Integrand size = 24, antiderivative size = 166 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}-\frac {4 \left (a^2-b^2 x^2\right )^{3/2}}{99 a^2 b (a+b x)^6}-\frac {4 \left (a^2-b^2 x^2\right )^{3/2}}{231 a^3 b (a+b x)^5}-\frac {8 \left (a^2-b^2 x^2\right )^{3/2}}{1155 a^4 b (a+b x)^4}-\frac {8 \left (a^2-b^2 x^2\right )^{3/2}}{3465 a^5 b (a+b x)^3} \]
-1/11*(-b^2*x^2+a^2)^(3/2)/a/b/(b*x+a)^7-4/99*(-b^2*x^2+a^2)^(3/2)/a^2/b/( b*x+a)^6-4/231*(-b^2*x^2+a^2)^(3/2)/a^3/b/(b*x+a)^5-8/1155*(-b^2*x^2+a^2)^ (3/2)/a^4/b/(b*x+a)^4-8/3465*(-b^2*x^2+a^2)^(3/2)/a^5/b/(b*x+a)^3
Time = 0.72 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=\frac {\sqrt {a^2-b^2 x^2} \left (-547 a^5+183 a^4 b x+184 a^3 b^2 x^2+124 a^2 b^3 x^3+48 a b^4 x^4+8 b^5 x^5\right )}{3465 a^5 b (a+b x)^6} \]
(Sqrt[a^2 - b^2*x^2]*(-547*a^5 + 183*a^4*b*x + 184*a^3*b^2*x^2 + 124*a^2*b ^3*x^3 + 48*a*b^4*x^4 + 8*b^5*x^5))/(3465*a^5*b*(a + b*x)^6)
Time = 0.28 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {461, 461, 461, 461, 460}
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.
\(\displaystyle \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6}dx}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \left (\frac {\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5}dx}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \left (\frac {\frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^4}dx}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \left (\frac {\frac {2 \left (\frac {\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3}dx}{5 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4}\right )}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {4 \left (\frac {\frac {2 \left (-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{15 a^2 b (a+b x)^3}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4}\right )}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\right )}{11 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{11 a b (a+b x)^7}\) |
-1/11*(a^2 - b^2*x^2)^(3/2)/(a*b*(a + b*x)^7) + (4*(-1/9*(a^2 - b^2*x^2)^( 3/2)/(a*b*(a + b*x)^6) + (-1/7*(a^2 - b^2*x^2)^(3/2)/(a*b*(a + b*x)^5) + ( 2*(-1/5*(a^2 - b^2*x^2)^(3/2)/(a*b*(a + b*x)^4) - (a^2 - b^2*x^2)^(3/2)/(1 5*a^2*b*(a + b*x)^3)))/(7*a))/(3*a)))/(11*a)
3.8.87.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(b*c*n)), x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && EqQ[n + 2*p + 2, 0]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-d)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*b*c*(n + p + 1))), x] + Simp[Simpl ify[n + 2*p + 2]/(2*c*(n + p + 1)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x ], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && ILtQ[Simp lify[n + 2*p + 2], 0] && (LtQ[n, -1] || GtQ[n + p, 0])
Time = 2.63 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.46
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (8 b^{4} x^{4}+56 a \,b^{3} x^{3}+180 a^{2} b^{2} x^{2}+364 a^{3} b x +547 a^{4}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3465 \left (b x +a \right )^{6} a^{5} b}\) | \(77\) |
trager | \(-\frac {\left (-8 b^{5} x^{5}-48 a \,b^{4} x^{4}-124 a^{2} b^{3} x^{3}-184 a^{3} b^{2} x^{2}-183 a^{4} b x +547 a^{5}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{3465 a^{5} \left (b x +a \right )^{6} b}\) | \(82\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{11 a b \left (x +\frac {a}{b}\right )^{7}}+\frac {4 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{6}}+\frac {b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{5}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{5 a b \left (x +\frac {a}{b}\right )^{4}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{15 a^{2} \left (x +\frac {a}{b}\right )^{3}}\right )}{7 a}\right )}{3 a}\right )}{11 a}}{b^{7}}\) | \(249\) |
-1/3465*(-b*x+a)*(8*b^4*x^4+56*a*b^3*x^3+180*a^2*b^2*x^2+364*a^3*b*x+547*a ^4)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^6/a^5/b
Time = 0.33 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.23 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=-\frac {547 \, b^{6} x^{6} + 3282 \, a b^{5} x^{5} + 8205 \, a^{2} b^{4} x^{4} + 10940 \, a^{3} b^{3} x^{3} + 8205 \, a^{4} b^{2} x^{2} + 3282 \, a^{5} b x + 547 \, a^{6} - {\left (8 \, b^{5} x^{5} + 48 \, a b^{4} x^{4} + 124 \, a^{2} b^{3} x^{3} + 184 \, a^{3} b^{2} x^{2} + 183 \, a^{4} b x - 547 \, a^{5}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{3465 \, {\left (a^{5} b^{7} x^{6} + 6 \, a^{6} b^{6} x^{5} + 15 \, a^{7} b^{5} x^{4} + 20 \, a^{8} b^{4} x^{3} + 15 \, a^{9} b^{3} x^{2} + 6 \, a^{10} b^{2} x + a^{11} b\right )}} \]
-1/3465*(547*b^6*x^6 + 3282*a*b^5*x^5 + 8205*a^2*b^4*x^4 + 10940*a^3*b^3*x ^3 + 8205*a^4*b^2*x^2 + 3282*a^5*b*x + 547*a^6 - (8*b^5*x^5 + 48*a*b^4*x^4 + 124*a^2*b^3*x^3 + 184*a^3*b^2*x^2 + 183*a^4*b*x - 547*a^5)*sqrt(-b^2*x^ 2 + a^2))/(a^5*b^7*x^6 + 6*a^6*b^6*x^5 + 15*a^7*b^5*x^4 + 20*a^8*b^4*x^3 + 15*a^9*b^3*x^2 + 6*a^10*b^2*x + a^11*b)
\[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=\int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{7}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (146) = 292\).
Time = 0.19 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=-\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{11 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{99 \, {\left (a b^{6} x^{5} + 5 \, a^{2} b^{5} x^{4} + 10 \, a^{3} b^{4} x^{3} + 10 \, a^{4} b^{3} x^{2} + 5 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {4 \, \sqrt {-b^{2} x^{2} + a^{2}}}{693 \, {\left (a^{2} b^{5} x^{4} + 4 \, a^{3} b^{4} x^{3} + 6 \, a^{4} b^{3} x^{2} + 4 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {4 \, \sqrt {-b^{2} x^{2} + a^{2}}}{1155 \, {\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {8 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3465 \, {\left (a^{4} b^{3} x^{2} + 2 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {8 \, \sqrt {-b^{2} x^{2} + a^{2}}}{3465 \, {\left (a^{5} b^{2} x + a^{6} b\right )}} \]
-2/11*sqrt(-b^2*x^2 + a^2)/(b^7*x^6 + 6*a*b^6*x^5 + 15*a^2*b^5*x^4 + 20*a^ 3*b^4*x^3 + 15*a^4*b^3*x^2 + 6*a^5*b^2*x + a^6*b) + 1/99*sqrt(-b^2*x^2 + a ^2)/(a*b^6*x^5 + 5*a^2*b^5*x^4 + 10*a^3*b^4*x^3 + 10*a^4*b^3*x^2 + 5*a^5*b ^2*x + a^6*b) + 4/693*sqrt(-b^2*x^2 + a^2)/(a^2*b^5*x^4 + 4*a^3*b^4*x^3 + 6*a^4*b^3*x^2 + 4*a^5*b^2*x + a^6*b) + 4/1155*sqrt(-b^2*x^2 + a^2)/(a^3*b^ 4*x^3 + 3*a^4*b^3*x^2 + 3*a^5*b^2*x + a^6*b) + 8/3465*sqrt(-b^2*x^2 + a^2) /(a^4*b^3*x^2 + 2*a^5*b^2*x + a^6*b) + 8/3465*sqrt(-b^2*x^2 + a^2)/(a^5*b^ 2*x + a^6*b)
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (146) = 292\).
Time = 0.29 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.11 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=\frac {2 \, {\left (\frac {2552 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {16225 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {42900 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {92730 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {122892 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {129822 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {87780 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {47355 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + \frac {13860 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{9}}{b^{18} x^{9}} + \frac {3465 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{10}}{b^{20} x^{10}} + 547\right )}}{3465 \, a^{5} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{11} {\left | b \right |}} \]
2/3465*(2552*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 16225*(a*b + sq rt(-b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 42900*(a*b + sqrt(-b^2*x^2 + a^2) *abs(b))^3/(b^6*x^3) + 92730*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^ 4) + 122892*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 129822*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6) + 87780*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^7/(b^14*x^7) + 47355*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/ (b^16*x^8) + 13860*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^9/(b^18*x^9) + 3465 *(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^10/(b^20*x^10) + 547)/(a^5*((a*b + sq rt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^11*abs(b))
Time = 10.21 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^7} \, dx=\frac {\sqrt {a^2-b^2\,x^2}}{99\,a\,b\,{\left (a+b\,x\right )}^5}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{11\,b\,{\left (a+b\,x\right )}^6}+\frac {4\,\sqrt {a^2-b^2\,x^2}}{693\,a^2\,b\,{\left (a+b\,x\right )}^4}+\frac {4\,\sqrt {a^2-b^2\,x^2}}{1155\,a^3\,b\,{\left (a+b\,x\right )}^3}+\frac {8\,\sqrt {a^2-b^2\,x^2}}{3465\,a^4\,b\,{\left (a+b\,x\right )}^2}+\frac {8\,\sqrt {a^2-b^2\,x^2}}{3465\,a^5\,b\,\left (a+b\,x\right )} \]
(a^2 - b^2*x^2)^(1/2)/(99*a*b*(a + b*x)^5) - (2*(a^2 - b^2*x^2)^(1/2))/(11 *b*(a + b*x)^6) + (4*(a^2 - b^2*x^2)^(1/2))/(693*a^2*b*(a + b*x)^4) + (4*( a^2 - b^2*x^2)^(1/2))/(1155*a^3*b*(a + b*x)^3) + (8*(a^2 - b^2*x^2)^(1/2)) /(3465*a^4*b*(a + b*x)^2) + (8*(a^2 - b^2*x^2)^(1/2))/(3465*a^5*b*(a + b*x ))