3.25.72 \(\int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx\) [2472]

3.25.72.1 Optimal result

Integrand size = 27, antiderivative size = 444 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {(B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}-\frac {\left (5 A e (2 c d-b e)-B \left (4 c d^2+e (b d-6 a e)\right )\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )-A e \left (44 c^2 d^2+15 b^2 e^2-4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}} \]

output

-1/16*(b^3*e^2*(5*A*e+B*d)-2*b^2*e*(9*A*c*d*e+3*B*a*e^2+2*B*c*d^2)+4*b*c*( 
-3*A*a*e^3+6*A*c*d^2*e+5*B*a*d*e^2+2*B*c*d^3)-8*c*(A*c*d*(-3*a*e^2+2*c*d^2 
)+a*B*e*(-a*e^2+4*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)^(7/2)+1/3*(-A*e 
+B*d)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^3-1/12*(5*A*e*(-b*e+ 
2*c*d)-B*(4*c*d^2+e*(-6*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)^2+1/24*(B*(8*c^2*d^3+2*c*d*e*(-26*a*e+5*b*d)-3*b*e^2*(-6*a*e+b* 
d))-A*e*(44*c^2*d^2+15*b^2*e^2-4*c*e*(4*a*e+11*b*d)))*(c*x^2+b*x+a)^(1/2)/ 
(a*e^2-b*d*e+c*d^2)^3/(e*x+d)
 

3.25.72.2 Mathematica [A] (verified)

Time = 10.56 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.98 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {2 (B d-A e) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^3}+\frac {\left (4 B c d^2+B e (b d-6 a e)+5 A e (-2 c d+b e)\right ) \sqrt {a+x (b+c x)}}{2 (d+e x)^2}+\frac {\left (B \left (8 c^2 d^3+2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)\right )+A e \left (-44 c^2 d^2-15 b^2 e^2+4 c e (11 b d+4 a e)\right )\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {3 \left (b^3 e^2 (B d+5 A e)-2 b^2 e \left (2 B c d^2+9 A c d e+3 a B e^2\right )+4 b c \left (2 B c d^3+6 A c d^2 e+5 a B d e^2-3 a A e^3\right )+8 c \left (a B e \left (-4 c d^2+a e^2\right )+A c d \left (-2 c d^2+3 a e^2\right )\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}}{6 \left (c d^2+e (-b d+a e)\right )^2} \]

input

Integrate[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
 

output

((2*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e*x 
)^3 + ((4*B*c*d^2 + B*e*(b*d - 6*a*e) + 5*A*e*(-2*c*d + b*e))*Sqrt[a + x*( 
b + c*x)])/(2*(d + e*x)^2) + ((B*(8*c^2*d^3 + 2*c*d*e*(5*b*d - 26*a*e) - 3 
*b*e^2*(b*d - 6*a*e)) + A*e*(-44*c^2*d^2 - 15*b^2*e^2 + 4*c*e*(11*b*d + 4* 
a*e)))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e*x)) + ( 
3*(b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9*A*c*d*e + 3*a*B*e^2) + 4 
*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^3) + 8*c*(a*B*e*(-4* 
c*d^2 + a*e^2) + A*c*d*(-2*c*d^2 + 3*a*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))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)
 

3.25.72.3 Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1237, 27, 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.

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

\(\Big \downarrow \) 1237

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1237

\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\int \frac {3 e (B d+5 A e) b^2-2 \left (4 B c d^2+17 A c e d+9 a B e^2\right ) b+8 c \left (3 A c d^2+5 a B e d-2 a A e^2\right )+2 c \left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) x}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\int \frac {3 e (B d+5 A e) b^2-2 \left (4 B c d^2+17 A c e d+9 a B e^2\right ) b+8 c \left (3 A c d^2+5 a B e d-2 a A e^2\right )+2 c \left (4 B c d^2+B e (b d-6 a e)-5 A e (2 c d-b e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1228

\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1154

\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\frac {3 \left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}+\frac {\sqrt {a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {\frac {\sqrt {a+b x+c x^2} \left (B \left (2 c d e (5 b d-26 a e)-3 b e^2 (b d-6 a e)+8 c^2 d^3\right )-A e \left (-4 c e (4 a e+11 b d)+15 b^2 e^2+44 c^2 d^2\right )\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac {3 \left (-2 b^2 e \left (3 a B e^2+9 A c d e+2 B c d^2\right )+4 b c \left (-3 a A e^3+5 a B d e^2+6 A c d^2 e+2 B c d^3\right )-8 c \left (A c d \left (2 c d^2-3 a e^2\right )+a B e \left (4 c d^2-a e^2\right )\right )+b^3 e^2 (5 A e+B d)\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (B e (b d-6 a e)-5 A e (2 c d-b e)+4 B c d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}\)

input

Int[(A + B*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
 

output

((B*d - A*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^3 
) - (-1/2*((4*B*c*d^2 + B*e*(b*d - 6*a*e) - 5*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)^2) - (((B*(8*c^2*d^3 + 2* 
c*d*e*(5*b*d - 26*a*e) - 3*b*e^2*(b*d - 6*a*e)) - A*e*(44*c^2*d^2 + 15*b^2 
*e^2 - 4*c*e*(11*b*d + 4*a*e)))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a 
*e^2)*(d + e*x)) - (3*(b^3*e^2*(B*d + 5*A*e) - 2*b^2*e*(2*B*c*d^2 + 9*A*c* 
d*e + 3*a*B*e^2) + 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 5*a*B*d*e^2 - 3*a*A*e^ 
3) - 8*c*(A*c*d*(2*c*d^2 - 3*a*e^2) + a*B*e*(4*c*d^2 - a*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)) 
)/(6*(c*d^2 - b*d*e + a*e^2))
 

3.25.72.3.1 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])
 

3.25.72.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1535\) vs. \(2(422)=844\).

Time = 0.77 (sec) , antiderivative size = 1536, normalized size of antiderivative = 3.46

method	result	size
default	\(\text {Expression too large to display}\)	\(1536\)

input

int((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

output

B/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)))+(A*e-B*d)/e^5*(-1/3/(a*e^2-b*d 
*e+c*d^2)*e^2/(x+d/e)^3*((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)-5/6*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(-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+...
 

3.25.72.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1821 vs. \(2 (422) = 844\).

Time = 54.93 (sec) , antiderivative size = 3684, normalized size of antiderivative = 8.30 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

input

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

output

[-1/96*(3*(8*(B*b*c^2 - 2*A*c^3)*d^6 - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2) 
*d^5*e + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^4*e^2 - (6*B*a* 
b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^3*e^3 + (8*(B*b*c^2 - 2*A*c^3)* 
d^3*e^3 - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^2*e^4 + (B*b^3 + 24*A*a*c^ 
2 + 2*(10*B*a*b - 9*A*b^2)*c)*d*e^5 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 
3*A*a*b)*c)*e^6)*x^3 + 3*(8*(B*b*c^2 - 2*A*c^3)*d^4*e^2 - 4*(B*b^2*c + 2*( 
4*B*a - 3*A*b)*c^2)*d^3*e^3 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2) 
*c)*d^2*e^4 - (6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d*e^5)*x^2 + 
 3*(8*(B*b*c^2 - 2*A*c^3)*d^5*e - 4*(B*b^2*c + 2*(4*B*a - 3*A*b)*c^2)*d^4* 
e^2 + (B*b^3 + 24*A*a*c^2 + 2*(10*B*a*b - 9*A*b^2)*c)*d^3*e^3 - (6*B*a*b^2 
 - 5*A*b^3 - 4*(2*B*a^2 - 3*A*a*b)*c)*d^2*e^4)*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*(24*B*c^3*d^7 - 8*A*a^3*e 
^7 - 36*(B*b*c^2 + 2*A*c^3)*d^6*e + (15*B*b^2*c - 2*(8*B*a - 81*A*b)*c^2)* 
d^5*e^2 - (3*B*b^3 + 92*A*a*c^2 - (44*B*a*b - 123*A*b^2)*c)*d^4*e^3 - (13* 
B*a*b^2 - 33*A*b^3 + 4*(11*B*a^2 - 34*A*a*b)*c)*d^3*e^4 + (20*B*a^2*b - 59 
*A*a*b^2 - 28*A*a^2*c)*d^2*e^5 - 2*(2*B*a^3 - 17*A*a^2*b)*d*e^6 + (8*B*c^3 
*d^5*e^2 + 2*(B*b*c^2 - 22*A*c^3)*d^4*e^3 - (13*B*b^2*c + 44*(B*a - 2*A...
 

3.25.72.6 Sympy [F]

\[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{4} \sqrt {a + b x + c x^{2}}}\, dx \]

input

integrate((B*x+A)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)
 

output

Integral((A + B*x)/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)
 

3.25.72.7 Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \]

input

integrate((B*x+A)/(e*x+d)^4/(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
 

3.25.72.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4368 vs. \(2 (422) = 844\).

Time = 0.37 (sec) , antiderivative size = 4368, normalized size of antiderivative = 9.84 \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]

input

integrate((B*x+A)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

output

-1/8*(8*B*b*c^2*d^3 - 16*A*c^3*d^3 - 4*B*b^2*c*d^2*e - 32*B*a*c^2*d^2*e + 
24*A*b*c^2*d^2*e + B*b^3*d*e^2 + 20*B*a*b*c*d*e^2 - 18*A*b^2*c*d*e^2 + 24* 
A*a*c^2*d*e^2 - 6*B*a*b^2*e^3 + 5*A*b^3*e^3 + 8*B*a^2*c*e^3 - 12*A*a*b*c*e 
^3)*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^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2* 
d^4*e^2 - b^3*d^3*e^3 - 6*a*b*c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^ 
4 - 3*a^2*b*d*e^5 + a^3*e^6)*sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/24*(24*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^5*B*b*c^2*d^3*e^3 - 48*(sqrt(c)*x - sqrt(c 
*x^2 + b*x + a))^5*A*c^3*d^3*e^3 - 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
5*B*b^2*c*d^2*e^4 - 96*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*c^2*d^2*e 
^4 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b*c^2*d^2*e^4 + 3*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^5*B*b^3*d*e^5 + 60*(sqrt(c)*x - sqrt(c*x^2 + b 
*x + a))^5*B*a*b*c*d*e^5 - 54*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^2* 
c*d*e^5 + 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*c^2*d*e^5 - 18*(sqr 
t(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a*b^2*e^6 + 15*(sqrt(c)*x - sqrt(c*x^2 
 + b*x + a))^5*A*b^3*e^6 + 24*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^2* 
c*e^6 - 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*b*c*e^6 + 120*(sqrt(c 
)*x - sqrt(c*x^2 + b*x + a))^4*B*b*c^(5/2)*d^4*e^2 - 240*(sqrt(c)*x - sqrt 
(c*x^2 + b*x + a))^4*A*c^(7/2)*d^4*e^2 - 60*(sqrt(c)*x - sqrt(c*x^2 + b*x 
+ a))^4*B*b^2*c^(3/2)*d^3*e^3 - 480*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))...
 

3.25.72.9 Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

input

int((A + B*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)),x)
 

output

int((A + B*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)), x)