3.25.71 \(\int \frac {A+B x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx\) [2471]

3.25.71.1 Optimal result

Integrand size = 27, antiderivative size = 271 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{2 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {\left (3 A e (2 c d-b e)-B \left (2 c d^2+e (b d-4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{4 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {\left (b^2 e (B d+3 A e)+4 c \left (2 A c d^2+3 a B d e-a A e^2\right )-4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{5/2}} \]

output

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

3.25.71.2 Mathematica [A] (verified)

Time = 10.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\frac {(B d-A e) \sqrt {a+x (b+c x)}}{2 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}+\frac {\left (2 B c d^2+B e (b d-4 a e)+3 A e (-2 c d+b e)\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}-\frac {\left (b^2 e (B d+3 A e)+4 c \left (2 A c d^2+3 a B d e-a A e^2\right )-4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{5/2}} \]

input

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

output

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

3.25.71.3 Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1237, 27, 1228, 1154, 219}

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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1228

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

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

input

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

output

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

3.25.71.3.1 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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1228

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

rule 1237

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

3.25.71.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(840\) vs. \(2(253)=506\).

Time = 0.60 (sec) , antiderivative size = 841, normalized size of antiderivative = 3.10

method	result	size
default	\(\frac {B \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{3}}+\frac {\left (A e -B d \right ) \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )^{2}}-\frac {3 \left (b e -2 c d \right ) e \left (-\frac {e^{2} \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{\left (e^{2} a -b d e +c \,d^{2}\right ) \left (x +\frac {d}{e}\right )}+\frac {\left (b e -2 c d \right ) e \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{4 \left (e^{2} a -b d e +c \,d^{2}\right )}+\frac {c \,e^{2} \ln \left (\frac {\frac {2 e^{2} a -2 b d e +2 c \,d^{2}}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{2 \left (e^{2} a -b d e +c \,d^{2}\right ) \sqrt {\frac {e^{2} a -b d e +c \,d^{2}}{e^{2}}}}\right )}{e^{4}}\)	\(841\)

input

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

output

B/e^3*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/ 
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/(( 
a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e* 
(x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/ 
e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))+(A*e-B*d)/e^4*(-1/2/(a*e^2-b* 
d*e+c*d^2)*e^2/(x+d/e)^2*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c 
*d^2)/e^2)^(1/2)-3/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-1/(a*e^2-b*d*e+c* 
d^2)*e^2/(x+d/e)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^ 
2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^( 
1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d*e+c 
*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^ 
2)^(1/2))/(x+d/e)))+1/2*c/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^2-b*d*e+c*d^2)/e^2 
)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*((a*e^2-b*d* 
e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2) 
/e^2)^(1/2))/(x+d/e)))
 

3.25.71.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 854 vs. \(2 (253) = 506\).

Time = 9.12 (sec) , antiderivative size = 1750, normalized size of antiderivative = 6.46 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

input

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

output

[1/16*((4*(B*b*c - 2*A*c^2)*d^4 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^3*e + (4 
*B*a*b - 3*A*b^2 + 4*A*a*c)*d^2*e^2 + (4*(B*b*c - 2*A*c^2)*d^2*e^2 - (B*b^ 
2 + 4*(3*B*a - 2*A*b)*c)*d*e^3 + (4*B*a*b - 3*A*b^2 + 4*A*a*c)*e^4)*x^2 + 
2*(4*(B*b*c - 2*A*c^2)*d^3*e - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^2*e^2 + (4* 
B*a*b - 3*A*b^2 + 4*A*a*c)*d*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a* 
b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4* 
a*c)*e^2)*x^2 + 4*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 
 2*a*e + (2*c*d - b*e)*x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e 
)*x)/(e^2*x^2 + 2*d*e*x + d^2)) + 4*(4*B*c^2*d^5 - 2*A*a^2*e^5 - (5*B*b*c 
+ 8*A*c^2)*d^4*e + (B*b^2 + (2*B*a + 13*A*b)*c)*d^3*e^2 + (B*a*b - 5*A*b^2 
 - 10*A*a*c)*d^2*e^3 - (2*B*a^2 - 7*A*a*b)*d*e^4 + (2*B*c^2*d^4*e - (B*b*c 
 + 6*A*c^2)*d^3*e^2 - (B*b^2 + (2*B*a - 9*A*b)*c)*d^2*e^3 + (5*B*a*b - 3*A 
*b^2 - 6*A*a*c)*d*e^4 - (4*B*a^2 - 3*A*a*b)*e^5)*x)*sqrt(c*x^2 + b*x + a)) 
/(c^3*d^8 - 3*b*c^2*d^7*e - 3*a^2*b*d^3*e^5 + a^3*d^2*e^6 + 3*(b^2*c + a*c 
^2)*d^6*e^2 - (b^3 + 6*a*b*c)*d^5*e^3 + 3*(a*b^2 + a^2*c)*d^4*e^4 + (c^3*d 
^6*e^2 - 3*b*c^2*d^5*e^3 - 3*a^2*b*d*e^7 + a^3*e^8 + 3*(b^2*c + a*c^2)*d^4 
*e^4 - (b^3 + 6*a*b*c)*d^3*e^5 + 3*(a*b^2 + a^2*c)*d^2*e^6)*x^2 + 2*(c^3*d 
^7*e - 3*b*c^2*d^6*e^2 - 3*a^2*b*d^2*e^6 + a^3*d*e^7 + 3*(b^2*c + a*c^2)*d 
^5*e^3 - (b^3 + 6*a*b*c)*d^4*e^4 + 3*(a*b^2 + a^2*c)*d^3*e^5)*x), -1/8*((4 
*(B*b*c - 2*A*c^2)*d^4 - (B*b^2 + 4*(3*B*a - 2*A*b)*c)*d^3*e + (4*B*a*b...
 

3.25.71.6 Sympy [F]

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

input

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

output

Integral((A + B*x)/((d + e*x)**3*sqrt(a + b*x + c*x**2)), x)
 

3.25.71.7 Maxima [F(-2)]

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

input

integrate((B*x+A)/(e*x+d)^3/(c*x^2+b*x+a)^(1/2),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(a*e^2-b*d*e>0)', see `assume?` f 
or more de
 

3.25.71.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1508 vs. \(2 (253) = 506\).

Time = 0.33 (sec) , antiderivative size = 1508, normalized size of antiderivative = 5.56 \[ \int \frac {A+B x}{(d+e x)^3 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

input

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

output

-1/4*(4*B*b*c*d^2 - 8*A*c^2*d^2 - B*b^2*d*e - 12*B*a*c*d*e + 8*A*b*c*d*e + 
 4*B*a*b*e^2 - 3*A*b^2*e^2 + 4*A*a*c*e^2)*arctan(-((sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^2*d^4 - 2*b* 
c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*sqrt(-c*d^2 
 + b*d*e - a*e^2)) + 1/4*(4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*b*c*d^ 
2*e^2 - 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*c^2*d^2*e^2 - (sqrt(c)*x 
 - sqrt(c*x^2 + b*x + a))^3*B*b^2*d*e^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^3*B*a*c*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*b*c*d*e^3 
 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a*b*e^4 - 3*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^3*A*b^2*e^4 + 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A 
*a*c*e^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*c^(5/2)*d^4 - 4*(sqrt 
(c)*x - sqrt(c*x^2 + b*x + a))^2*B*b*c^(3/2)*d^3*e - 24*(sqrt(c)*x - sqrt( 
c*x^2 + b*x + a))^2*A*c^(5/2)*d^3*e + 5*(sqrt(c)*x - sqrt(c*x^2 + b*x + a) 
)^2*B*b^2*sqrt(c)*d^2*e^2 - 20*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*c 
^(3/2)*d^2*e^2 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b*c^(3/2)*d^2* 
e^2 - 4*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2*B*a*b*sqrt(c)*d*e^3 - 9*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^2*A*b^2*sqrt(c)*d*e^3 + 12*(sqrt(c)*x - sq 
rt(c*x^2 + b*x + a))^2*A*a*c^(3/2)*d*e^3 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x 
 + a))^2*B*a^2*sqrt(c)*e^4 + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*b*c^2 
*d^4 - 16*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a*c^2*d^3*e - 24*(sqrt(...
 

3.25.71.9 Mupad [F(-1)]

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

input

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

output

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