Integrand size = 26, antiderivative size = 133 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=-\frac {A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}-\frac {(3 b B-2 A c) \left (b x^2+c x^4\right )^{3/2}}{21 b^2 x^{10}}+\frac {4 c (3 b B-2 A c) \left (b x^2+c x^4\right )^{3/2}}{105 b^3 x^8}-\frac {8 c^2 (3 b B-2 A c) \left (b x^2+c x^4\right )^{3/2}}{315 b^4 x^6} \]
-1/9*A*(c*x^4+b*x^2)^(3/2)/b/x^12-1/21*(-2*A*c+3*B*b)*(c*x^4+b*x^2)^(3/2)/ b^2/x^10+4/105*c*(-2*A*c+3*B*b)*(c*x^4+b*x^2)^(3/2)/b^3/x^8-8/315*c^2*(-2* A*c+3*B*b)*(c*x^4+b*x^2)^(3/2)/b^4/x^6
Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=-\frac {\left (x^2 \left (b+c x^2\right )\right )^{3/2} \left (3 b B x^2 \left (15 b^2-12 b c x^2+8 c^2 x^4\right )+A \left (35 b^3-30 b^2 c x^2+24 b c^2 x^4-16 c^3 x^6\right )\right )}{315 b^4 x^{12}} \]
-1/315*((x^2*(b + c*x^2))^(3/2)*(3*b*B*x^2*(15*b^2 - 12*b*c*x^2 + 8*c^2*x^ 4) + A*(35*b^3 - 30*b^2*c*x^2 + 24*b*c^2*x^4 - 16*c^3*x^6)))/(b^4*x^12)
Time = 0.29 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {1940, 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.
| \(\displaystyle \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx\) |
| \(\Big \downarrow \) 1940 |
| \(\displaystyle \frac {1}{2} \int \frac {\left (B x^2+A\right ) \sqrt {c x^4+b x^2}}{x^{12}}dx^2\) |
| \(\Big \downarrow \) 1220 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 b B-2 A c) \int \frac {\sqrt {c x^4+b x^2}}{x^{10}}dx^2}{3 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 b B-2 A c) \left (-\frac {4 c \int \frac {\sqrt {c x^4+b x^2}}{x^8}dx^2}{7 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}\right )}{3 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )\) |
| \(\Big \downarrow \) 1129 |
| \(\displaystyle \frac {1}{2} \left (\frac {(3 b B-2 A c) \left (-\frac {4 c \left (-\frac {2 c \int \frac {\sqrt {c x^4+b x^2}}{x^6}dx^2}{5 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}\right )}{7 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}\right )}{3 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )\) |
| \(\Big \downarrow \) 1123 |
| \(\displaystyle \frac {1}{2} \left (\frac {\left (-\frac {4 c \left (\frac {4 c \left (b x^2+c x^4\right )^{3/2}}{15 b^2 x^6}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{5 b x^8}\right )}{7 b}-\frac {2 \left (b x^2+c x^4\right )^{3/2}}{7 b x^{10}}\right ) (3 b B-2 A c)}{3 b}-\frac {2 A \left (b x^2+c x^4\right )^{3/2}}{9 b x^{12}}\right )\) |
((-2*A*(b*x^2 + c*x^4)^(3/2))/(9*b*x^12) + ((3*b*B - 2*A*c)*((-2*(b*x^2 + c*x^4)^(3/2))/(7*b*x^10) - (4*c*((-2*(b*x^2 + c*x^4)^(3/2))/(5*b*x^8) + (4 *c*(b*x^2 + c*x^4)^(3/2))/(15*b^2*x^6)))/(7*b)))/(3*b))/2
3.1.98.3.1 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 ]
Int[(x_)^(m_.)*((b_.)*(x_)^(k_.) + (a_.)*(x_)^(j_))^(p_)*((c_) + (d_.)*(x_) ^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1) *(a*x^Simplify[j/n] + b*x^Simplify[k/n])^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, j, k, m, n, p, q}, x] && !IntegerQ[p] && NeQ[k, j] && I ntegerQ[Simplify[j/n]] && IntegerQ[Simplify[k/n]] && IntegerQ[Simplify[(m + 1)/n]] && NeQ[n^2, 1]
Time = 1.80 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.63
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\left (\frac {9 x^{2} B}{7}+A \right ) b^{3}-\frac {6 x^{2} c \left (\frac {6 x^{2} B}{5}+A \right ) b^{2}}{7}+\frac {24 c^{2} x^{4} \left (x^{2} B +A \right ) b}{35}-\frac {16 A \,c^{3} x^{6}}{35}\right ) \sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (c \,x^{2}+b \right )}{9 b^{4} x^{10}}\) | \(84\) |
| gosper | \(-\frac {\left (c \,x^{2}+b \right ) \left (-16 A \,c^{3} x^{6}+24 x^{6} B b \,c^{2}+24 A b \,c^{2} x^{4}-36 x^{4} B \,b^{2} c -30 A \,b^{2} c \,x^{2}+45 b^{3} B \,x^{2}+35 b^{3} A \right ) \sqrt {x^{4} c +b \,x^{2}}}{315 b^{4} x^{10}}\) | \(94\) |
| default | \(-\frac {\left (c \,x^{2}+b \right ) \left (-16 A \,c^{3} x^{6}+24 x^{6} B b \,c^{2}+24 A b \,c^{2} x^{4}-36 x^{4} B \,b^{2} c -30 A \,b^{2} c \,x^{2}+45 b^{3} B \,x^{2}+35 b^{3} A \right ) \sqrt {x^{4} c +b \,x^{2}}}{315 b^{4} x^{10}}\) | \(94\) |
| trager | \(-\frac {\left (-16 A \,x^{8} c^{4}+24 B \,x^{8} b \,c^{3}+8 A \,x^{6} b \,c^{3}-12 B \,x^{6} b^{2} c^{2}-6 A \,b^{2} c^{2} x^{4}+9 B \,b^{3} c \,x^{4}+5 A \,x^{2} b^{3} c +45 B \,x^{2} b^{4}+35 A \,b^{4}\right ) \sqrt {x^{4} c +b \,x^{2}}}{315 b^{4} x^{10}}\) | \(111\) |
| risch | \(-\frac {\sqrt {x^{2} \left (c \,x^{2}+b \right )}\, \left (-16 A \,x^{8} c^{4}+24 B \,x^{8} b \,c^{3}+8 A \,x^{6} b \,c^{3}-12 B \,x^{6} b^{2} c^{2}-6 A \,b^{2} c^{2} x^{4}+9 B \,b^{3} c \,x^{4}+5 A \,x^{2} b^{3} c +45 B \,x^{2} b^{4}+35 A \,b^{4}\right )}{315 x^{10} b^{4}}\) | \(111\) |
-1/9*((9/7*x^2*B+A)*b^3-6/7*x^2*c*(6/5*x^2*B+A)*b^2+24/35*c^2*x^4*(B*x^2+A )*b-16/35*A*c^3*x^6)*(x^2*(c*x^2+b))^(1/2)*(c*x^2+b)/b^4/x^10
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.82 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=-\frac {{\left (8 \, {\left (3 \, B b c^{3} - 2 \, A c^{4}\right )} x^{8} - 4 \, {\left (3 \, B b^{2} c^{2} - 2 \, A b c^{3}\right )} x^{6} + 35 \, A b^{4} + 3 \, {\left (3 \, B b^{3} c - 2 \, A b^{2} c^{2}\right )} x^{4} + 5 \, {\left (9 \, B b^{4} + A b^{3} c\right )} x^{2}\right )} \sqrt {c x^{4} + b x^{2}}}{315 \, b^{4} x^{10}} \]
-1/315*(8*(3*B*b*c^3 - 2*A*c^4)*x^8 - 4*(3*B*b^2*c^2 - 2*A*b*c^3)*x^6 + 35 *A*b^4 + 3*(3*B*b^3*c - 2*A*b^2*c^2)*x^4 + 5*(9*B*b^4 + A*b^3*c)*x^2)*sqrt (c*x^4 + b*x^2)/(b^4*x^10)
\[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=\int \frac {\sqrt {x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )}{x^{11}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.57 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=-\frac {1}{105} \, B {\left (\frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{2}} - \frac {4 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{4}} + \frac {3 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{6}} + \frac {15 \, \sqrt {c x^{4} + b x^{2}}}{x^{8}}\right )} + \frac {1}{315} \, A {\left (\frac {16 \, \sqrt {c x^{4} + b x^{2}} c^{4}}{b^{4} x^{2}} - \frac {8 \, \sqrt {c x^{4} + b x^{2}} c^{3}}{b^{3} x^{4}} + \frac {6 \, \sqrt {c x^{4} + b x^{2}} c^{2}}{b^{2} x^{6}} - \frac {5 \, \sqrt {c x^{4} + b x^{2}} c}{b x^{8}} - \frac {35 \, \sqrt {c x^{4} + b x^{2}}}{x^{10}}\right )} \]
-1/105*B*(8*sqrt(c*x^4 + b*x^2)*c^3/(b^3*x^2) - 4*sqrt(c*x^4 + b*x^2)*c^2/ (b^2*x^4) + 3*sqrt(c*x^4 + b*x^2)*c/(b*x^6) + 15*sqrt(c*x^4 + b*x^2)/x^8) + 1/315*A*(16*sqrt(c*x^4 + b*x^2)*c^4/(b^4*x^2) - 8*sqrt(c*x^4 + b*x^2)*c^ 3/(b^3*x^4) + 6*sqrt(c*x^4 + b*x^2)*c^2/(b^2*x^6) - 5*sqrt(c*x^4 + b*x^2)* c/(b*x^8) - 35*sqrt(c*x^4 + b*x^2)/x^10)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (117) = 234\).
Time = 1.40 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.78 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=\frac {16 \, {\left (210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{12} B c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} B b c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 630 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{10} A c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} B b^{2} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 378 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{8} A b c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 42 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} B b^{3} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 168 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{6} A b^{2} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 108 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} B b^{4} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 72 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{4} A b^{3} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) - 27 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} B b^{5} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) + 18 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} A b^{4} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right ) + 3 \, B b^{6} c^{\frac {7}{2}} \mathrm {sgn}\left (x\right ) - 2 \, A b^{5} c^{\frac {9}{2}} \mathrm {sgn}\left (x\right )\right )}}{315 \, {\left ({\left (\sqrt {c} x - \sqrt {c x^{2} + b}\right )}^{2} - b\right )}^{9}} \]
16/315*(210*(sqrt(c)*x - sqrt(c*x^2 + b))^12*B*c^(7/2)*sgn(x) - 315*(sqrt( c)*x - sqrt(c*x^2 + b))^10*B*b*c^(7/2)*sgn(x) + 630*(sqrt(c)*x - sqrt(c*x^ 2 + b))^10*A*c^(9/2)*sgn(x) + 63*(sqrt(c)*x - sqrt(c*x^2 + b))^8*B*b^2*c^( 7/2)*sgn(x) + 378*(sqrt(c)*x - sqrt(c*x^2 + b))^8*A*b*c^(9/2)*sgn(x) - 42* (sqrt(c)*x - sqrt(c*x^2 + b))^6*B*b^3*c^(7/2)*sgn(x) + 168*(sqrt(c)*x - sq rt(c*x^2 + b))^6*A*b^2*c^(9/2)*sgn(x) + 108*(sqrt(c)*x - sqrt(c*x^2 + b))^ 4*B*b^4*c^(7/2)*sgn(x) - 72*(sqrt(c)*x - sqrt(c*x^2 + b))^4*A*b^3*c^(9/2)* sgn(x) - 27*(sqrt(c)*x - sqrt(c*x^2 + b))^2*B*b^5*c^(7/2)*sgn(x) + 18*(sqr t(c)*x - sqrt(c*x^2 + b))^2*A*b^4*c^(9/2)*sgn(x) + 3*B*b^6*c^(7/2)*sgn(x) - 2*A*b^5*c^(9/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 - b)^9
Time = 9.76 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.58 \[ \int \frac {\left (A+B x^2\right ) \sqrt {b x^2+c x^4}}{x^{11}} \, dx=\frac {2\,A\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b^2\,x^6}-\frac {B\,\sqrt {c\,x^4+b\,x^2}}{7\,x^8}-\frac {A\,c\,\sqrt {c\,x^4+b\,x^2}}{63\,b\,x^8}-\frac {B\,c\,\sqrt {c\,x^4+b\,x^2}}{35\,b\,x^6}-\frac {A\,\sqrt {c\,x^4+b\,x^2}}{9\,x^{10}}-\frac {8\,A\,c^3\,\sqrt {c\,x^4+b\,x^2}}{315\,b^3\,x^4}+\frac {16\,A\,c^4\,\sqrt {c\,x^4+b\,x^2}}{315\,b^4\,x^2}+\frac {4\,B\,c^2\,\sqrt {c\,x^4+b\,x^2}}{105\,b^2\,x^4}-\frac {8\,B\,c^3\,\sqrt {c\,x^4+b\,x^2}}{105\,b^3\,x^2} \]
(2*A*c^2*(b*x^2 + c*x^4)^(1/2))/(105*b^2*x^6) - (B*(b*x^2 + c*x^4)^(1/2))/ (7*x^8) - (A*c*(b*x^2 + c*x^4)^(1/2))/(63*b*x^8) - (B*c*(b*x^2 + c*x^4)^(1 /2))/(35*b*x^6) - (A*(b*x^2 + c*x^4)^(1/2))/(9*x^10) - (8*A*c^3*(b*x^2 + c *x^4)^(1/2))/(315*b^3*x^4) + (16*A*c^4*(b*x^2 + c*x^4)^(1/2))/(315*b^4*x^2 ) + (4*B*c^2*(b*x^2 + c*x^4)^(1/2))/(105*b^2*x^4) - (8*B*c^3*(b*x^2 + c*x^ 4)^(1/2))/(105*b^3*x^2)