqrt {-d^{2} e + c d f}} + \frac {2 \, {\left (f x + e\right )} b d^{2} e f g - 2 \, b d^{2} e^{2} f g + {\left (f x + e\right )} b c d f^{2} g - 3 \, {\left (f x + e\right )} a d^{2} f^{2} g + 2 \, b c d e f^{2} g + 2 \, a d^{2} e f^{2} g - 2 \, a c d f^{3} g - 2 \, {\left (f x + e\right )} b d^{2} e^{2} h + 2 \, b d^{2} e^{3} h + 2 \, {\left (f x + e\right )} a d^{2} e f h - 2 \, b c d e^{2} f h - 2 \, a d^{2} e^{2} f h - {\left (f x + e\right )} b c^{2} f^{2} h + {\left (f x + e\right )} a c d f^{2} h + 2 \, a c d e f^{2} h}{{\left (d^{3} e^{2} f - 2 \, c d^{2} e f^{2} + c^{2} d f^{3}\right )} {\left ({\left (f x + e\right )}^{\frac {3}{2}} d - \sqrt {f x + e} d e + \sqrt {f x + e} c f\right )}} \] Input:

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

Output:

(2*b*d^2*e*g + b*c*d*f*g - 3*a*d^2*f*g - 4*b*c*d*e*h + 2*a*d^2*e*h + b*c^2 
*f*h + a*c*d*f*h)*arctan(sqrt(f*x + e)*d/sqrt(-d^2*e + c*d*f))/((d^3*e^2 - 
 2*c*d^2*e*f + c^2*d*f^2)*sqrt(-d^2*e + c*d*f)) + (2*(f*x + e)*b*d^2*e*f*g 
 - 2*b*d^2*e^2*f*g + (f*x + e)*b*c*d*f^2*g - 3*(f*x + e)*a*d^2*f^2*g + 2*b 
*c*d*e*f^2*g + 2*a*d^2*e*f^2*g - 2*a*c*d*f^3*g - 2*(f*x + e)*b*d^2*e^2*h + 
 2*b*d^2*e^3*h + 2*(f*x + e)*a*d^2*e*f*h - 2*b*c*d*e^2*f*h - 2*a*d^2*e^2*f 
*h - (f*x + e)*b*c^2*f^2*h + (f*x + e)*a*c*d*f^2*h + 2*a*c*d*e*f^2*h)/((d^ 
3*e^2*f - 2*c*d^2*e*f^2 + c^2*d*f^3)*((f*x + e)^(3/2)*d - sqrt(f*x + e)*d* 
e + sqrt(f*x + e)*c*f))
 

Mupad [B] (verification not implemented)

Time = 2.73 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.49 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx=\frac {\mathrm {atan}\left (\frac {\sqrt {e+f\,x}\,\left (c^2\,d\,f^2-2\,c\,d^2\,e\,f+d^3\,e^2\right )}{\sqrt {d}\,{\left (c\,f-d\,e\right )}^{5/2}}\right )\,\left (2\,a\,d^2\,e\,h-3\,a\,d^2\,f\,g+2\,b\,d^2\,e\,g+b\,c^2\,f\,h+a\,c\,d\,f\,h-4\,b\,c\,d\,e\,h+b\,c\,d\,f\,g\right )}{d^{3/2}\,{\left (c\,f-d\,e\right )}^{5/2}}-\frac {\frac {2\,\left (a\,f^2\,g+b\,e^2\,h-a\,e\,f\,h-b\,e\,f\,g\right )}{c\,f-d\,e}-\frac {\left (e+f\,x\right )\,\left (a\,c\,d\,f^2\,h-b\,c^2\,f^2\,h-2\,b\,d^2\,e^2\,h-3\,a\,d^2\,f^2\,g+b\,c\,d\,f^2\,g+2\,a\,d^2\,e\,f\,h+2\,b\,d^2\,e\,f\,g\right )}{d\,{\left (c\,f-d\,e\right )}^2}}{\sqrt {e+f\,x}\,\left (c\,f^2-d\,e\,f\right )+d\,f\,{\left (e+f\,x\right )}^{3/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 1089, normalized size of antiderivative = 6.02 \[ \int \frac {(a+b x) (g+h x)}{(c+d x)^2 (e+f x)^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqr 
t(c*f - d*e)))*a*c**2*d*f**2*h + 2*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*a 
tan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**2*e*f*h - 3*sqrt(d 
)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
 d*e)))*a*c*d**2*f**2*g + sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt 
(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**2*f**2*h*x + 2*sqrt(d)*sqrt 
(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)) 
)*a*d**3*e*f*h*x - 3*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + 
f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*d**3*f**2*g*x + sqrt(d)*sqrt(e + f*x) 
*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**3* 
f**2*h - 4*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(s 
qrt(d)*sqrt(c*f - d*e)))*b*c**2*d*e*f*h + sqrt(d)*sqrt(e + f*x)*sqrt(c*f - 
 d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**2*d*f**2*g + 
sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt 
(c*f - d*e)))*b*c**2*d*f**2*h*x + 2*sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)* 
atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c*d**2*e*f*g - 4*sqrt( 
d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f 
- d*e)))*b*c*d**2*e*f*h*x + sqrt(d)*sqrt(e + f*x)*sqrt(c*f - d*e)*atan((sq 
rt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c*d**2*f**2*g*x + 2*sqrt(d)*sq 
rt(e + f*x)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - ...