\(\int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx\) [152]

Optimal result

Integrand size = 23, antiderivative size = 231 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}+\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{24 a^2 x^3}-\frac {\left (35 A b^2-40 a b B-36 a A c\right ) \sqrt {a+b x+c x^2}}{96 a^3 x^2}+\frac {\left (105 A b^3-120 a b^2 B-220 a A b c+128 a^2 B c\right ) \sqrt {a+b x+c x^2}}{192 a^4 x}+\frac {\left (8 a b B \left (5 b^2-12 a c\right )-A \left (35 b^4-120 a b^2 c+48 a^2 c^2\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{128 a^{9/2}} \] Output:

-1/4*A*(c*x^2+b*x+a)^(1/2)/a/x^4+1/24*(7*A*b-8*B*a)*(c*x^2+b*x+a)^(1/2)/a^ 
2/x^3-1/96*(-36*A*a*c+35*A*b^2-40*B*a*b)*(c*x^2+b*x+a)^(1/2)/a^3/x^2+1/192 
*(-220*A*a*b*c+105*A*b^3+128*B*a^2*c-120*B*a*b^2)*(c*x^2+b*x+a)^(1/2)/a^4/ 
x+1/128*(8*a*b*B*(-12*a*c+5*b^2)-A*(48*a^2*c^2-120*a*b^2*c+35*b^4))*arctan 
h(1/2*(b*x+2*a)/a^(1/2)/(c*x^2+b*x+a)^(1/2))/a^(9/2)
 

Mathematica [A] (verified)

Time = 1.63 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {\sqrt {a} \sqrt {a+x (b+c x)} \left (105 A b^3 x^3-16 a^3 (3 A+4 B x)-10 a b x^2 (7 A b+12 b B x+22 A c x)+8 a^2 x (2 B x (5 b+8 c x)+A (7 b+9 c x))\right )}{x^4}+105 A b^4 \text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )-24 a \left (-5 b^3 B-15 A b^2 c+12 a b B c+6 a A c^2\right ) \text {arctanh}\left (\frac {-\sqrt {c} x+\sqrt {a+x (b+c x)}}{\sqrt {a}}\right )}{192 a^{9/2}} \] Input:

Integrate[(A + B*x)/(x^5*Sqrt[a + b*x + c*x^2]),x]
 

Output:

((Sqrt[a]*Sqrt[a + x*(b + c*x)]*(105*A*b^3*x^3 - 16*a^3*(3*A + 4*B*x) - 10 
*a*b*x^2*(7*A*b + 12*b*B*x + 22*A*c*x) + 8*a^2*x*(2*B*x*(5*b + 8*c*x) + A* 
(7*b + 9*c*x))))/x^4 + 105*A*b^4*ArcTanh[(Sqrt[c]*x - Sqrt[a + x*(b + c*x) 
])/Sqrt[a]] - 24*a*(-5*b^3*B - 15*A*b^2*c + 12*a*b*B*c + 6*a*A*c^2)*ArcTan 
h[(-(Sqrt[c]*x) + Sqrt[a + x*(b + c*x)])/Sqrt[a]])/(192*a^(9/2))
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {1237, 27, 1237, 27, 1237, 27, 25, 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.

\(\displaystyle \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {\int \frac {7 A b-8 a B+6 A c x}{2 x^4 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {7 A b-8 a B+6 A c x}{x^4 \sqrt {c x^2+b x+a}}dx}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {\int \frac {35 A b^2-40 a B b-36 a A c+4 (7 A b-8 a B) c x}{2 x^3 \sqrt {c x^2+b x+a}}dx}{3 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {35 A b^2-40 a B b-36 a A c+4 (7 A b-8 a B) c x}{x^3 \sqrt {c x^2+b x+a}}dx}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 1237

\(\displaystyle -\frac {-\frac {-\frac {\int -\frac {8 a B \left (15 b^2-16 a c\right )-5 A \left (21 b^3-44 a b c\right )-2 c \left (35 A b^2-40 a B b-36 a A c\right ) x}{2 x^2 \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{2 a x^2}}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {\int -\frac {105 A b^3-120 a B b^2-220 a A c b+128 a^2 B c+2 c \left (35 A b^2-40 a B b-36 a A c\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{2 a x^2}}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {105 A b^3-120 a B b^2-220 a A c b+128 a^2 B c+2 c \left (35 A b^2-40 a B b-36 a A c\right ) x}{x^2 \sqrt {c x^2+b x+a}}dx}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{2 a x^2}}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 1228

\(\displaystyle -\frac {-\frac {-\frac {\frac {3 \left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \int \frac {1}{x \sqrt {c x^2+b x+a}}dx}{2 a}-\frac {\sqrt {a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{2 a x^2}}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 1154

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \int \frac {1}{4 a-\frac {(2 a+b x)^2}{c x^2+b x+a}}d\frac {2 a+b x}{\sqrt {c x^2+b x+a}}}{a}-\frac {\sqrt {a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{a x}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{2 a x^2}}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\sqrt {a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{a x}-\frac {3 \left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \text {arctanh}\left (\frac {2 a+b x}{2 \sqrt {a} \sqrt {a+b x+c x^2}}\right )}{2 a^{3/2}}}{4 a}-\frac {\sqrt {a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{2 a x^2}}{6 a}-\frac {(7 A b-8 a B) \sqrt {a+b x+c x^2}}{3 a x^3}}{8 a}-\frac {A \sqrt {a+b x+c x^2}}{4 a x^4}\)

Input:

Int[(A + B*x)/(x^5*Sqrt[a + b*x + c*x^2]),x]
 

Output:

-1/4*(A*Sqrt[a + b*x + c*x^2])/(a*x^4) - (-1/3*((7*A*b - 8*a*B)*Sqrt[a + b 
*x + c*x^2])/(a*x^3) - (-1/2*((35*A*b^2 - 40*a*b*B - 36*a*A*c)*Sqrt[a + b* 
x + c*x^2])/(a*x^2) - (-(((105*A*b^3 - 120*a*b^2*B - 220*a*A*b*c + 128*a^2 
*B*c)*Sqrt[a + b*x + c*x^2])/(a*x)) - (3*(8*a*b*B*(5*b^2 - 12*a*c) - A*(35 
*b^4 - 120*a*b^2*c + 48*a^2*c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + 
b*x + c*x^2])])/(2*a^(3/2)))/(4*a))/(6*a))/(8*a)
 

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 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])
 

Maple [A] (verified)

Time = 1.35 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.80

method	result	size
risch	\(-\frac {\sqrt {c \,x^{2}+b x +a}\, \left (220 A a b c \,x^{3}-105 A \,b^{3} x^{3}-128 B \,a^{2} c \,x^{3}+120 B a \,b^{2} x^{3}-72 A \,a^{2} c \,x^{2}+70 A a \,b^{2} x^{2}-80 B \,a^{2} b \,x^{2}-56 A \,a^{2} b x +64 B \,a^{3} x +48 a^{3} A \right )}{192 a^{4} x^{4}}-\frac {\left (48 a^{2} A \,c^{2}-120 A a \,b^{2} c +35 A \,b^{4}+96 a^{2} b B c -40 B a \,b^{3}\right ) \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{128 a^{\frac {9}{2}}}\)	\(185\)
default	\(A \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{4 a \,x^{4}}-\frac {7 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )}{8 a}-\frac {3 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}\right )+B \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{3 a \,x^{3}}-\frac {5 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{2 a \,x^{2}}-\frac {3 b \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{4 a}+\frac {c \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{6 a}-\frac {2 c \left (-\frac {\sqrt {c \,x^{2}+b x +a}}{a x}+\frac {b \ln \left (\frac {2 a +b x +2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{3 a}\right )\)	\(571\)

Input:

int((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/192*(c*x^2+b*x+a)^(1/2)*(220*A*a*b*c*x^3-105*A*b^3*x^3-128*B*a^2*c*x^3+ 
120*B*a*b^2*x^3-72*A*a^2*c*x^2+70*A*a*b^2*x^2-80*B*a^2*b*x^2-56*A*a^2*b*x+ 
64*B*a^3*x+48*A*a^3)/a^4/x^4-1/128*(48*A*a^2*c^2-120*A*a*b^2*c+35*A*b^4+96 
*B*a^2*b*c-40*B*a*b^3)/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/ 
x)
 

Fricas [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.84 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=\left [-\frac {3 \, {\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} c\right )} \sqrt {a} x^{4} \log \left (-\frac {8 \, a b x + {\left (b^{2} + 4 \, a c\right )} x^{2} - 4 \, \sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {a} + 8 \, a^{2}}{x^{2}}\right ) + 4 \, {\left (48 \, A a^{4} + {\left (120 \, B a^{2} b^{2} - 105 \, A a b^{3} - 4 \, {\left (32 \, B a^{3} - 55 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (40 \, B a^{3} b - 35 \, A a^{2} b^{2} + 36 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{768 \, a^{5} x^{4}}, -\frac {3 \, {\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \, {\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} c\right )} \sqrt {-a} x^{4} \arctan \left (\frac {\sqrt {c x^{2} + b x + a} {\left (b x + 2 \, a\right )} \sqrt {-a}}{2 \, {\left (a c x^{2} + a b x + a^{2}\right )}}\right ) + 2 \, {\left (48 \, A a^{4} + {\left (120 \, B a^{2} b^{2} - 105 \, A a b^{3} - 4 \, {\left (32 \, B a^{3} - 55 \, A a^{2} b\right )} c\right )} x^{3} - 2 \, {\left (40 \, B a^{3} b - 35 \, A a^{2} b^{2} + 36 \, A a^{3} c\right )} x^{2} + 8 \, {\left (8 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {c x^{2} + b x + a}}{384 \, a^{5} x^{4}}\right ] \] Input:

integrate((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/768*(3*(40*B*a*b^3 - 35*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - 5*A*a*b 
^2)*c)*sqrt(a)*x^4*log(-(8*a*b*x + (b^2 + 4*a*c)*x^2 - 4*sqrt(c*x^2 + b*x 
+ a)*(b*x + 2*a)*sqrt(a) + 8*a^2)/x^2) + 4*(48*A*a^4 + (120*B*a^2*b^2 - 10 
5*A*a*b^3 - 4*(32*B*a^3 - 55*A*a^2*b)*c)*x^3 - 2*(40*B*a^3*b - 35*A*a^2*b^ 
2 + 36*A*a^3*c)*x^2 + 8*(8*B*a^4 - 7*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a 
^5*x^4), -1/384*(3*(40*B*a*b^3 - 35*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - 
 5*A*a*b^2)*c)*sqrt(-a)*x^4*arctan(1/2*sqrt(c*x^2 + b*x + a)*(b*x + 2*a)*s 
qrt(-a)/(a*c*x^2 + a*b*x + a^2)) + 2*(48*A*a^4 + (120*B*a^2*b^2 - 105*A*a* 
b^3 - 4*(32*B*a^3 - 55*A*a^2*b)*c)*x^3 - 2*(40*B*a^3*b - 35*A*a^2*b^2 + 36 
*A*a^3*c)*x^2 + 8*(8*B*a^4 - 7*A*a^3*b)*x)*sqrt(c*x^2 + b*x + a))/(a^5*x^4 
)]
 

Sympy [F]

\[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{x^{5} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((B*x+A)/x**5/(c*x**2+b*x+a)**(1/2),x)
 

Output:

Integral((A + B*x)/(x**5*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*x+A)/x^5/(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(4*a*c-b^2>0)', see `assume?` for 
 more deta
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 884 vs. \(2 (205) = 410\).

Time = 0.26 (sec) , antiderivative size = 884, normalized size of antiderivative = 3.83 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
 

Output:

-1/64*(40*B*a*b^3 - 35*A*b^4 - 96*B*a^2*b*c + 120*A*a*b^2*c - 48*A*a^2*c^2 
)*arctan(-(sqrt(c)*x - sqrt(c*x^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^4) + 1 
/192*(120*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*B*a*b^3 - 105*(sqrt(c)*x - 
 sqrt(c*x^2 + b*x + a))^7*A*b^4 - 288*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 
7*B*a^2*b*c + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a*b^2*c - 144*(s 
qrt(c)*x - sqrt(c*x^2 + b*x + a))^7*A*a^2*c^2 - 440*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^5*B*a^2*b^3 + 385*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a* 
b^4 + 1056*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*B*a^3*b*c - 1320*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^5*A*a^2*b^2*c + 528*(sqrt(c)*x - sqrt(c*x^2 + 
b*x + a))^5*A*a^3*c^2 + 768*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^4*c^ 
(3/2) + 584*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*B*a^3*b^3 - 511*(sqrt(c) 
*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b^4 - 480*(sqrt(c)*x - sqrt(c*x^2 + b* 
x + a))^3*B*a^4*b*c + 1752*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^3*b^2 
*c + 528*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^4*c^2 + 384*(sqrt(c)*x 
- sqrt(c*x^2 + b*x + a))^2*B*a^4*b^2*sqrt(c) - 1024*(sqrt(c)*x - sqrt(c*x^ 
2 + b*x + a))^2*B*a^5*c^(3/2) + 2048*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 
*A*a^4*b*c^(3/2) - 264*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^4*b^3 + 279 
*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b^4 - 288*(sqrt(c)*x - sqrt(c*x 
^2 + b*x + a))*B*a^5*b*c + 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^4*b 
^2*c - 144*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^5*c^2 - 384*B*a^5*b^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{x^5\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int((A + B*x)/(x^5*(a + b*x + c*x^2)^(1/2)),x)
 

Output:

int((A + B*x)/(x^5*(a + b*x + c*x^2)^(1/2)), x)
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.19 \[ \int \frac {A+B x}{x^5 \sqrt {a+b x+c x^2}} \, dx=\frac {-96 \sqrt {c \,x^{2}+b x +a}\, a^{4}-16 \sqrt {c \,x^{2}+b x +a}\, a^{3} b x +144 \sqrt {c \,x^{2}+b x +a}\, a^{3} c \,x^{2}+20 \sqrt {c \,x^{2}+b x +a}\, a^{2} b^{2} x^{2}-184 \sqrt {c \,x^{2}+b x +a}\, a^{2} b c \,x^{3}-30 \sqrt {c \,x^{2}+b x +a}\, a \,b^{3} x^{3}+144 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a^{2} c^{2} x^{4}-72 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) a \,b^{2} c \,x^{4}-15 \sqrt {a}\, \mathrm {log}\left (2 \sqrt {a}\, \sqrt {c \,x^{2}+b x +a}-2 a -b x \right ) b^{4} x^{4}-144 \sqrt {a}\, \mathrm {log}\left (x \right ) a^{2} c^{2} x^{4}+72 \sqrt {a}\, \mathrm {log}\left (x \right ) a \,b^{2} c \,x^{4}+15 \sqrt {a}\, \mathrm {log}\left (x \right ) b^{4} x^{4}}{384 a^{4} x^{4}} \] Input:

int((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x)
 

Output:

( - 96*sqrt(a + b*x + c*x**2)*a**4 - 16*sqrt(a + b*x + c*x**2)*a**3*b*x + 
144*sqrt(a + b*x + c*x**2)*a**3*c*x**2 + 20*sqrt(a + b*x + c*x**2)*a**2*b* 
*2*x**2 - 184*sqrt(a + b*x + c*x**2)*a**2*b*c*x**3 - 30*sqrt(a + b*x + c*x 
**2)*a*b**3*x**3 + 144*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 2*a 
- b*x)*a**2*c**2*x**4 - 72*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) - 
2*a - b*x)*a*b**2*c*x**4 - 15*sqrt(a)*log(2*sqrt(a)*sqrt(a + b*x + c*x**2) 
 - 2*a - b*x)*b**4*x**4 - 144*sqrt(a)*log(x)*a**2*c**2*x**4 + 72*sqrt(a)*l 
og(x)*a*b**2*c*x**4 + 15*sqrt(a)*log(x)*b**4*x**4)/(384*a**4*x**4)