Integrand size = 22, antiderivative size = 90 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}-\frac {2 (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{99 b^2 x^8}+\frac {4 c (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{693 b^3 x^7} \]
-2/11*A*(c*x^2+b*x)^(7/2)/b/x^9-2/99*(-4*A*c+11*B*b)*(c*x^2+b*x)^(7/2)/b^2 /x^8+4/693*c*(-4*A*c+11*B*b)*(c*x^2+b*x)^(7/2)/b^3/x^7
Time = 0.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.70 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=-\frac {2 (b+c x) (x (b+c x))^{5/2} \left (63 A b^2+77 b^2 B x-28 A b c x-22 b B c x^2+8 A c^2 x^2\right )}{693 b^3 x^8} \]
(-2*(b + c*x)*(x*(b + c*x))^(5/2)*(63*A*b^2 + 77*b^2*B*x - 28*A*b*c*x - 22 *b*B*c*x^2 + 8*A*c^2*x^2))/(693*b^3*x^8)
Time = 0.23 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1220, 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.
\(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx\) |
\(\Big \downarrow \) 1220 |
\(\displaystyle \frac {(11 b B-4 A c) \int \frac {\left (c x^2+b x\right )^{5/2}}{x^8}dx}{11 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}\) |
\(\Big \downarrow \) 1129 |
\(\displaystyle \frac {(11 b B-4 A c) \left (-\frac {2 c \int \frac {\left (c x^2+b x\right )^{5/2}}{x^7}dx}{9 b}-\frac {2 \left (b x+c x^2\right )^{7/2}}{9 b x^8}\right )}{11 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}\) |
\(\Big \downarrow \) 1123 |
\(\displaystyle \frac {\left (\frac {4 c \left (b x+c x^2\right )^{7/2}}{63 b^2 x^7}-\frac {2 \left (b x+c x^2\right )^{7/2}}{9 b x^8}\right ) (11 b B-4 A c)}{11 b}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}\) |
(-2*A*(b*x + c*x^2)^(7/2))/(11*b*x^9) + ((11*b*B - 4*A*c)*((-2*(b*x + c*x^ 2)^(7/2))/(9*b*x^8) + (4*c*(b*x + c*x^2)^(7/2))/(63*b^2*x^7)))/(11*b)
3.2.5.3.1 Defintions of rubi rules used
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]
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]
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 ]
Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62
method | result | size |
pseudoelliptic | \(-\frac {2 \left (\left (\frac {11 B x}{9}+A \right ) b^{2}-\frac {4 c \left (\frac {11 B x}{14}+A \right ) x b}{9}+\frac {8 A \,c^{2} x^{2}}{63}\right ) \left (c x +b \right )^{3} \sqrt {x \left (c x +b \right )}}{11 x^{6} b^{3}}\) | \(56\) |
gosper | \(-\frac {2 \left (c x +b \right ) \left (8 A \,c^{2} x^{2}-22 B b c \,x^{2}-28 A b c x +77 b^{2} B x +63 A \,b^{2}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{693 x^{8} b^{3}}\) | \(62\) |
default | \(B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )+A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{11 b \,x^{9}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )}{11 b}\right )\) | \(112\) |
trager | \(-\frac {2 \left (8 A \,c^{5} x^{5}-22 B b \,c^{4} x^{5}-4 A b \,c^{4} x^{4}+11 B \,b^{2} c^{3} x^{4}+3 A \,b^{2} c^{3} x^{3}+165 B \,b^{3} c^{2} x^{3}+113 A \,b^{3} c^{2} x^{2}+209 B \,b^{4} c \,x^{2}+161 A \,b^{4} c x +77 B \,b^{5} x +63 A \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{693 b^{3} x^{6}}\) | \(129\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (8 A \,c^{5} x^{5}-22 B b \,c^{4} x^{5}-4 A b \,c^{4} x^{4}+11 B \,b^{2} c^{3} x^{4}+3 A \,b^{2} c^{3} x^{3}+165 B \,b^{3} c^{2} x^{3}+113 A \,b^{3} c^{2} x^{2}+209 B \,b^{4} c \,x^{2}+161 A \,b^{4} c x +77 B \,b^{5} x +63 A \,b^{5}\right )}{693 x^{5} \sqrt {x \left (c x +b \right )}\, b^{3}}\) | \(132\) |
-2/11*((11/9*B*x+A)*b^2-4/9*c*(11/14*B*x+A)*x*b+8/63*A*c^2*x^2)*(c*x+b)^3* (x*(c*x+b))^(1/2)/x^6/b^3
Time = 0.30 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=-\frac {2 \, {\left (63 \, A b^{5} - 2 \, {\left (11 \, B b c^{4} - 4 \, A c^{5}\right )} x^{5} + {\left (11 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} x^{4} + 3 \, {\left (55 \, B b^{3} c^{2} + A b^{2} c^{3}\right )} x^{3} + {\left (209 \, B b^{4} c + 113 \, A b^{3} c^{2}\right )} x^{2} + 7 \, {\left (11 \, B b^{5} + 23 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{693 \, b^{3} x^{6}} \]
-2/693*(63*A*b^5 - 2*(11*B*b*c^4 - 4*A*c^5)*x^5 + (11*B*b^2*c^3 - 4*A*b*c^ 4)*x^4 + 3*(55*B*b^3*c^2 + A*b^2*c^3)*x^3 + (209*B*b^4*c + 113*A*b^3*c^2)* x^2 + 7*(11*B*b^5 + 23*A*b^4*c)*x)*sqrt(c*x^2 + b*x)/(b^3*x^6)
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{9}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (78) = 156\).
Time = 0.19 (sec) , antiderivative size = 304, normalized size of antiderivative = 3.38 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=\frac {4 \, \sqrt {c x^{2} + b x} B c^{4}}{63 \, b^{2} x} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{5}}{693 \, b^{3} x} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{3}}{63 \, b x^{2}} + \frac {8 \, \sqrt {c x^{2} + b x} A c^{4}}{693 \, b^{2} x^{2}} + \frac {\sqrt {c x^{2} + b x} B c^{2}}{42 \, x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{231 \, b x^{3}} - \frac {5 \, \sqrt {c x^{2} + b x} B b c}{252 \, x^{4}} + \frac {5 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, x^{4}} - \frac {5 \, \sqrt {c x^{2} + b x} B b^{2}}{36 \, x^{5}} - \frac {5 \, \sqrt {c x^{2} + b x} A b c}{792 \, x^{5}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{12 \, x^{6}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{88 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{2 \, x^{7}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{24 \, x^{7}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{3 \, x^{8}} \]
4/63*sqrt(c*x^2 + b*x)*B*c^4/(b^2*x) - 16/693*sqrt(c*x^2 + b*x)*A*c^5/(b^3 *x) - 2/63*sqrt(c*x^2 + b*x)*B*c^3/(b*x^2) + 8/693*sqrt(c*x^2 + b*x)*A*c^4 /(b^2*x^2) + 1/42*sqrt(c*x^2 + b*x)*B*c^2/x^3 - 2/231*sqrt(c*x^2 + b*x)*A* c^3/(b*x^3) - 5/252*sqrt(c*x^2 + b*x)*B*b*c/x^4 + 5/693*sqrt(c*x^2 + b*x)* A*c^2/x^4 - 5/36*sqrt(c*x^2 + b*x)*B*b^2/x^5 - 5/792*sqrt(c*x^2 + b*x)*A*b *c/x^5 + 5/12*(c*x^2 + b*x)^(3/2)*B*b/x^6 - 5/88*sqrt(c*x^2 + b*x)*A*b^2/x ^6 - 1/2*(c*x^2 + b*x)^(5/2)*B/x^7 + 5/24*(c*x^2 + b*x)^(3/2)*A*b/x^7 - 1/ 3*(c*x^2 + b*x)^(5/2)*A/x^8
Leaf count of result is larger than twice the leaf count of optimal. 491 vs. \(2 (78) = 156\).
Time = 0.28 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.46 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=\frac {2 \, {\left (693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B c^{\frac {7}{2}} + 3003 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b c^{3} + 924 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A c^{4} + 6237 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{2} c^{\frac {5}{2}} + 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b c^{\frac {7}{2}} + 7623 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{3} c^{2} + 11781 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{2} c^{3} + 5775 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{4} c^{\frac {3}{2}} + 16863 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{3} c^{\frac {5}{2}} + 2673 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{5} c + 15345 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{4} c^{2} + 693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{6} \sqrt {c} + 9009 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{5} c^{\frac {3}{2}} + 77 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{7} + 3311 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{6} c + 693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{7} \sqrt {c} + 63 \, A b^{8}\right )}}{693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \]
2/693*(693*(sqrt(c)*x - sqrt(c*x^2 + b*x))^9*B*c^(7/2) + 3003*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*B*b*c^3 + 924*(sqrt(c)*x - sqrt(c*x^2 + b*x))^8*A*c^ 4 + 6237*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*B*b^2*c^(5/2) + 4851*(sqrt(c)*x - sqrt(c*x^2 + b*x))^7*A*b*c^(7/2) + 7623*(sqrt(c)*x - sqrt(c*x^2 + b*x)) ^6*B*b^3*c^2 + 11781*(sqrt(c)*x - sqrt(c*x^2 + b*x))^6*A*b^2*c^3 + 5775*(s qrt(c)*x - sqrt(c*x^2 + b*x))^5*B*b^4*c^(3/2) + 16863*(sqrt(c)*x - sqrt(c* x^2 + b*x))^5*A*b^3*c^(5/2) + 2673*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*B*b^5 *c + 15345*(sqrt(c)*x - sqrt(c*x^2 + b*x))^4*A*b^4*c^2 + 693*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^6*sqrt(c) + 9009*(sqrt(c)*x - sqrt(c*x^2 + b*x))^ 3*A*b^5*c^(3/2) + 77*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*b^7 + 3311*(sqrt( c)*x - sqrt(c*x^2 + b*x))^2*A*b^6*c + 693*(sqrt(c)*x - sqrt(c*x^2 + b*x))* A*b^7*sqrt(c) + 63*A*b^8)/(sqrt(c)*x - sqrt(c*x^2 + b*x))^11
Time = 12.48 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.60 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^9} \, dx=\frac {8\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^2}-\frac {226\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,x^4}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {10\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{21\,x^3}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^3}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {16\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{693\,b^3\,x}-\frac {2\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{63\,b\,x^2}+\frac {4\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{63\,b^2\,x}-\frac {46\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{99\,x^5}-\frac {38\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{63\,x^4} \]
(8*A*c^4*(b*x + c*x^2)^(1/2))/(693*b^2*x^2) - (226*A*c^2*(b*x + c*x^2)^(1/ 2))/(693*x^4) - (2*B*b^2*(b*x + c*x^2)^(1/2))/(9*x^5) - (10*B*c^2*(b*x + c *x^2)^(1/2))/(21*x^3) - (2*A*c^3*(b*x + c*x^2)^(1/2))/(231*b*x^3) - (2*A*b ^2*(b*x + c*x^2)^(1/2))/(11*x^6) - (16*A*c^5*(b*x + c*x^2)^(1/2))/(693*b^3 *x) - (2*B*c^3*(b*x + c*x^2)^(1/2))/(63*b*x^2) + (4*B*c^4*(b*x + c*x^2)^(1 /2))/(63*b^2*x) - (46*A*b*c*(b*x + c*x^2)^(1/2))/(99*x^5) - (38*B*b*c*(b*x + c*x^2)^(1/2))/(63*x^4)