\(\int \frac {(A+B x) \sqrt {d+e x}}{(b x+c x^2)^3} \, dx\) [90]

Optimal result

Integrand size = 26, antiderivative size = 355 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=-\frac {c (6 b B d-12 A c d+A b e) \sqrt {d+e x}}{4 b^3 d (b+c x)^2}-\frac {A \sqrt {d+e x}}{2 b x^2 (b+c x)^2}-\frac {(4 b B d-8 A c d+A b e) \sqrt {d+e x}}{4 b^2 d x (b+c x)^2}+\frac {c \left (24 A c^2 d^2+b^2 e (11 B d+A e)-12 b c d (B d+2 A e)\right ) \sqrt {d+e x}}{4 b^4 d (c d-b e) (b+c x)}-\frac {\left (48 A c^2 d^2+b^2 e (4 B d-A e)-12 b c d (2 B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{3/2}}+\frac {\sqrt {c} \left (48 A c^3 d^2-15 b^3 B e^2-12 b c^2 d (2 B d+7 A e)+5 b^2 c e (8 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{3/2}} \] Output:

-1/4*c*(A*b*e-12*A*c*d+6*B*b*d)*(e*x+d)^(1/2)/b^3/d/(c*x+b)^2-1/2*A*(e*x+d 
)^(1/2)/b/x^2/(c*x+b)^2-1/4*(A*b*e-8*A*c*d+4*B*b*d)*(e*x+d)^(1/2)/b^2/d/x/ 
(c*x+b)^2+1/4*c*(24*A*c^2*d^2+b^2*e*(A*e+11*B*d)-12*b*c*d*(2*A*e+B*d))*(e* 
x+d)^(1/2)/b^4/d/(-b*e+c*d)/(c*x+b)-1/4*(48*A*c^2*d^2+b^2*e*(-A*e+4*B*d)-1 
2*b*c*d*(A*e+2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^(3/2)+1/4*c^(1/2 
)*(48*A*c^3*d^2-15*b^3*B*e^2-12*b*c^2*d*(7*A*e+2*B*d)+5*b^2*c*e*(7*A*e+8*B 
*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^5/(-b*e+c*d)^(3/2)
 

Mathematica [A] (verified)

Time = 4.09 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.96 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\frac {-\frac {b \sqrt {d+e x} \left (b B d x \left (-4 b^3 e+12 c^3 d x^2+b^2 c (4 d-17 e x)+b c^2 x (18 d-11 e x)\right )-A \left (24 c^4 d^2 x^3+12 b c^3 d x^2 (3 d-2 e x)+b^4 e (2 d+e x)+b^2 c^2 x \left (8 d^2-37 d e x+e^2 x^2\right )+b^3 c \left (-2 d^2-9 d e x+2 e^2 x^2\right )\right )\right )}{d (c d-b e) x^2 (b+c x)^2}+\frac {\sqrt {c} \left (48 A c^3 d^2-15 b^3 B e^2-12 b c^2 d (2 B d+7 A e)+5 b^2 c e (8 B d+7 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{3/2}}+\frac {\left (-48 A c^2 d^2+b^2 e (-4 B d+A e)+12 b c d (2 B d+A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{3/2}}}{4 b^5} \] Input:

Integrate[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^3,x]
 

Output:

(-((b*Sqrt[d + e*x]*(b*B*d*x*(-4*b^3*e + 12*c^3*d*x^2 + b^2*c*(4*d - 17*e* 
x) + b*c^2*x*(18*d - 11*e*x)) - A*(24*c^4*d^2*x^3 + 12*b*c^3*d*x^2*(3*d - 
2*e*x) + b^4*e*(2*d + e*x) + b^2*c^2*x*(8*d^2 - 37*d*e*x + e^2*x^2) + b^3* 
c*(-2*d^2 - 9*d*e*x + 2*e^2*x^2))))/(d*(c*d - b*e)*x^2*(b + c*x)^2)) + (Sq 
rt[c]*(48*A*c^3*d^2 - 15*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 7*A*e) + 5*b^2*c* 
e*(8*B*d + 7*A*e))*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-( 
c*d) + b*e)^(3/2) + ((-48*A*c^2*d^2 + b^2*e*(-4*B*d + A*e) + 12*b*c*d*(2*B 
*d + A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(3/2))/(4*b^5)
 

Rubi [A] (verified)

Time = 1.39 (sec) , antiderivative size = 352, normalized size of antiderivative = 0.99, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {1234, 27, 25, 1235, 27, 1197, 27, 1480, 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.

\(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1234

\(\displaystyle -\frac {\int \frac {12 A c d-b (6 B d+A e)-5 (b B-2 A c) e x}{2 \sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{2 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -\frac {6 b B d-12 A c d+A b e+5 (b B-2 A c) e x}{\sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {6 b B d-12 A c d+A b e+5 (b B-2 A c) e x}{\sqrt {d+e x} \left (c x^2+b x\right )^2}dx}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1235

\(\displaystyle \frac {-\frac {\int -\frac {(c d-b e) \left (e (4 B d-A e) b^2-12 c d (2 B d+A e) b+48 A c^2 d^2\right )+c e \left (e (11 B d+A e) b^2-12 c d (B d+2 A e) b+24 A c^2 d^2\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {(c d-b e) \left (e (4 B d-A e) b^2-12 c d (2 B d+A e) b+48 A c^2 d^2\right )+c e \left (e (11 B d+A e) b^2-12 c d (B d+2 A e) b+24 A c^2 d^2\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle \frac {\frac {\int \frac {e \left (-e^2 (4 B d-A e) b^3+c d e (17 B d+10 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3+c \left (e (11 B d+A e) b^2-12 c d (B d+2 A e) b+24 A c^2 d^2\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {e \int \frac {-e^2 (4 B d-A e) b^3+c d e (17 B d+10 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3+c \left (e (11 B d+A e) b^2-12 c d (B d+2 A e) b+24 A c^2 d^2\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {\frac {e \left (\frac {c (c d-b e) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c d \left (5 b^2 c e (7 A e+8 B d)-12 b c^2 d (7 A e+2 B d)+48 A c^3 d^2-15 b^3 B e^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {e \left (\frac {\sqrt {c} d \left (5 b^2 c e (7 A e+8 B d)-12 b c^2 d (7 A e+2 B d)+48 A c^3 d^2-15 b^3 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e) \left (b^2 e (4 B d-A e)-12 b c d (A e+2 B d)+48 A c^2 d^2\right )}{b \sqrt {d} e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) (A b e-12 A c d+6 b B d)-c x \left (b^2 e (A e+11 B d)-12 b c d (2 A e+B d)+24 A c^2 d^2\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2}-\frac {\sqrt {d+e x} (A b-x (b B-2 A c))}{2 b^2 \left (b x+c x^2\right )^2}\)

Input:

Int[((A + B*x)*Sqrt[d + e*x])/(b*x + c*x^2)^3,x]
 

Output:

-1/2*((A*b - (b*B - 2*A*c)*x)*Sqrt[d + e*x])/(b^2*(b*x + c*x^2)^2) + (-((S 
qrt[d + e*x]*(b*(c*d - b*e)*(6*b*B*d - 12*A*c*d + A*b*e) - c*(24*A*c^2*d^2 
 + b^2*e*(11*B*d + A*e) - 12*b*c*d*(B*d + 2*A*e))*x))/(b^2*d*(c*d - b*e)*( 
b*x + c*x^2))) + (e*(-(((c*d - b*e)*(48*A*c^2*d^2 + b^2*e*(4*B*d - A*e) - 
12*b*c*d*(2*B*d + A*e))*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + ( 
Sqrt[c]*d*(48*A*c^3*d^2 - 15*b^3*B*e^2 - 12*b*c^2*d*(2*B*d + 7*A*e) + 5*b^ 
2*c*e*(8*B*d + 7*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/( 
b*e*Sqrt[c*d - b*e])))/(b^2*d*(c*d - b*e)))/(4*b^2)
 

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 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]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

rule 1234

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*( 
(f*b - 2*a*g + (2*c*f - b*g)*x)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 
 1)*(b^2 - 4*a*c))   Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g 
*(2*a*e*m + b*d*(2*p + 3)) - f*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)* 
(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1 
] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 1235

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 
*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a 
+ b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] 
 + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))   Int[(d + e*x)^m 
*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 
 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* 
m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - 
f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] 
)
 

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 1.36 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.95

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \left (A b e x -12 c x A d +4 B b d x +2 A b d \right )}{4 d \,b^{4} x^{2}}-\frac {e \left (-\frac {\left (A \,b^{2} e^{2}+12 A b c d e -48 A \,c^{2} d^{2}-4 B \,b^{2} d e +24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}-\frac {8 c d \left (\frac {\frac {c e b \left (11 A c e b -12 A \,c^{2} d -7 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b e -8 c d}+\frac {e b \left (13 A c e b -12 A \,c^{2} d -9 b^{2} B e +8 B b c d \right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (35 A \,b^{2} e^{2} c -84 A b \,c^{2} d e +48 A \,c^{3} d^{2}-15 b^{3} B \,e^{2}+40 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \left (b e -c d \right ) \sqrt {c \left (b e -c d \right )}}\right )}{b e}\right )}{4 b^{4} d}\)	\(337\)
pseudoelliptic	\(\frac {12 c \,x^{2} \left (c x +b \right )^{2} \left (\frac {35 \left (A c -\frac {3 B b}{7}\right ) e^{2} b^{2} d^{\frac {5}{2}}}{48}+c \left (\frac {5 b^{2} B e}{6}-\frac {7 c \left (A e +\frac {2 B d}{7}\right ) b}{4}+A \,c^{2} d \right ) d^{\frac {7}{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\frac {\sqrt {c \left (b e -c d \right )}\, \left (\frac {\left (\left (A \,e^{2}-4 B d e \right ) b^{2}+12 b c d \left (A e +2 B d \right )-48 A \,c^{2} d^{2}\right ) d \,x^{2} \left (c x +b \right )^{2} \left (b e -c d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\sqrt {e x +d}\, \left (-e \left (-12 A \,c^{3} x^{3}-\frac {37 \left (-\frac {11 B x}{37}+A \right ) x^{2} b \,c^{2}}{2}-\frac {9 x \,b^{2} \left (-\frac {17 B x}{9}+A \right ) c}{2}+b^{3} \left (2 B x +A \right )\right ) b \,d^{\frac {5}{2}}-\frac {A \,b^{2} e^{2} x \left (c x +b \right )^{2} d^{\frac {3}{2}}}{2}+\left (-12 A \,c^{3} x^{3}-18 c^{2} x^{2} \left (-\frac {B x}{3}+A \right ) b -4 \left (-\frac {9 B x}{4}+A \right ) x \,b^{2} c +b^{3} \left (2 B x +A \right )\right ) c \,d^{\frac {7}{2}}\right ) b \right )}{2}}{\sqrt {c \left (b e -c d \right )}\, d^{\frac {5}{2}} x^{2} b^{5} \left (c x +b \right )^{2} \left (b e -c d \right )}\)	\(351\)
derivativedivides	\(2 e^{4} \left (\frac {c \left (\frac {\frac {c e b \left (11 A c e b -12 A \,c^{2} d -7 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b e -8 c d}+\frac {e b \left (13 A c e b -12 A \,c^{2} d -9 b^{2} B e +8 B b c d \right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (35 A \,b^{2} e^{2} c -84 A b \,c^{2} d e +48 A \,c^{3} d^{2}-15 b^{3} B \,e^{2}+40 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \left (b e -c d \right ) \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{4}}-\frac {\frac {\frac {b e \left (A b e -12 A c d +4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d}+\left (\frac {3}{2} A b c d e -\frac {1}{2} B \,b^{2} d e +\frac {1}{8} A \,b^{2} e^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}-\frac {\left (A \,b^{2} e^{2}+12 A b c d e -48 A \,c^{2} d^{2}-4 B \,b^{2} d e +24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{e^{4} b^{5}}\right )\)	\(362\)
default	\(2 e^{4} \left (\frac {c \left (\frac {\frac {c e b \left (11 A c e b -12 A \,c^{2} d -7 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b e -8 c d}+\frac {e b \left (13 A c e b -12 A \,c^{2} d -9 b^{2} B e +8 B b c d \right ) \sqrt {e x +d}}{8}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (35 A \,b^{2} e^{2} c -84 A b \,c^{2} d e +48 A \,c^{3} d^{2}-15 b^{3} B \,e^{2}+40 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \left (b e -c d \right ) \sqrt {c \left (b e -c d \right )}}\right )}{b^{5} e^{4}}-\frac {\frac {\frac {b e \left (A b e -12 A c d +4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d}+\left (\frac {3}{2} A b c d e -\frac {1}{2} B \,b^{2} d e +\frac {1}{8} A \,b^{2} e^{2}\right ) \sqrt {e x +d}}{e^{2} x^{2}}-\frac {\left (A \,b^{2} e^{2}+12 A b c d e -48 A \,c^{2} d^{2}-4 B \,b^{2} d e +24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {3}{2}}}}{e^{4} b^{5}}\right )\)	\(362\)

Input:

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

Output:

-1/4*(e*x+d)^(1/2)*(A*b*e*x-12*A*c*d*x+4*B*b*d*x+2*A*b*d)/d/b^4/x^2-1/4/b^ 
4/d*e*(-1/b/e*(A*b^2*e^2+12*A*b*c*d*e-48*A*c^2*d^2-4*B*b^2*d*e+24*B*b*c*d^ 
2)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2))-8*c*d/b/e*((1/8*c*e*b*(11*A*b*c* 
e-12*A*c^2*d-7*B*b^2*e+8*B*b*c*d)/(b*e-c*d)*(e*x+d)^(3/2)+1/8*e*b*(13*A*b* 
c*e-12*A*c^2*d-9*B*b^2*e+8*B*b*c*d)*(e*x+d)^(1/2))/((e*x+d)*c+b*e-c*d)^2+1 
/8*(35*A*b^2*c*e^2-84*A*b*c^2*d*e+48*A*c^3*d^2-15*B*b^3*e^2+40*B*b^2*c*d*e 
-24*B*b*c^2*d^2)/(b*e-c*d)/(c*(b*e-c*d))^(1/2)*arctan(c*(e*x+d)^(1/2)/(c*( 
b*e-c*d))^(1/2))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 819 vs. \(2 (319) = 638\).

Time = 3.87 (sec) , antiderivative size = 3353, normalized size of antiderivative = 9.45 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^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(b*e-c*d>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. 813 vs. \(2 (319) = 638\).

Time = 0.28 (sec) , antiderivative size = 813, normalized size of antiderivative = 2.29 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

1/4*(24*B*b*c^3*d^2 - 48*A*c^4*d^2 - 40*B*b^2*c^2*d*e + 84*A*b*c^3*d*e + 1 
5*B*b^3*c*e^2 - 35*A*b^2*c^2*e^2)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + b*c 
*e))/((b^5*c*d - b^6*e)*sqrt(-c^2*d + b*c*e)) - 1/4*(12*(e*x + d)^(7/2)*B* 
b*c^3*d^2*e - 24*(e*x + d)^(7/2)*A*c^4*d^2*e - 36*(e*x + d)^(5/2)*B*b*c^3* 
d^3*e + 72*(e*x + d)^(5/2)*A*c^4*d^3*e + 36*(e*x + d)^(3/2)*B*b*c^3*d^4*e 
- 72*(e*x + d)^(3/2)*A*c^4*d^4*e - 12*sqrt(e*x + d)*B*b*c^3*d^5*e + 24*sqr 
t(e*x + d)*A*c^4*d^5*e - 11*(e*x + d)^(7/2)*B*b^2*c^2*d*e^2 + 24*(e*x + d) 
^(7/2)*A*b*c^3*d*e^2 + 51*(e*x + d)^(5/2)*B*b^2*c^2*d^2*e^2 - 108*(e*x + d 
)^(5/2)*A*b*c^3*d^2*e^2 - 69*(e*x + d)^(3/2)*B*b^2*c^2*d^3*e^2 + 144*(e*x 
+ d)^(3/2)*A*b*c^3*d^3*e^2 + 29*sqrt(e*x + d)*B*b^2*c^2*d^4*e^2 - 60*sqrt( 
e*x + d)*A*b*c^3*d^4*e^2 - (e*x + d)^(7/2)*A*b^2*c^2*e^3 - 17*(e*x + d)^(5 
/2)*B*b^3*c*d*e^3 + 40*(e*x + d)^(5/2)*A*b^2*c^2*d*e^3 + 38*(e*x + d)^(3/2 
)*B*b^3*c*d^2*e^3 - 85*(e*x + d)^(3/2)*A*b^2*c^2*d^2*e^3 - 21*sqrt(e*x + d 
)*B*b^3*c*d^3*e^3 + 46*sqrt(e*x + d)*A*b^2*c^2*d^3*e^3 - 2*(e*x + d)^(5/2) 
*A*b^3*c*e^4 - 4*(e*x + d)^(3/2)*B*b^4*d*e^4 + 13*(e*x + d)^(3/2)*A*b^3*c* 
d*e^4 + 4*sqrt(e*x + d)*B*b^4*d^2*e^4 - 9*sqrt(e*x + d)*A*b^3*c*d^2*e^4 - 
(e*x + d)^(3/2)*A*b^4*e^5 - sqrt(e*x + d)*A*b^4*d*e^5)/((b^4*c*d^2 - b^5*d 
*e)*((e*x + d)^2*c - 2*(e*x + d)*c*d + c*d^2 + (e*x + d)*b*e - b*d*e)^2) - 
 1/4*(24*B*b*c*d^2 - 48*A*c^2*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e + A*b^2*e^2 
)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^5*sqrt(-d)*d)
 

Mupad [B] (verification not implemented)

Time = 15.45 (sec) , antiderivative size = 8411, normalized size of antiderivative = 23.69 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

(((d + e*x)^(3/2)*(A*b^4*e^5 + 72*A*c^4*d^4*e + 4*B*b^4*d*e^4 - 144*A*b*c^ 
3*d^3*e^2 - 38*B*b^3*c*d^2*e^3 + 85*A*b^2*c^2*d^2*e^3 + 69*B*b^2*c^2*d^3*e 
^2 - 13*A*b^3*c*d*e^4 - 36*B*b*c^3*d^4*e))/(4*b^4*(c*d^2 - b*d*e)) - ((d + 
 e*x)^(1/2)*(A*b^3*e^4 + 24*A*c^3*d^3*e - 4*B*b^3*d*e^3 - 36*A*b*c^2*d^2*e 
^2 + 17*B*b^2*c*d^2*e^2 + 10*A*b^2*c*d*e^3 - 12*B*b*c^2*d^3*e))/(4*b^4) + 
((d + e*x)^(5/2)*(2*A*b^3*c*e^4 - 72*A*c^4*d^3*e + 108*A*b*c^3*d^2*e^2 - 4 
0*A*b^2*c^2*d*e^3 - 51*B*b^2*c^2*d^2*e^2 + 36*B*b*c^3*d^3*e + 17*B*b^3*c*d 
*e^3))/(4*b^4*(c*d^2 - b*d*e)) + (c*(d + e*x)^(7/2)*(A*b^2*c*e^3 + 24*A*c^ 
3*d^2*e - 24*A*b*c^2*d*e^2 - 12*B*b*c^2*d^2*e + 11*B*b^2*c*d*e^2))/(4*b^4* 
(c*d^2 - b*d*e)))/(c^2*(d + e*x)^4 - (d + e*x)*(4*c^2*d^3 + 2*b^2*d*e^2 - 
6*b*c*d^2*e) - (4*c^2*d - 2*b*c*e)*(d + e*x)^3 + (d + e*x)^2*(b^2*e^2 + 6* 
c^2*d^2 - 6*b*c*d*e) + c^2*d^4 + b^2*d^2*e^2 - 2*b*c*d^3*e) - (atan((((-c* 
(b*e - c*d)^3)^(1/2)*(((d + e*x)^(1/2)*(A^2*b^6*c^3*e^8 + 4608*A^2*c^9*d^6 
*e^2 + 15072*A^2*b^2*c^7*d^4*e^4 - 7104*A^2*b^3*c^6*d^3*e^5 + 1226*A^2*b^4 
*c^5*d^2*e^6 + 1152*B^2*b^2*c^7*d^6*e^2 - 3264*B^2*b^3*c^6*d^5*e^3 + 3296* 
B^2*b^4*c^5*d^4*e^4 - 1424*B^2*b^5*c^4*d^3*e^5 + 241*B^2*b^6*c^3*d^2*e^6 - 
 13824*A^2*b*c^8*d^5*e^3 + 22*A^2*b^5*c^4*d*e^7 - 4608*A*B*b*c^8*d^6*e^2 - 
 8*A*B*b^6*c^3*d*e^7 + 13440*A*B*b^2*c^7*d^5*e^3 - 14112*A*B*b^3*c^6*d^4*e 
^4 + 6368*A*B*b^4*c^5*d^3*e^5 - 1082*A*B*b^5*c^4*d^2*e^6))/(8*(b^8*c^2*d^4 
 + b^10*d^2*e^2 - 2*b^9*c*d^3*e)) + (((A*b^14*c^2*d*e^7 - 24*A*b^10*c^6...
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 2990, normalized size of antiderivative = 8.42 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(70*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d 
)))*a*b**4*c*d**2*e**2*x**2 - 168*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e 
*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**3*c**2*d**3*e*x**2 + 140*sqrt(c)*sq 
rt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**3*c** 
2*d**2*e**2*x**3 + 96*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt 
(c)*sqrt(b*e - c*d)))*a*b**2*c**3*d**4*x**2 - 336*sqrt(c)*sqrt(b*e - c*d)* 
atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**2*c**3*d**3*e*x**3 
+ 70*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c* 
d)))*a*b**2*c**3*d**2*e**2*x**4 + 192*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d 
 + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b*c**4*d**4*x**3 - 168*sqrt(c)*sqr 
t(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b*c**4*d* 
*3*e*x**4 + 96*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqr 
t(b*e - c*d)))*a*c**5*d**4*x**4 - 30*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d 
+ e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**6*d**2*e**2*x**2 + 80*sqrt(c)*sqrt 
(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**5*c*d**3* 
e*x**2 - 60*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b 
*e - c*d)))*b**5*c*d**2*e**2*x**3 - 48*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt( 
d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c**2*d**4*x**2 + 160*sqrt(c)*s 
qrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**4*c**2 
*d**3*e*x**3 - 30*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(...