\(\int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx\) [111]

Optimal result

Integrand size = 22, antiderivative size = 125 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}-\frac {2 (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{21 b^2 x^5}+\frac {8 c (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{105 b^3 x^4}-\frac {16 c^2 (3 b B-2 A c) \left (b x+c x^2\right )^{3/2}}{315 b^4 x^3} \] Output:

-2/9*A*(c*x^2+b*x)^(3/2)/b/x^6-2/21*(-2*A*c+3*B*b)*(c*x^2+b*x)^(3/2)/b^2/x 
^5+8/105*c*(-2*A*c+3*B*b)*(c*x^2+b*x)^(3/2)/b^3/x^4-16/315*c^2*(-2*A*c+3*B 
*b)*(c*x^2+b*x)^(3/2)/b^4/x^3
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.62 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=-\frac {2 (x (b+c x))^{3/2} \left (3 b B x \left (15 b^2-12 b c x+8 c^2 x^2\right )+A \left (35 b^3-30 b^2 c x+24 b c^2 x^2-16 c^3 x^3\right )\right )}{315 b^4 x^6} \] Input:

Integrate[((A + B*x)*Sqrt[b*x + c*x^2])/x^6,x]
 

Output:

(-2*(x*(b + c*x))^(3/2)*(3*b*B*x*(15*b^2 - 12*b*c*x + 8*c^2*x^2) + A*(35*b 
^3 - 30*b^2*c*x + 24*b*c^2*x^2 - 16*c^3*x^3)))/(315*b^4*x^6)
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {1220, 1129, 1129, 1123}

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.

\(\displaystyle \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(3 b B-2 A c) \int \frac {\sqrt {c x^2+b x}}{x^5}dx}{3 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(3 b B-2 A c) \left (-\frac {4 c \int \frac {\sqrt {c x^2+b x}}{x^4}dx}{7 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}\right )}{3 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}\)

\(\Big \downarrow \) 1129

\(\displaystyle \frac {(3 b B-2 A c) \left (-\frac {4 c \left (-\frac {2 c \int \frac {\sqrt {c x^2+b x}}{x^3}dx}{5 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}\right )}{7 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}\right )}{3 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}\)

\(\Big \downarrow \) 1123

\(\displaystyle \frac {\left (-\frac {4 c \left (\frac {4 c \left (b x+c x^2\right )^{3/2}}{15 b^2 x^3}-\frac {2 \left (b x+c x^2\right )^{3/2}}{5 b x^4}\right )}{7 b}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}\right ) (3 b B-2 A c)}{3 b}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{9 b x^6}\)

Input:

Int[((A + B*x)*Sqrt[b*x + c*x^2])/x^6,x]
 

Output:

(-2*A*(b*x + c*x^2)^(3/2))/(9*b*x^6) + ((3*b*B - 2*A*c)*((-2*(b*x + c*x^2) 
^(3/2))/(7*b*x^5) - (4*c*((-2*(b*x + c*x^2)^(3/2))/(5*b*x^4) + (4*c*(b*x + 
 c*x^2)^(3/2))/(15*b^2*x^3)))/(7*b)))/(3*b)
 

Defintions of rubi rules used

rule 1123

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[e*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(2*c*d - b 
*e))), x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 
0] && EqQ[m + 2*p + 2, 0]
 

rule 1129

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(-e)*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/((m + p + 1)*(2* 
c*d - b*e))), x] + Simp[c*(Simplify[m + 2*p + 2]/((m + p + 1)*(2*c*d - b*e) 
))   Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d 
, e, m, p}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && ILtQ[Simplify[m + 2*p + 
2], 0]
 

rule 1220

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d*g - e*f)*(d + e*x)^m*((a + b*x + c*x 
^2)^(p + 1)/((2*c*d - b*e)*(m + p + 1))), x] + Simp[(m*(g*(c*d - b*e) + c*e 
*f) + e*(p + 1)*(2*c*f - b*g))/(e*(2*c*d - b*e)*(m + p + 1))   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[c*d^2 - b*d*e + a*e^2, 0] && ((LtQ[m, -1] &&  !IGtQ[m + p + 1, 0 
]) || (LtQ[m, 0] && LtQ[p, -1]) || EqQ[m + 2*p + 2, 0]) && NeQ[m + p + 1, 0 
]
 

Maple [A] (verified)

Time = 0.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.56

method	result	size
pseudoelliptic	\(-\frac {2 \left (c x +b \right ) \sqrt {x \left (c x +b \right )}\, \left (\left (\frac {9 B x}{7}+A \right ) b^{3}-\frac {6 c x \left (\frac {6 B x}{5}+A \right ) b^{2}}{7}+\frac {24 c^{2} x^{2} \left (B x +A \right ) b}{35}-\frac {16 A \,c^{3} x^{3}}{35}\right )}{9 x^{5} b^{4}}\)	\(70\)
gosper	\(-\frac {2 \left (c x +b \right ) \left (-16 A \,c^{3} x^{3}+24 x^{3} B b \,c^{2}+24 A b \,c^{2} x^{2}-36 x^{2} B \,b^{2} c -30 A \,b^{2} c x +45 x B \,b^{3}+35 A \,b^{3}\right ) \sqrt {c \,x^{2}+b x}}{315 b^{4} x^{5}}\)	\(86\)
orering	\(-\frac {2 \left (c x +b \right ) \left (-16 A \,c^{3} x^{3}+24 x^{3} B b \,c^{2}+24 A b \,c^{2} x^{2}-36 x^{2} B \,b^{2} c -30 A \,b^{2} c x +45 x B \,b^{3}+35 A \,b^{3}\right ) \sqrt {c \,x^{2}+b x}}{315 b^{4} x^{5}}\)	\(86\)
trager	\(-\frac {2 \left (-16 A \,c^{4} x^{4}+24 B b \,c^{3} x^{4}+8 A b \,c^{3} x^{3}-12 B \,b^{2} c^{2} x^{3}-6 A \,b^{2} c^{2} x^{2}+9 B \,b^{3} c \,x^{2}+5 A \,b^{3} c x +45 B \,b^{4} x +35 A \,b^{4}\right ) \sqrt {c \,x^{2}+b x}}{315 b^{4} x^{5}}\)	\(105\)
risch	\(-\frac {2 \left (c x +b \right ) \left (-16 A \,c^{4} x^{4}+24 B b \,c^{3} x^{4}+8 A b \,c^{3} x^{3}-12 B \,b^{2} c^{2} x^{3}-6 A \,b^{2} c^{2} x^{2}+9 B \,b^{3} c \,x^{2}+5 A \,b^{3} c x +45 B \,b^{4} x +35 A \,b^{4}\right )}{315 x^{4} \sqrt {x \left (c x +b \right )}\, b^{4}}\)	\(108\)
default	\(A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{9 b \,x^{6}}-\frac {2 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )}{3 b}\right )+B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{7 b \,x^{5}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{5 b \,x^{4}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{15 b^{2} x^{3}}\right )}{7 b}\right )\)	\(164\)

Input:

int((B*x+A)*(c*x^2+b*x)^(1/2)/x^6,x,method=_RETURNVERBOSE)
 

Output:

-2/9*(c*x+b)*(x*(c*x+b))^(1/2)*((9/7*B*x+A)*b^3-6/7*c*x*(6/5*B*x+A)*b^2+24 
/35*c^2*x^2*(B*x+A)*b-16/35*A*c^3*x^3)/x^5/b^4
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.84 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=-\frac {2 \, {\left (35 \, A b^{4} + 8 \, {\left (3 \, B b c^{3} - 2 \, A c^{4}\right )} x^{4} - 4 \, {\left (3 \, B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{3} + 3 \, {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{2} + 5 \, {\left (9 \, B b^{4} + A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, b^{4} x^{5}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^6,x, algorithm="fricas")
 

Output:

-2/315*(35*A*b^4 + 8*(3*B*b*c^3 - 2*A*c^4)*x^4 - 4*(3*B*b^2*c^2 - 2*A*b*c^ 
3)*x^3 + 3*(3*B*b^3*c - 2*A*b^2*c^2)*x^2 + 5*(9*B*b^4 + A*b^3*c)*x)*sqrt(c 
*x^2 + b*x)/(b^4*x^5)
 

Sympy [F]

\[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=\int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{6}}\, dx \] Input:

integrate((B*x+A)*(c*x**2+b*x)**(1/2)/x**6,x)
 

Output:

Integral(sqrt(x*(b + c*x))*(A + B*x)/x**6, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=-\frac {16 \, \sqrt {c x^{2} + b x} B c^{3}}{105 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{4}}{315 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{3}}{315 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B c}{35 \, b x^{3}} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{2}}{105 \, b^{2} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{7 \, x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A c}{63 \, b x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{9 \, x^{5}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^6,x, algorithm="maxima")
 

Output:

-16/105*sqrt(c*x^2 + b*x)*B*c^3/(b^3*x) + 32/315*sqrt(c*x^2 + b*x)*A*c^4/( 
b^4*x) + 8/105*sqrt(c*x^2 + b*x)*B*c^2/(b^2*x^2) - 16/315*sqrt(c*x^2 + b*x 
)*A*c^3/(b^3*x^2) - 2/35*sqrt(c*x^2 + b*x)*B*c/(b*x^3) + 4/105*sqrt(c*x^2 
+ b*x)*A*c^2/(b^2*x^3) - 2/7*sqrt(c*x^2 + b*x)*B/x^4 - 2/63*sqrt(c*x^2 + b 
*x)*A*c/(b*x^4) - 2/9*sqrt(c*x^2 + b*x)*A/x^5
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 311 vs. \(2 (109) = 218\).

Time = 0.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.49 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=\frac {2 \, {\left (420 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B c^{2} + 945 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b c^{\frac {3}{2}} + 630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A c^{\frac {5}{2}} + 819 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{2} c + 1764 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b c^{2} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{3} \sqrt {c} + 1995 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{2} c^{\frac {3}{2}} + 45 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{4} + 1125 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{3} c + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{4} \sqrt {c} + 35 \, A b^{5}\right )}}{315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9}} \] Input:

integrate((B*x+A)*(c*x^2+b*x)^(1/2)/x^6,x, algorithm="giac")
 

Output:

2/315*(420*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*B*c^2 + 945*(sqrt(c)*x - sqrt 
(c*x^2 + b*x))^5*B*b*c^(3/2) + 630*(sqrt(c)*x - sqrt(c*x^2 + b*x))^5*A*c^( 
5/2) + 819*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^2*c + 1764*(sqrt(c)*x - s 
qrt(c*x^2 + b*x))^4*A*b*c^2 + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^3* 
sqrt(c) + 1995*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^2*c^(3/2) + 45*(sqrt( 
c)*x - sqrt(c*x^2 + b*x))^2*B*b^4 + 1125*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2 
*A*b^3*c + 315*(sqrt(c)*x - sqrt(c*x^2 + b*x))*A*b^4*sqrt(c) + 35*A*b^5)/( 
sqrt(c)*x - sqrt(c*x^2 + b*x))^9
 

Mupad [B] (verification not implemented)

Time = 5.85 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.54 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=\frac {4\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^2\,x^3}-\frac {2\,B\,\sqrt {c\,x^2+b\,x}}{7\,x^4}-\frac {2\,A\,c\,\sqrt {c\,x^2+b\,x}}{63\,b\,x^4}-\frac {2\,B\,c\,\sqrt {c\,x^2+b\,x}}{35\,b\,x^3}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {16\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^3\,x^2}+\frac {32\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^4\,x}+\frac {8\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^2\,x^2}-\frac {16\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x} \] Input:

int(((b*x + c*x^2)^(1/2)*(A + B*x))/x^6,x)
 

Output:

(4*A*c^2*(b*x + c*x^2)^(1/2))/(105*b^2*x^3) - (2*B*(b*x + c*x^2)^(1/2))/(7 
*x^4) - (2*A*c*(b*x + c*x^2)^(1/2))/(63*b*x^4) - (2*B*c*(b*x + c*x^2)^(1/2 
))/(35*b*x^3) - (2*A*(b*x + c*x^2)^(1/2))/(9*x^5) - (16*A*c^3*(b*x + c*x^2 
)^(1/2))/(315*b^3*x^2) + (32*A*c^4*(b*x + c*x^2)^(1/2))/(315*b^4*x) + (8*B 
*c^2*(b*x + c*x^2)^(1/2))/(105*b^2*x^2) - (16*B*c^3*(b*x + c*x^2)^(1/2))/( 
105*b^3*x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.50 \[ \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^6} \, dx=\frac {-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{4}}{9}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{3} c x}{63}+\frac {4 \sqrt {x}\, \sqrt {c x +b}\, a \,b^{2} c^{2} x^{2}}{105}-\frac {16 \sqrt {x}\, \sqrt {c x +b}\, a b \,c^{3} x^{3}}{315}+\frac {32 \sqrt {x}\, \sqrt {c x +b}\, a \,c^{4} x^{4}}{315}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, b^{5} x}{7}-\frac {2 \sqrt {x}\, \sqrt {c x +b}\, b^{4} c \,x^{2}}{35}+\frac {8 \sqrt {x}\, \sqrt {c x +b}\, b^{3} c^{2} x^{3}}{105}-\frac {16 \sqrt {x}\, \sqrt {c x +b}\, b^{2} c^{3} x^{4}}{105}-\frac {32 \sqrt {c}\, a \,c^{4} x^{5}}{315}+\frac {16 \sqrt {c}\, b^{2} c^{3} x^{5}}{105}}{b^{4} x^{5}} \] Input:

int((B*x+A)*(c*x^2+b*x)^(1/2)/x^6,x)
 

Output:

(2*( - 35*sqrt(x)*sqrt(b + c*x)*a*b**4 - 5*sqrt(x)*sqrt(b + c*x)*a*b**3*c* 
x + 6*sqrt(x)*sqrt(b + c*x)*a*b**2*c**2*x**2 - 8*sqrt(x)*sqrt(b + c*x)*a*b 
*c**3*x**3 + 16*sqrt(x)*sqrt(b + c*x)*a*c**4*x**4 - 45*sqrt(x)*sqrt(b + c* 
x)*b**5*x - 9*sqrt(x)*sqrt(b + c*x)*b**4*c*x**2 + 12*sqrt(x)*sqrt(b + c*x) 
*b**3*c**2*x**3 - 24*sqrt(x)*sqrt(b + c*x)*b**2*c**3*x**4 - 16*sqrt(c)*a*c 
**4*x**5 + 24*sqrt(c)*b**2*c**3*x**5))/(315*b**4*x**5)