\(\int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx\) [32]

Optimal result

Integrand size = 25, antiderivative size = 205 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\frac {(d e-c f) \sqrt {a+b x}}{2 c d (c+d x)^2}-\frac {\left (4 a d^2 e-b c (3 d e+c f)\right ) \sqrt {a+b x}}{4 c^2 d (b c-a d) (c+d x)}-\frac {\left (12 a b c d^2 e-8 a^2 d^3 e-b^2 c^2 (3 d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{4 c^3 d^{3/2} (b c-a d)^{3/2}}-\frac {2 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{c^3} \] Output:

1/2*(-c*f+d*e)*(b*x+a)^(1/2)/c/d/(d*x+c)^2-1/4*(4*a*d^2*e-b*c*(c*f+3*d*e)) 
*(b*x+a)^(1/2)/c^2/d/(-a*d+b*c)/(d*x+c)-1/4*(12*a*b*c*d^2*e-8*a^2*d^3*e-b^ 
2*c^2*(c*f+3*d*e))*arctan(d^(1/2)*(b*x+a)^(1/2)/(-a*d+b*c)^(1/2))/c^3/d^(3 
/2)/(-a*d+b*c)^(3/2)-2*a^(1/2)*e*arctanh((b*x+a)^(1/2)/a^(1/2))/c^3
 

Mathematica [A] (verified)

Time = 0.84 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\frac {\frac {c \sqrt {a+b x} \left (2 a d \left (3 c d e-c^2 f+2 d^2 e x\right )+b c \left (c^2 f-3 d^2 e x-c d (5 e+f x)\right )\right )}{d (-b c+a d) (c+d x)^2}+\frac {\left (-12 a b c d^2 e+8 a^2 d^3 e+b^2 c^2 (3 d e+c f)\right ) \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{d^{3/2} (b c-a d)^{3/2}}-8 \sqrt {a} e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 c^3} \] Input:

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

Output:

((c*Sqrt[a + b*x]*(2*a*d*(3*c*d*e - c^2*f + 2*d^2*e*x) + b*c*(c^2*f - 3*d^ 
2*e*x - c*d*(5*e + f*x))))/(d*(-(b*c) + a*d)*(c + d*x)^2) + ((-12*a*b*c*d^ 
2*e + 8*a^2*d^3*e + b^2*c^2*(3*d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/ 
Sqrt[b*c - a*d]])/(d^(3/2)*(b*c - a*d)^(3/2)) - 8*Sqrt[a]*e*ArcTanh[Sqrt[a 
 + b*x]/Sqrt[a]])/(4*c^3)
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.15, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {166, 27, 168, 27, 174, 73, 218, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 168

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 174

\(\displaystyle \frac {\frac {\frac {8 a d e (b c-a d) \int \frac {1}{x \sqrt {a+b x}}dx}{c}-\frac {\left (-8 a^2 d^3 e+12 a b c d^2 e-b^2 c^2 (c f+3 d e)\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)}dx}{c}}{2 c (b c-a d)}-\frac {\sqrt {a+b x} \left (4 a d^2 e-b c (c f+3 d e)\right )}{c (c+d x) (b c-a d)}}{4 c d}+\frac {\sqrt {a+b x} (d e-c f)}{2 c d (c+d x)^2}\)

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {\frac {\frac {16 a d e (b c-a d) \int \frac {1}{\frac {a+b x}{b}-\frac {a}{b}}d\sqrt {a+b x}}{b c}-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (-8 a^2 d^3 e+12 a b c d^2 e-b^2 c^2 (c f+3 d e)\right )}{c \sqrt {d} \sqrt {b c-a d}}}{2 c (b c-a d)}-\frac {\sqrt {a+b x} \left (4 a d^2 e-b c (c f+3 d e)\right )}{c (c+d x) (b c-a d)}}{4 c d}+\frac {\sqrt {a+b x} (d e-c f)}{2 c d (c+d x)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {-\frac {2 \arctan \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right ) \left (-8 a^2 d^3 e+12 a b c d^2 e-b^2 c^2 (c f+3 d e)\right )}{c \sqrt {d} \sqrt {b c-a d}}-\frac {16 \sqrt {a} d e \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right ) (b c-a d)}{c}}{2 c (b c-a d)}-\frac {\sqrt {a+b x} \left (4 a d^2 e-b c (c f+3 d e)\right )}{c (c+d x) (b c-a d)}}{4 c d}+\frac {\sqrt {a+b x} (d e-c f)}{2 c d (c+d x)^2}\)

Input:

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

Output:

((d*e - c*f)*Sqrt[a + b*x])/(2*c*d*(c + d*x)^2) + (-(((4*a*d^2*e - b*c*(3* 
d*e + c*f))*Sqrt[a + b*x])/(c*(b*c - a*d)*(c + d*x))) + ((-2*(12*a*b*c*d^2 
*e - 8*a^2*d^3*e - b^2*c^2*(3*d*e + c*f))*ArcTan[(Sqrt[d]*Sqrt[a + b*x])/S 
qrt[b*c - a*d]])/(c*Sqrt[d]*Sqrt[b*c - a*d]) - (16*Sqrt[a]*d*(b*c - a*d)*e 
*ArcTanh[Sqrt[a + b*x]/Sqrt[a]])/c)/(2*c*(b*c - a*d)))/(4*c*d)
 

Defintions of rubi rules used

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 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

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 = 0.41 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.04

method	result	size
pseudoelliptic	\(-\frac {2 \left (-\left (x d +c \right )^{2} \left (a^{2} d^{3} e -\frac {3}{2} a b c \,d^{2} e +\frac {1}{8} b^{2} c^{3} f +\frac {3}{8} b^{2} c^{2} d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )+\left (\left (x d +c \right )^{2} \left (a^{\frac {3}{2}} d -b c \sqrt {a}\right ) d e \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {c \left (-2 a \,d^{3} e x -3 \left (-\frac {b x}{2}+a \right ) c e \,d^{2}+\left (\frac {\left (f x +5 e \right ) b}{2}+a f \right ) c^{2} d -\frac {b \,c^{3} f}{2}\right ) \sqrt {b x +a}}{4}\right ) \sqrt {\left (a d -b c \right ) d}\right )}{\sqrt {\left (a d -b c \right ) d}\, \left (a d -b c \right ) d \left (x d +c \right )^{2} c^{3}}\)	\(214\)
derivativedivides	\(2 b^{2} \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b^{2} c^{3}}+\frac {\frac {\frac {b c \left (4 a \,d^{2} e -b \,c^{2} f -3 b c d e \right ) \left (b x +a \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\left (4 a \,d^{2} e +b \,c^{2} f -5 b c d e \right ) b c \sqrt {b x +a}}{8 d}}{\left (-d \left (b x +a \right )+a d -b c \right )^{2}}+\frac {\left (8 a^{2} d^{3} e -12 a b c \,d^{2} e +b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{8 \left (a d -b c \right ) d \sqrt {\left (a d -b c \right ) d}}}{c^{3} b^{2}}\right )\)	\(221\)
default	\(2 b^{2} \left (-\frac {e \sqrt {a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{b^{2} c^{3}}+\frac {\frac {\frac {b c \left (4 a \,d^{2} e -b \,c^{2} f -3 b c d e \right ) \left (b x +a \right )^{\frac {3}{2}}}{8 a d -8 b c}-\frac {\left (4 a \,d^{2} e +b \,c^{2} f -5 b c d e \right ) b c \sqrt {b x +a}}{8 d}}{\left (-d \left (b x +a \right )+a d -b c \right )^{2}}+\frac {\left (8 a^{2} d^{3} e -12 a b c \,d^{2} e +b^{2} c^{3} f +3 b^{2} c^{2} d e \right ) \operatorname {arctanh}\left (\frac {d \sqrt {b x +a}}{\sqrt {\left (a d -b c \right ) d}}\right )}{8 \left (a d -b c \right ) d \sqrt {\left (a d -b c \right ) d}}}{c^{3} b^{2}}\right )\)	\(221\)

Input:

int((b*x+a)^(1/2)*(f*x+e)/x/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (179) = 358\).

Time = 0.52 (sec) , antiderivative size = 2205, normalized size of antiderivative = 10.76 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate((b*x+a)^(1/2)*(f*x+e)/x/(d*x+c)^3,x, algorithm="fricas")
 

Output:

[1/8*((b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12*a*b*c*d^4 + 8*a^2* 
d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 2*(b^2*c^ 
4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*sqrt(-b*c*d + 
 a*d^2)*log((b*d*x - b*c + 2*a*d + 2*sqrt(-b*c*d + a*d^2)*sqrt(b*x + a))/( 
d*x + c)) + 8*((b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*e*x^2 + 2*(b^2*c^3*d^ 
3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e*x + (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^ 
2*d^4)*e)*sqrt(a)*log((b*x - 2*sqrt(b*x + a)*sqrt(a) + 2*a)/x) + 2*((5*b^2 
*c^4*d^2 - 11*a*b*c^3*d^3 + 6*a^2*c^2*d^4)*e - (b^2*c^5*d - 3*a*b*c^4*d^2 
+ 2*a^2*c^3*d^3)*f + ((3*b^2*c^3*d^3 - 7*a*b*c^2*d^4 + 4*a^2*c*d^5)*e + (b 
^2*c^4*d^2 - a*b*c^3*d^3)*f)*x)*sqrt(b*x + a))/(b^2*c^7*d^2 - 2*a*b*c^6*d^ 
3 + a^2*c^5*d^4 + (b^2*c^5*d^4 - 2*a*b*c^4*d^5 + a^2*c^3*d^6)*x^2 + 2*(b^2 
*c^6*d^3 - 2*a*b*c^5*d^4 + a^2*c^4*d^5)*x), 1/8*(16*((b^2*c^2*d^4 - 2*a*b* 
c*d^5 + a^2*d^6)*e*x^2 + 2*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*e*x + 
 (b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*e)*sqrt(-a)*arctan(sqrt(-a)/s 
qrt(b*x + a)) + (b^2*c^5*f + (b^2*c^3*d^2*f + (3*b^2*c^2*d^3 - 12*a*b*c*d^ 
4 + 8*a^2*d^5)*e)*x^2 + (3*b^2*c^4*d - 12*a*b*c^3*d^2 + 8*a^2*c^2*d^3)*e + 
 2*(b^2*c^4*d*f + (3*b^2*c^3*d^2 - 12*a*b*c^2*d^3 + 8*a^2*c*d^4)*e)*x)*sqr 
t(-b*c*d + a*d^2)*log((b*d*x - b*c + 2*a*d + 2*sqrt(-b*c*d + a*d^2)*sqrt(b 
*x + a))/(d*x + c)) + 2*((5*b^2*c^4*d^2 - 11*a*b*c^3*d^3 + 6*a^2*c^2*d^4)* 
e - (b^2*c^5*d - 3*a*b*c^4*d^2 + 2*a^2*c^3*d^3)*f + ((3*b^2*c^3*d^3 - 7...
 

Sympy [F(-1)]

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

integrate((b*x+a)**(1/2)*(f*x+e)/x/(d*x+c)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((b*x+a)^(1/2)*(f*x+e)/x/(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*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\frac {{\left (3 \, b^{2} c^{2} d e - 12 \, a b c d^{2} e + 8 \, a^{2} d^{3} e + b^{2} c^{3} f\right )} \arctan \left (\frac {\sqrt {b x + a} d}{\sqrt {b c d - a d^{2}}}\right )}{4 \, {\left (b c^{4} d - a c^{3} d^{2}\right )} \sqrt {b c d - a d^{2}}} + \frac {2 \, a e \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} c^{3}} + \frac {5 \, \sqrt {b x + a} b^{3} c^{2} d e + 3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} c d^{2} e - 9 \, \sqrt {b x + a} a b^{2} c d^{2} e - 4 \, {\left (b x + a\right )}^{\frac {3}{2}} a b d^{3} e + 4 \, \sqrt {b x + a} a^{2} b d^{3} e - \sqrt {b x + a} b^{3} c^{3} f + {\left (b x + a\right )}^{\frac {3}{2}} b^{2} c^{2} d f + \sqrt {b x + a} a b^{2} c^{2} d f}{4 \, {\left (b c^{3} d - a c^{2} d^{2}\right )} {\left (b c + {\left (b x + a\right )} d - a d\right )}^{2}} \] Input:

integrate((b*x+a)^(1/2)*(f*x+e)/x/(d*x+c)^3,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 3.25 (sec) , antiderivative size = 4839, normalized size of antiderivative = 23.60 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx=\text {Too large to display} \] Input:

int(((e + f*x)*(a + b*x)^(1/2))/(x*(c + d*x)^3),x)
                                                                                    
                                                                                    
 

Output:

(atan((((d^3*(a*d - b*c)^3)^(1/2)*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4 
*b^2*d^6*e^2 + 9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e 
^2 - 72*a*b^5*c^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f 
 - 24*a*b^5*c^4*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) - 
((d^3*(a*d - b*c)^3)^(1/2)*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b 
^4*c^7*d^4*e + 4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c 
^6*d^3 - 2*a*b*c^7*d^2) - ((d^3*(a*d - b*c)^3)^(1/2)*(a + b*x)^(1/2)*(8*a^ 
2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e)*(64*b^5*c^9*d^3 - 25 
6*a*b^4*c^8*d^4 + 320*a^2*b^3*c^7*d^5 - 128*a^3*b^2*c^6*d^6))/(64*(b^2*c^6 
*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5 
*d^4 - 3*a^2*b*c^4*d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a* 
b*c*d^2*e))/(8*(a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4* 
d^5)))*(8*a^2*d^3*e + b^2*c^3*f + 3*b^2*c^2*d*e - 12*a*b*c*d^2*e)*1i)/(8*( 
a^3*c^3*d^6 - b^3*c^6*d^3 + 3*a*b^2*c^5*d^4 - 3*a^2*b*c^4*d^5)) + ((d^3*(a 
*d - b*c)^3)^(1/2)*(((a + b*x)^(1/2)*(b^6*c^6*f^2 + 128*a^4*b^2*d^6*e^2 + 
9*b^6*c^4*d^2*e^2 + 6*b^6*c^5*d*e*f + 256*a^2*b^4*c^2*d^4*e^2 - 72*a*b^5*c 
^3*d^3*e^2 - 320*a^3*b^3*c*d^5*e^2 + 16*a^2*b^4*c^3*d^3*e*f - 24*a*b^5*c^4 
*d^2*e*f))/(8*(b^2*c^6*d + a^2*c^4*d^3 - 2*a*b*c^5*d^2)) + ((d^3*(a*d - b* 
c)^3)^(1/2)*((5*a*b^5*c^8*d^3*e - a*b^5*c^9*d^2*f - 9*a^2*b^4*c^7*d^4*e + 
4*a^3*b^3*c^6*d^5*e + a^2*b^4*c^8*d^3*f)/(b^2*c^8*d + a^2*c^6*d^3 - 2*a...
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1311, normalized size of antiderivative = 6.40 \[ \int \frac {\sqrt {a+b x} (e+f x)}{x (c+d x)^3} \, dx =\text {Too large to display} \] Input:

int((b*x+a)^(1/2)*(f*x+e)/x/(d*x+c)^3,x)
 

Output:

(8*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d 
+ b*c)))*a**2*c**2*d**3*e + 16*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b 
*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a**2*c*d**4*e*x + 8*sqrt(d)*sqrt( - a 
*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a**2*d**5*e 
*x**2 - 12*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt 
( - a*d + b*c)))*a*b*c**3*d**2*e - 24*sqrt(d)*sqrt( - a*d + b*c)*atan((sqr 
t(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a*b*c**2*d**3*e*x - 12*sqrt(d) 
*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*a 
*b*c*d**4*e*x**2 + sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt 
(d)*sqrt( - a*d + b*c)))*b**2*c**5*f + 3*sqrt(d)*sqrt( - a*d + b*c)*atan(( 
sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*b**2*c**4*d*e + 2*sqrt(d)*s 
qrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*b** 
2*c**4*d*f*x + 6*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d 
)*sqrt( - a*d + b*c)))*b**2*c**3*d**2*e*x + sqrt(d)*sqrt( - a*d + b*c)*ata 
n((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + b*c)))*b**2*c**3*d**2*f*x**2 + 
3*sqrt(d)*sqrt( - a*d + b*c)*atan((sqrt(a + b*x)*d)/(sqrt(d)*sqrt( - a*d + 
 b*c)))*b**2*c**2*d**3*e*x**2 - 2*sqrt(a + b*x)*a**2*c**3*d**3*f + 6*sqrt( 
a + b*x)*a**2*c**2*d**4*e + 4*sqrt(a + b*x)*a**2*c*d**5*e*x + 3*sqrt(a + b 
*x)*a*b*c**4*d**2*f - 11*sqrt(a + b*x)*a*b*c**3*d**3*e - sqrt(a + b*x)*a*b 
*c**3*d**3*f*x - 7*sqrt(a + b*x)*a*b*c**2*d**4*e*x - sqrt(a + b*x)*b**2...