\(\int \frac {A+B x}{\sqrt {d+e x} (b x+c x^2)^3} \, dx\) [91]

Optimal result

Integrand size = 26, antiderivative size = 418 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\frac {c \left (12 A c^2 d^2+b^2 e (4 B d-3 A e)-b c d (6 B d+7 A e)\right ) \sqrt {d+e x}}{4 b^3 d^2 (c d-b e) (b+c x)^2}-\frac {A \sqrt {d+e x}}{2 b d x^2 (b+c x)^2}-\frac {(4 b B d-8 A c d-3 A b e) \sqrt {d+e x}}{4 b^2 d^2 x (b+c x)^2}+\frac {c \left (24 A c^3 d^3-b^3 e^2 (4 B d-3 A e)-12 b c^2 d^2 (B d+3 A e)+b^2 c d e (19 B d+6 A e)\right ) \sqrt {d+e x}}{4 b^4 d^2 (c d-b e)^2 (b+c x)}-\frac {\left (48 A c^2 d^2-b^2 e (4 B d-3 A e)-12 b c d (2 B d-A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{4 b^5 d^{5/2}}+\frac {c^{3/2} \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{4 b^5 (c d-b e)^{5/2}} \] Output:

1/4*c*(12*A*c^2*d^2+b^2*e*(-3*A*e+4*B*d)-b*c*d*(7*A*e+6*B*d))*(e*x+d)^(1/2 
)/b^3/d^2/(-b*e+c*d)/(c*x+b)^2-1/2*A*(e*x+d)^(1/2)/b/d/x^2/(c*x+b)^2-1/4*( 
-3*A*b*e-8*A*c*d+4*B*b*d)*(e*x+d)^(1/2)/b^2/d^2/x/(c*x+b)^2+1/4*c*(24*A*c^ 
3*d^3-b^3*e^2*(-3*A*e+4*B*d)-12*b*c^2*d^2*(3*A*e+B*d)+b^2*c*d*e*(6*A*e+19* 
B*d))*(e*x+d)^(1/2)/b^4/d^2/(-b*e+c*d)^2/(c*x+b)-1/4*(48*A*c^2*d^2-b^2*e*( 
-3*A*e+4*B*d)-12*b*c*d*(-A*e+2*B*d))*arctanh((e*x+d)^(1/2)/d^(1/2))/b^5/d^ 
(5/2)+1/4*c^(3/2)*(48*A*c^3*d^2-35*b^3*B*e^2-12*b*c^2*d*(9*A*e+2*B*d)+7*b^ 
2*c*e*(9*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)^(5/2)
 

Mathematica [A] (verified)

Time = 4.68 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=-\frac {\frac {b \sqrt {d+e x} \left (b B d x \left (4 b^4 e^2+12 c^4 d^2 x^2+b c^3 d x (18 d-19 e x)+8 b^3 c e (-d+e x)+b^2 c^2 \left (4 d^2-29 d e x+4 e^2 x^2\right )\right )+A \left (-24 c^5 d^3 x^3+b^5 e^2 (2 d-3 e x)-36 b c^4 d^2 x^2 (d-e x)+b^2 c^3 d x \left (-8 d^2+55 d e x-6 e^2 x^2\right )-2 b^4 c e \left (2 d^2+d e x+3 e^2 x^2\right )+b^3 c^2 \left (2 d^3+13 d^2 e x-10 d e^2 x^2-3 e^3 x^3\right )\right )\right )}{d^2 (c d-b e)^2 x^2 (b+c x)^2}+\frac {c^{3/2} \left (48 A c^3 d^2-35 b^3 B e^2-12 b c^2 d (2 B d+9 A e)+7 b^2 c e (8 B d+9 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}+\frac {\left (48 A c^2 d^2+12 b c d (-2 B d+A e)+b^2 e (-4 B d+3 A e)\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}}{4 b^5} \] Input:

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

Output:

-1/4*((b*Sqrt[d + e*x]*(b*B*d*x*(4*b^4*e^2 + 12*c^4*d^2*x^2 + b*c^3*d*x*(1 
8*d - 19*e*x) + 8*b^3*c*e*(-d + e*x) + b^2*c^2*(4*d^2 - 29*d*e*x + 4*e^2*x 
^2)) + A*(-24*c^5*d^3*x^3 + b^5*e^2*(2*d - 3*e*x) - 36*b*c^4*d^2*x^2*(d - 
e*x) + b^2*c^3*d*x*(-8*d^2 + 55*d*e*x - 6*e^2*x^2) - 2*b^4*c*e*(2*d^2 + d* 
e*x + 3*e^2*x^2) + b^3*c^2*(2*d^3 + 13*d^2*e*x - 10*d*e^2*x^2 - 3*e^3*x^3) 
)))/(d^2*(c*d - b*e)^2*x^2*(b + c*x)^2) + (c^(3/2)*(48*A*c^3*d^2 - 35*b^3* 
B*e^2 - 12*b*c^2*d*(2*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d + 9*A*e))*ArcTan[(Sq 
rt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(5/2) + ((48*A*c^ 
2*d^2 + 12*b*c*d*(-2*B*d + A*e) + b^2*e*(-4*B*d + 3*A*e))*ArcTanh[Sqrt[d + 
 e*x]/Sqrt[d]])/d^(5/2))/b^5
 

Rubi [A] (verified)

Time = 1.78 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1235, 27, 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.

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {-\frac {\int \frac {\left (-e (4 B d-3 A e) b^2-12 c d (2 B d-A e) b+48 A c^2 d^2\right ) (c d-b e)^2+c e \left (-e^2 (4 B d-3 A e) b^3+c d e (19 B d+6 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x\right )}dx}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\left (-e (4 B d-3 A e) b^2-12 c d (2 B d-A e) b+48 A c^2 d^2\right ) (c d-b e)^2+c e \left (-e^2 (4 B d-3 A e) b^3+c d e (19 B d+6 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) x}{\sqrt {d+e x} \left (c x^2+b x\right )}dx}{2 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {-\frac {\int \frac {e \left (-e^3 (4 B d-3 A e) b^4-3 c d e^2 (4 B d-A e) b^3+c^2 d^2 e (25 B d+21 A e) b^2-12 c^3 d^3 (B d+4 A e) b+24 A c^4 d^4+c \left (-e^2 (4 B d-3 A e) b^3+c d e (19 B d+6 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) (d+e x)\right )}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {e \int \frac {-e^3 (4 B d-3 A e) b^4-3 c d e^2 (4 B d-A e) b^3+c^2 d^2 e (25 B d+21 A e) b^2-12 c^3 d^3 (B d+4 A e) b+24 A c^4 d^4+c \left (-e^2 (4 B d-3 A e) b^3+c d e (19 B d+6 A e) b^2-12 c^2 d^2 (B d+3 A e) b+24 A c^3 d^3\right ) (d+e x)}{c (d+e x)^2-(2 c d-b e) (d+e x)+d (c d-b e)}d\sqrt {d+e x}}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {-\frac {e \left (\frac {c (c d-b e)^2 \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right ) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}-\frac {c^2 d^2 \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {-\frac {e \left (\frac {c^{3/2} d^2 \left (7 b^2 c e (9 A e+8 B d)-12 b c^2 d (9 A e+2 B d)+48 A c^3 d^2-35 b^3 B e^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}-\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^2 \left (b^2 (-e) (4 B d-3 A e)-12 b c d (2 B d-A e)+48 A c^2 d^2\right )}{b \sqrt {d} e}\right )}{b^2 d (c d-b e)}-\frac {\sqrt {d+e x} \left (b (c d-b e) \left (b^2 e (4 B d-3 A e)-b c d (7 A e+6 B d)+12 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (4 B d-3 A e)+b^2 c d e (6 A e+19 B d)-12 b c^2 d^2 (3 A e+B d)+24 A c^3 d^3\right )\right )}{b^2 d \left (b x+c x^2\right ) (c d-b e)}}{4 b^2 d (c d-b e)}-\frac {\sqrt {d+e x} (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{2 b^2 d \left (b x+c x^2\right )^2 (c d-b e)}\)

Input:

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

Output:

-1/2*(Sqrt[d + e*x]*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(b^ 
2*d*(c*d - b*e)*(b*x + c*x^2)^2) - (-((Sqrt[d + e*x]*(b*(c*d - b*e)*(12*A* 
c^2*d^2 + b^2*e*(4*B*d - 3*A*e) - b*c*d*(6*B*d + 7*A*e)) + c*(24*A*c^3*d^3 
 - b^3*e^2*(4*B*d - 3*A*e) - 12*b*c^2*d^2*(B*d + 3*A*e) + b^2*c*d*e*(19*B* 
d + 6*A*e))*x))/(b^2*d*(c*d - b*e)*(b*x + c*x^2))) - (e*(-(((c*d - b*e)^2* 
(48*A*c^2*d^2 - b^2*e*(4*B*d - 3*A*e) - 12*b*c*d*(2*B*d - A*e))*ArcTanh[Sq 
rt[d + e*x]/Sqrt[d]])/(b*Sqrt[d]*e)) + (c^(3/2)*d^2*(48*A*c^3*d^2 - 35*b^3 
*B*e^2 - 12*b*c^2*d*(2*B*d + 9*A*e) + 7*b^2*c*e*(8*B*d + 9*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*d*(c*d - b*e))
 

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 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 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 = 1.43 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.91

method	result	size
risch	\(-\frac {\sqrt {e x +d}\, \left (-3 A b e x -12 c x A d +4 B b d x +2 A b d \right )}{4 d^{2} b^{4} x^{2}}+\frac {e \left (-\frac {\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}-\frac {8 c^{2} d^{2} \left (\frac {\frac {b c e \left (15 A c e b -12 A \,c^{2} d -11 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b^{2} e^{2}-16 b c d e +8 c^{2} d^{2}}+\frac {\left (17 A c e b -12 A \,c^{2} d -13 b^{2} B e +8 B b c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (63 A \,b^{2} e^{2} c -108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c \left (b e -c d \right )}}\right )}{b e}\right )}{4 b^{4} d^{2}}\)	\(379\)
derivativedivides	\(2 e^{4} \left (-\frac {\frac {-\frac {b e \left (3 A b e +12 A c d -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 A b e +12 A c d -4 B b d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{b^{5} e^{4}}-\frac {c^{2} \left (\frac {\frac {b c e \left (15 A c e b -12 A \,c^{2} d -11 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b^{2} e^{2}-16 b c d e +8 c^{2} d^{2}}+\frac {\left (17 A c e b -12 A \,c^{2} d -13 b^{2} B e +8 B b c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (63 A \,b^{2} e^{2} c -108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c \left (b e -c d \right )}}\right )}{e^{4} b^{5}}\right )\)	\(400\)
default	\(2 e^{4} \left (-\frac {\frac {-\frac {b e \left (3 A b e +12 A c d -4 B b d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 d^{2}}+\frac {b e \left (5 A b e +12 A c d -4 B b d \right ) \sqrt {e x +d}}{8 d}}{e^{2} x^{2}}+\frac {\left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 B b c \,d^{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{8 d^{\frac {5}{2}}}}{b^{5} e^{4}}-\frac {c^{2} \left (\frac {\frac {b c e \left (15 A c e b -12 A \,c^{2} d -11 b^{2} B e +8 B b c d \right ) \left (e x +d \right )^{\frac {3}{2}}}{8 b^{2} e^{2}-16 b c d e +8 c^{2} d^{2}}+\frac {\left (17 A c e b -12 A \,c^{2} d -13 b^{2} B e +8 B b c d \right ) b e \sqrt {e x +d}}{8 b e -8 c d}}{\left (\left (e x +d \right ) c +b e -c d \right )^{2}}+\frac {\left (63 A \,b^{2} e^{2} c -108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{8 \left (b^{2} e^{2}-2 b c d e +c^{2} d^{2}\right ) \sqrt {c \left (b e -c d \right )}}\right )}{e^{4} b^{5}}\right )\)	\(400\)
pseudoelliptic	\(\frac {-\frac {c^{2} \left (b e -c d \right ) d^{\frac {9}{2}} x^{2} \left (c x +b \right )^{2} \left (63 A \,b^{2} e^{2} c -108 A b \,c^{2} d e +48 A \,c^{3} d^{2}-35 b^{3} B \,e^{2}+56 B \,b^{2} c d e -24 B b \,c^{2} d^{2}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2}+\left (-\frac {d^{2} \left (3 A \,b^{2} e^{2}+12 A b c d e +48 A \,c^{2} d^{2}-4 B \,b^{2} d e -24 B b c \,d^{2}\right ) x^{2} \left (c x +b \right )^{2} \left (b e -c d \right )^{3} \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2}+\left (-12 A \,c^{5} d^{3} x^{3}-18 d^{2} x^{2} b \left (\left (-\frac {B x}{3}+A \right ) d -A e x \right ) c^{4}-4 d \left (\left (-\frac {9 B x}{4}+A \right ) d^{2}-\frac {55 e x \left (-\frac {19 B x}{55}+A \right ) d}{8}+\frac {3 A \,e^{2} x^{2}}{4}\right ) x \,b^{2} c^{3}+\left (\left (2 B x +A \right ) d^{3}+\frac {13 e x \left (-\frac {29 B x}{13}+A \right ) d^{2}}{2}-5 \left (-\frac {2 B x}{5}+A \right ) e^{2} x^{2} d -\frac {3 A \,e^{3} x^{3}}{2}\right ) b^{3} c^{2}-2 e \left (\left (2 B x +A \right ) d^{2}+\frac {e x \left (-4 B x +A \right ) d}{2}+\frac {3 A \,e^{2} x^{2}}{2}\right ) b^{4} c +\left (\left (2 B x +A \right ) d -\frac {3 A e x}{2}\right ) e^{2} b^{5}\right ) \left (-b e +c d \right ) d^{\frac {5}{2}} b \sqrt {e x +d}\right ) \sqrt {c \left (b e -c d \right )}}{2 \sqrt {c \left (b e -c d \right )}\, d^{\frac {9}{2}} x^{2} \left (c x +b \right )^{2} \left (b e -c d \right )^{3} b^{5}}\)	\(454\)

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)*(-3*A*b*e*x-12*A*c*d*x+4*B*b*d*x+2*A*b*d)/d^2/b^4/x^2+1 
/4/b^4/d^2*e*(-1/b/e*(3*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^2*d^2/b/e*((1/8*b*c 
*e*(15*A*b*c*e-12*A*c^2*d-11*B*b^2*e+8*B*b*c*d)/(b^2*e^2-2*b*c*d*e+c^2*d^2 
)*(e*x+d)^(3/2)+1/8*(17*A*b*c*e-12*A*c^2*d-13*B*b^2*e+8*B*b*c*d)*b*e/(b*e- 
c*d)*(e*x+d)^(1/2))/((e*x+d)*c+b*e-c*d)^2+1/8*(63*A*b^2*c*e^2-108*A*b*c^2* 
d*e+48*A*c^3*d^2-35*B*b^3*e^2+56*B*b^2*c*d*e-24*B*b*c^2*d^2)/(b^2*e^2-2*b* 
c*d*e+c^2*d^2)/(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. 1047 vs. \(2 (382) = 764\).

Time = 16.73 (sec) , antiderivative size = 4265, normalized size of antiderivative = 10.20 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] 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. 1020 vs. \(2 (382) = 764\).

Time = 0.26 (sec) , antiderivative size = 1020, normalized size of antiderivative = 2.44 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] 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^4*d^2 - 48*A*c^5*d^2 - 56*B*b^2*c^3*d*e + 108*A*b*c^4*d*e + 
35*B*b^3*c^2*e^2 - 63*A*b^2*c^3*e^2)*arctan(sqrt(e*x + d)*c/sqrt(-c^2*d + 
b*c*e))/((b^5*c^2*d^2 - 2*b^6*c*d*e + b^7*e^2)*sqrt(-c^2*d + b*c*e)) - 1/4 
*(12*(e*x + d)^(7/2)*B*b*c^4*d^3*e - 24*(e*x + d)^(7/2)*A*c^5*d^3*e - 36*( 
e*x + d)^(5/2)*B*b*c^4*d^4*e + 72*(e*x + d)^(5/2)*A*c^5*d^4*e + 36*(e*x + 
d)^(3/2)*B*b*c^4*d^5*e - 72*(e*x + d)^(3/2)*A*c^5*d^5*e - 12*sqrt(e*x + d) 
*B*b*c^4*d^6*e + 24*sqrt(e*x + d)*A*c^5*d^6*e - 19*(e*x + d)^(7/2)*B*b^2*c 
^3*d^2*e^2 + 36*(e*x + d)^(7/2)*A*b*c^4*d^2*e^2 + 75*(e*x + d)^(5/2)*B*b^2 
*c^3*d^3*e^2 - 144*(e*x + d)^(5/2)*A*b*c^4*d^3*e^2 - 93*(e*x + d)^(3/2)*B* 
b^2*c^3*d^4*e^2 + 180*(e*x + d)^(3/2)*A*b*c^4*d^4*e^2 + 37*sqrt(e*x + d)*B 
*b^2*c^3*d^5*e^2 - 72*sqrt(e*x + d)*A*b*c^4*d^5*e^2 + 4*(e*x + d)^(7/2)*B* 
b^3*c^2*d*e^3 - 6*(e*x + d)^(7/2)*A*b^2*c^3*d*e^3 - 41*(e*x + d)^(5/2)*B*b 
^3*c^2*d^2*e^3 + 73*(e*x + d)^(5/2)*A*b^2*c^3*d^2*e^3 + 74*(e*x + d)^(3/2) 
*B*b^3*c^2*d^3*e^3 - 136*(e*x + d)^(3/2)*A*b^2*c^3*d^3*e^3 - 37*sqrt(e*x + 
 d)*B*b^3*c^2*d^4*e^3 + 69*sqrt(e*x + d)*A*b^2*c^3*d^4*e^3 - 3*(e*x + d)^( 
7/2)*A*b^3*c^2*e^4 + 8*(e*x + d)^(5/2)*B*b^4*c*d*e^4 - (e*x + d)^(5/2)*A*b 
^3*c^2*d*e^4 - 24*(e*x + d)^(3/2)*B*b^4*c*d^2*e^4 + 24*(e*x + d)^(3/2)*A*b 
^3*c^2*d^2*e^4 + 16*sqrt(e*x + d)*B*b^4*c*d^3*e^4 - 18*sqrt(e*x + d)*A*b^3 
*c^2*d^3*e^4 - 6*(e*x + d)^(5/2)*A*b^4*c*e^5 + 4*(e*x + d)^(3/2)*B*b^5*d*e 
^5 + 10*(e*x + d)^(3/2)*A*b^4*c*d*e^5 - 4*sqrt(e*x + d)*B*b^5*d^2*e^5 -...
 

Mupad [B] (verification not implemented)

Time = 16.90 (sec) , antiderivative size = 11338, normalized size of antiderivative = 27.12 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

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

Output:

log((((((c^2*e^3*(3*A*b^4*e^4 + 24*A*c^4*d^4 - 12*B*b*c^3*d^4 - 4*B*b^4*d* 
e^3 + 25*B*b^2*c^2*d^3*e - 12*B*b^3*c*d^2*e^2 + 21*A*b^2*c^2*d^2*e^2 - 48* 
A*b*c^3*d^3*e + 3*A*b^3*c*d*e^3))/(b^2*d^2*(b*e - c*d)^2) - b^2*c^2*e^2*(b 
*e - 2*c*d)*(d + e*x)^(1/2)*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 
4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))*((3*A*b^2*e^2 + 48*A*c^2* 
d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b*c*d*e)^2/(b^10*d^5))^(1/2))/8 - 
((d + e*x)^(1/2)*(9*A^2*b^8*c^3*e^10 + 4608*A^2*c^11*d^8*e^2 + 27360*A^2*b 
^2*c^9*d^6*e^4 - 17568*A^2*b^3*c^8*d^5*e^5 + 3978*A^2*b^4*c^7*d^4*e^6 - 18 
0*A^2*b^5*c^6*d^3*e^7 + 198*A^2*b^6*c^5*d^2*e^8 + 1152*B^2*b^2*c^9*d^8*e^2 
 - 4800*B^2*b^3*c^8*d^7*e^3 + 7520*B^2*b^4*c^7*d^6*e^4 - 5136*B^2*b^5*c^6* 
d^5*e^5 + 1129*B^2*b^6*c^5*d^4*e^6 + 128*B^2*b^7*c^4*d^3*e^7 + 16*B^2*b^8* 
c^3*d^2*e^8 - 18432*A^2*b*c^10*d^7*e^3 + 36*A^2*b^7*c^4*d*e^9 - 4608*A*B*b 
*c^10*d^8*e^2 - 24*A*B*b^8*c^3*d*e^9 + 18816*A*B*b^2*c^9*d^7*e^3 - 28704*A 
*B*b^3*c^8*d^6*e^4 + 19008*A*B*b^4*c^7*d^5*e^5 - 4218*A*B*b^5*c^6*d^4*e^6 
- 144*A*B*b^6*c^5*d^3*e^7 - 144*A*B*b^7*c^4*d^2*e^8))/(8*b^8*d^4*(b*e - c* 
d)^4))*((3*A*b^2*e^2 + 48*A*c^2*d^2 - 24*B*b*c*d^2 - 4*B*b^2*d*e + 12*A*b* 
c*d*e)^2/(b^10*d^5))^(1/2))/8 - (567*A^3*b^7*c^5*e^10 + 55296*A^3*c^12*d^7 
*e^3 + 224640*A^3*b^2*c^10*d^5*e^5 - 77760*A^3*b^3*c^9*d^4*e^6 - 13608*A^3 
*b^4*c^8*d^3*e^7 + 1404*A^3*b^5*c^7*d^2*e^8 - 6912*B^3*b^3*c^9*d^7*e^3 + 2 
5920*B^3*b^4*c^8*d^6*e^4 - 33408*B^3*b^5*c^7*d^5*e^5 + 15016*B^3*b^6*c^...
 

Reduce [B] (verification not implemented)

Time = 0.32 (sec) , antiderivative size = 3571, normalized size of antiderivative = 8.54 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (b x+c x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 126*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - 
 c*d)))*a*b**4*c**2*d**3*e**2*x**2 + 216*sqrt(c)*sqrt(b*e - c*d)*atan((sqr 
t(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**3*c**3*d**4*e*x**2 - 252*sqr 
t(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b 
**3*c**3*d**3*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**4*d**5*x**2 + 432*sqrt(c)*sqrt(b*e 
- c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b**2*c**4*d**4* 
e*x**3 - 126*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt( 
b*e - c*d)))*a*b**2*c**4*d**3*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**5*d**5*x**3 + 216*sqr 
t(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b 
*c**5*d**4*e*x**4 - 96*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqr 
t(c)*sqrt(b*e - c*d)))*a*c**6*d**5*x**4 + 70*sqrt(c)*sqrt(b*e - c*d)*atan( 
(sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**6*c*d**3*e**2*x**2 - 112*s 
qrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b 
**5*c**2*d**4*e*x**2 + 140*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/ 
(sqrt(c)*sqrt(b*e - c*d)))*b**5*c**2*d**3*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**3*d**5*x* 
*2 - 224*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e 
- c*d)))*b**4*c**3*d**4*e*x**3 + 70*sqrt(c)*sqrt(b*e - c*d)*atan((sqrt(...