\(\int \frac {A+B x}{(d+e x)^{3/2} (b x+c x^2)^2} \, dx\) [83]

Optimal result

Integrand size = 26, antiderivative size = 263 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx=-\frac {e \left (2 A c^2 d^2-b^2 e (2 B d-3 A e)-b c d (B d+2 A e)\right )}{b^2 d^2 (c d-b e)^2 \sqrt {d+e x}}+\frac {c (b B d-2 A c d+A b e)}{b^2 d (c d-b e) (b+c x) \sqrt {d+e x}}-\frac {A}{b d x (b+c x) \sqrt {d+e x}}-\frac {(2 b B d-4 A c d-3 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{b^3 d^{5/2}}-\frac {c^{3/2} \left (4 A c^2 d+5 b^2 B e-b c (2 B d+7 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{5/2}} \] Output:

-e*(2*A*c^2*d^2-b^2*e*(-3*A*e+2*B*d)-b*c*d*(2*A*e+B*d))/b^2/d^2/(-b*e+c*d) 
^2/(e*x+d)^(1/2)+c*(A*b*e-2*A*c*d+B*b*d)/b^2/d/(-b*e+c*d)/(c*x+b)/(e*x+d)^ 
(1/2)-A/b/d/x/(c*x+b)/(e*x+d)^(1/2)-(-3*A*b*e-4*A*c*d+2*B*b*d)*arctanh((e* 
x+d)^(1/2)/d^(1/2))/b^3/d^(5/2)-c^(3/2)*(4*A*c^2*d+5*b^2*B*e-b*c*(7*A*e+2* 
B*d))*arctanh(c^(1/2)*(e*x+d)^(1/2)/(-b*e+c*d)^(1/2))/b^3/(-b*e+c*d)^(5/2)
 

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx=\frac {\frac {b \left (b B d x \left (2 b^2 e^2+2 b c e^2 x+c^2 d (d+e x)\right )-A \left (2 c^3 d^2 x (d+e x)+b^3 e^2 (d+3 e x)+b c^2 d \left (d^2-d e x-2 e^2 x^2\right )+b^2 c e \left (-2 d^2-d e x+3 e^2 x^2\right )\right )\right )}{d^2 (c d-b e)^2 x (b+c x) \sqrt {d+e x}}+\frac {c^{3/2} \left (4 A c^2 d+5 b^2 B e-b c (2 B d+7 A e)\right ) \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {-c d+b e}}\right )}{(-c d+b e)^{5/2}}+\frac {(-2 b B d+4 A c d+3 A b e) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{d^{5/2}}}{b^3} \] Input:

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

Output:

((b*(b*B*d*x*(2*b^2*e^2 + 2*b*c*e^2*x + c^2*d*(d + e*x)) - A*(2*c^3*d^2*x* 
(d + e*x) + b^3*e^2*(d + 3*e*x) + b*c^2*d*(d^2 - d*e*x - 2*e^2*x^2) + b^2* 
c*e*(-2*d^2 - d*e*x + 3*e^2*x^2))))/(d^2*(c*d - b*e)^2*x*(b + c*x)*Sqrt[d 
+ e*x]) + (c^(3/2)*(4*A*c^2*d + 5*b^2*B*e - b*c*(2*B*d + 7*A*e))*ArcTan[(S 
qrt[c]*Sqrt[d + e*x])/Sqrt[-(c*d) + b*e]])/(-(c*d) + b*e)^(5/2) + ((-2*b*B 
*d + 4*A*c*d + 3*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/d^(5/2))/b^3
 

Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.17, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1235, 27, 1198, 1197, 25, 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.

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

\(\Big \downarrow \) 1235

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1198

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

\(\Big \downarrow \) 1197

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1480

\(\displaystyle \frac {-\frac {2 e \left (\frac {c^2 d^2 \left (7 A b c e-4 A c^2 d-5 b^2 B e+2 b B c d\right ) \int \frac {1}{-c d+b e+c (d+e x)}d\sqrt {d+e x}}{b e}-\frac {c (c d-b e)^2 (-3 A b e-4 A c d+2 b B d) \int \frac {1}{c (d+e x)-c d}d\sqrt {d+e x}}{b e}\right )}{d (c d-b e)}-\frac {2 e \left (b^2 (-e) (2 B d-3 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {2 e \left (\frac {\text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (c d-b e)^2 (-3 A b e-4 A c d+2 b B d)}{b \sqrt {d} e}-\frac {c^{3/2} d^2 \left (7 A b c e-4 A c^2 d-5 b^2 B e+2 b B c d\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b e \sqrt {c d-b e}}\right )}{d (c d-b e)}-\frac {2 e \left (b^2 (-e) (2 B d-3 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{d \sqrt {d+e x} (c d-b e)}}{2 b^2 d (c d-b e)}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}\)

Input:

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

Output:

-((A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x)/(b^2*d*(c*d - b*e)*Sqr 
t[d + e*x]*(b*x + c*x^2))) + ((-2*e*(2*A*c^2*d^2 - b^2*e*(2*B*d - 3*A*e) - 
 b*c*d*(B*d + 2*A*e)))/(d*(c*d - b*e)*Sqrt[d + e*x]) - (2*e*(((c*d - b*e)^ 
2*(2*b*B*d - 4*A*c*d - 3*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b*Sqrt[d] 
*e) - (c^(3/2)*d^2*(2*b*B*c*d - 4*A*c^2*d - 5*b^2*B*e + 7*A*b*c*e)*ArcTanh 
[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b*e*Sqrt[c*d - b*e])))/(d*(c*d 
 - b*e)))/(2*b^2*d*(c*d - b*e))
 

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 1198

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

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 = 1.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(2 e^{2} \left (\frac {-\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (3 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{2} e^{2}}-\frac {A e -B d}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}-\frac {c^{2} \left (\frac {\left (\frac {1}{2} A c e b -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}+\frac {\left (7 A c e b -4 A \,c^{2} d -5 b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{2} e^{2} b^{3}}\right )\)	\(219\)
default	\(2 e^{2} \left (\frac {-\frac {A b \sqrt {e x +d}}{2 x}+\frac {\left (3 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{b^{3} d^{2} e^{2}}-\frac {A e -B d}{d^{2} \left (b e -c d \right )^{2} \sqrt {e x +d}}-\frac {c^{2} \left (\frac {\left (\frac {1}{2} A c e b -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}+\frac {\left (7 A c e b -4 A \,c^{2} d -5 b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{2} e^{2} b^{3}}\right )\)	\(219\)
risch	\(-\frac {A \sqrt {e x +d}}{d^{2} b^{2} x}-\frac {e \left (-\frac {\left (3 A b e +4 A c d -2 B b d \right ) \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b e \sqrt {d}}+\frac {2 c^{2} d^{2} \left (\frac {\left (\frac {1}{2} A c e b -\frac {1}{2} b^{2} B e \right ) \sqrt {e x +d}}{\left (e x +d \right ) c +b e -c d}+\frac {\left (7 A c e b -4 A \,c^{2} d -5 b^{2} B e +2 B b c d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )}{2 \sqrt {c \left (b e -c d \right )}}\right )}{\left (b e -c d \right )^{2} b e}+\frac {2 e \,b^{2} \left (A e -B d \right )}{\left (b e -c d \right )^{2} \sqrt {e x +d}}\right )}{b^{2} d^{2}}\)	\(228\)
pseudoelliptic	\(-\frac {-4 c^{2} \left (A \,c^{2} d -\frac {7 b \left (A e +\frac {2 B d}{7}\right ) c}{4}+\frac {5 b^{2} B e}{4}\right ) x \left (c x +b \right ) d^{\frac {5}{2}} \sqrt {e x +d}\, \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {c \left (b e -c d \right )}}\right )+\left (-3 \left (\frac {4 A c d}{3}+b \left (A e -\frac {2 B d}{3}\right )\right ) x \left (c x +b \right ) \left (b e -c d \right )^{2} \sqrt {e x +d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )+\left (2 A \,d^{2} x \left (e x +d \right ) c^{3}+d \left (e x +d \right ) \left (d \left (-B x +A \right )-2 A e x \right ) b \,c^{2}-2 e \,b^{2} \left (A \,d^{2}+\frac {e x \left (2 B x +A \right ) d}{2}-\frac {3 A \,e^{2} x^{2}}{2}\right ) c +e^{2} \left (\left (-2 B x +A \right ) d +3 A e x \right ) b^{3}\right ) \sqrt {d}\, b \right ) \sqrt {c \left (b e -c d \right )}}{\sqrt {c \left (b e -c d \right )}\, \sqrt {e x +d}\, x \,d^{\frac {5}{2}} b^{3} \left (c x +b \right ) \left (b e -c d \right )^{2}}\)	\(285\)

Input:

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

Output:

2*e^2*(1/b^3/d^2/e^2*(-1/2*A*b*(e*x+d)^(1/2)/x+1/2*(3*A*b*e+4*A*c*d-2*B*b* 
d)/d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))-(A*e-B*d)/d^2/(b*e-c*d)^2/(e*x+ 
d)^(1/2)-c^2/(b*e-c*d)^2/e^2/b^3*((1/2*A*c*e*b-1/2*b^2*B*e)*(e*x+d)^(1/2)/ 
((e*x+d)*c+b*e-c*d)+1/2*(7*A*b*c*e-4*A*c^2*d-5*B*b^2*e+2*B*b*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. 779 vs. \(2 (241) = 482\).

Time = 9.05 (sec) , antiderivative size = 3196, normalized size of antiderivative = 12.15 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^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(b*e-c*d>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.83 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx=-\frac {{\left (2 \, B b c^{3} d - 4 \, A c^{4} d - 5 \, B b^{2} c^{2} e + 7 \, A b c^{3} e\right )} \arctan \left (\frac {\sqrt {e x + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}\right )} \sqrt {-c^{2} d + b c e}} + \frac {{\left (e x + d\right )}^{2} B b c^{2} d^{2} e - 2 \, {\left (e x + d\right )}^{2} A c^{3} d^{2} e - {\left (e x + d\right )} B b c^{2} d^{3} e + 2 \, {\left (e x + d\right )} A c^{3} d^{3} e + 2 \, {\left (e x + d\right )}^{2} B b^{2} c d e^{2} + 2 \, {\left (e x + d\right )}^{2} A b c^{2} d e^{2} - 4 \, {\left (e x + d\right )} B b^{2} c d^{2} e^{2} - 3 \, {\left (e x + d\right )} A b c^{2} d^{2} e^{2} + 2 \, B b^{2} c d^{3} e^{2} - 3 \, {\left (e x + d\right )}^{2} A b^{2} c e^{3} + 2 \, {\left (e x + d\right )} B b^{3} d e^{3} + 7 \, {\left (e x + d\right )} A b^{2} c d e^{3} - 2 \, B b^{3} d^{2} e^{3} - 2 \, A b^{2} c d^{2} e^{3} - 3 \, {\left (e x + d\right )} A b^{3} e^{4} + 2 \, A b^{3} d e^{4}}{{\left (b^{2} c^{2} d^{4} - 2 \, b^{3} c d^{3} e + b^{4} d^{2} e^{2}\right )} {\left ({\left (e x + d\right )}^{\frac {5}{2}} c - 2 \, {\left (e x + d\right )}^{\frac {3}{2}} c d + \sqrt {e x + d} c d^{2} + {\left (e x + d\right )}^{\frac {3}{2}} b e - \sqrt {e x + d} b d e\right )}} + \frac {{\left (2 \, B b d - 4 \, A c d - 3 \, A b e\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{2}} \] Input:

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

Output:

-(2*B*b*c^3*d - 4*A*c^4*d - 5*B*b^2*c^2*e + 7*A*b*c^3*e)*arctan(sqrt(e*x + 
 d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^2*d^2 - 2*b^4*c*d*e + b^5*e^2)*sqrt(-c 
^2*d + b*c*e)) + ((e*x + d)^2*B*b*c^2*d^2*e - 2*(e*x + d)^2*A*c^3*d^2*e - 
(e*x + d)*B*b*c^2*d^3*e + 2*(e*x + d)*A*c^3*d^3*e + 2*(e*x + d)^2*B*b^2*c* 
d*e^2 + 2*(e*x + d)^2*A*b*c^2*d*e^2 - 4*(e*x + d)*B*b^2*c*d^2*e^2 - 3*(e*x 
 + d)*A*b*c^2*d^2*e^2 + 2*B*b^2*c*d^3*e^2 - 3*(e*x + d)^2*A*b^2*c*e^3 + 2* 
(e*x + d)*B*b^3*d*e^3 + 7*(e*x + d)*A*b^2*c*d*e^3 - 2*B*b^3*d^2*e^3 - 2*A* 
b^2*c*d^2*e^3 - 3*(e*x + d)*A*b^3*e^4 + 2*A*b^3*d*e^4)/((b^2*c^2*d^4 - 2*b 
^3*c*d^3*e + b^4*d^2*e^2)*((e*x + d)^(5/2)*c - 2*(e*x + d)^(3/2)*c*d + sqr 
t(e*x + d)*c*d^2 + (e*x + d)^(3/2)*b*e - sqrt(e*x + d)*b*d*e)) + (2*B*b*d 
- 4*A*c*d - 3*A*b*e)*arctan(sqrt(e*x + d)/sqrt(-d))/(b^3*sqrt(-d)*d^2)
 

Mupad [B] (verification not implemented)

Time = 15.04 (sec) , antiderivative size = 8946, normalized size of antiderivative = 34.02 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

(atan(((((d + e*x)^(1/2)*(64*A^2*b^6*c^15*d^18*e^2 - 576*A^2*b^7*c^14*d^17 
*e^3 + 2228*A^2*b^8*c^13*d^16*e^4 - 4768*A^2*b^9*c^12*d^15*e^5 + 5960*A^2* 
b^10*c^11*d^14*e^6 - 3976*A^2*b^11*c^10*d^13*e^7 + 578*A^2*b^12*c^9*d^12*e 
^8 + 1004*A^2*b^13*c^8*d^11*e^9 - 442*A^2*b^14*c^7*d^10*e^10 - 320*A^2*b^1 
5*c^6*d^9*e^11 + 362*A^2*b^16*c^5*d^8*e^12 - 132*A^2*b^17*c^4*d^7*e^13 + 1 
8*A^2*b^18*c^3*d^6*e^14 + 16*B^2*b^8*c^13*d^18*e^2 - 168*B^2*b^9*c^12*d^17 
*e^3 + 770*B^2*b^10*c^11*d^16*e^4 - 2020*B^2*b^11*c^10*d^15*e^5 + 3350*B^2 
*b^12*c^9*d^14*e^6 - 3664*B^2*b^13*c^8*d^13*e^7 + 2678*B^2*b^14*c^7*d^12*e 
^8 - 1300*B^2*b^15*c^6*d^11*e^9 + 410*B^2*b^16*c^5*d^10*e^10 - 80*B^2*b^17 
*c^4*d^9*e^11 + 8*B^2*b^18*c^3*d^8*e^12 - 64*A*B*b^7*c^14*d^18*e^2 + 624*A 
*B*b^8*c^13*d^17*e^3 - 2636*A*B*b^9*c^12*d^16*e^4 + 6280*A*B*b^10*c^11*d^1 
5*e^5 - 9140*A*B*b^11*c^10*d^14*e^6 + 8056*A*B*b^12*c^9*d^13*e^7 - 3620*A* 
B*b^13*c^8*d^12*e^8 - 224*A*B*b^14*c^7*d^11*e^9 + 1300*A*B*b^15*c^6*d^10*e 
^10 - 760*A*B*b^16*c^5*d^9*e^11 + 208*A*B*b^17*c^4*d^8*e^12 - 24*A*B*b^18* 
c^3*d^7*e^13) - ((-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b 
*c*e - 2*B*b*c*d)*(((-c^3*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d 
+ 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*(16*b^12*c^13*d^21*e^2 - 168*b^13*c^1 
2*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5 + 4320*b^16* 
c^9*d^17*e^6 - 5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19 
*c^6*d^14*e^9 + 1680*b^20*c^5*d^13*e^10 - 520*b^21*c^4*d^12*e^11 + 96*b...
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 1965, normalized size of antiderivative = 7.47 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x+c x^2\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 14*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt( 
c)*sqrt(b*e - c*d)))*a*b**2*c**2*d**3*e*x + 8*sqrt(c)*sqrt(d + e*x)*sqrt(b 
*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*b*c**3*d**4* 
x - 14*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt( 
c)*sqrt(b*e - c*d)))*a*b*c**3*d**3*e*x**2 + 8*sqrt(c)*sqrt(d + e*x)*sqrt(b 
*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*a*c**4*d**4*x* 
*2 + 10*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt 
(c)*sqrt(b*e - c*d)))*b**4*c*d**3*e*x - 4*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - 
 c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**3*c**2*d**4*x + 
 10*sqrt(c)*sqrt(d + e*x)*sqrt(b*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)* 
sqrt(b*e - c*d)))*b**3*c**2*d**3*e*x**2 - 4*sqrt(c)*sqrt(d + e*x)*sqrt(b*e 
 - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(b*e - c*d)))*b**2*c**3*d**4*x 
**2 - 3*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**5*e**4*x + 
 5*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**4*c*d*e**3*x - 
3*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**4*c*e**4*x**2 + 
3*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**3*c**2*d**2*e**2 
*x + 5*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**3*c**2*d*e* 
*3*x**2 - 9*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**2*c**3 
*d**3*e*x + 3*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d))*a*b**2*c* 
*3*d**2*e**2*x**2 + 4*sqrt(d)*sqrt(d + e*x)*log(sqrt(d + e*x) - sqrt(d)...