Integrand size = 24, antiderivative size = 100 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{63 a^2 b (a+b x)^6}-\frac {2 \left (a^2-b^2 x^2\right )^{5/2}}{315 a^3 b (a+b x)^5} \]
-1/9*(-b^2*x^2+a^2)^(5/2)/a/b/(b*x+a)^7-2/63*(-b^2*x^2+a^2)^(5/2)/a^2/b/(b *x+a)^6-2/315*(-b^2*x^2+a^2)^(5/2)/a^3/b/(b*x+a)^5
Time = 0.65 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.60 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {(a-b x)^2 \sqrt {a^2-b^2 x^2} \left (47 a^2+14 a b x+2 b^2 x^2\right )}{315 a^3 b (a+b x)^5} \]
-1/315*((a - b*x)^2*Sqrt[a^2 - b^2*x^2]*(47*a^2 + 14*a*b*x + 2*b^2*x^2))/( a^3*b*(a + b*x)^5)
Time = 0.22 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.08, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {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.
\(\displaystyle \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {2 \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^6}dx}{9 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^5}dx}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}\right )}{9 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {2 \left (-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{35 a^2 b (a+b x)^5}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{7 a b (a+b x)^6}\right )}{9 a}-\frac {\left (a^2-b^2 x^2\right )^{5/2}}{9 a b (a+b x)^7}\) |
-1/9*(a^2 - b^2*x^2)^(5/2)/(a*b*(a + b*x)^7) + (2*(-1/7*(a^2 - b^2*x^2)^(5 /2)/(a*b*(a + b*x)^6) - (a^2 - b^2*x^2)^(5/2)/(35*a^2*b*(a + b*x)^5)))/(9* a)
3.8.97.3.1 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 = 2.78 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (2 b^{2} x^{2}+14 a b x +47 a^{2}\right ) \left (-b^{2} x^{2}+a^{2}\right )^{\frac {3}{2}}}{315 \left (b x +a \right )^{6} a^{3} b}\) | \(55\) |
trager | \(-\frac {\left (2 b^{4} x^{4}+10 a \,b^{3} x^{3}+21 a^{2} b^{2} x^{2}-80 a^{3} b x +47 a^{4}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{315 a^{3} \left (b x +a \right )^{5} b}\) | \(71\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{7}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{6}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {5}{2}}}{35 a^{2} \left (x +\frac {a}{b}\right )^{5}}\right )}{9 a}}{b^{7}}\) | \(145\) |
Time = 0.29 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.70 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {47 \, b^{5} x^{5} + 235 \, a b^{4} x^{4} + 470 \, a^{2} b^{3} x^{3} + 470 \, a^{3} b^{2} x^{2} + 235 \, a^{4} b x + 47 \, a^{5} + {\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} - 80 \, a^{3} b x + 47 \, a^{4}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{6} x^{5} + 5 \, a^{4} b^{5} x^{4} + 10 \, a^{5} b^{4} x^{3} + 10 \, a^{6} b^{3} x^{2} + 5 \, a^{7} b^{2} x + a^{8} b\right )}} \]
-1/315*(47*b^5*x^5 + 235*a*b^4*x^4 + 470*a^2*b^3*x^3 + 470*a^3*b^2*x^2 + 2 35*a^4*b*x + 47*a^5 + (2*b^4*x^4 + 10*a*b^3*x^3 + 21*a^2*b^2*x^2 - 80*a^3* b*x + 47*a^4)*sqrt(-b^2*x^2 + a^2))/(a^3*b^6*x^5 + 5*a^4*b^5*x^4 + 10*a^5* b^4*x^3 + 10*a^6*b^3*x^2 + 5*a^7*b^2*x + a^8*b)
\[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=\int \frac {\left (- \left (- a + b x\right ) \left (a + b x\right )\right )^{\frac {3}{2}}}{\left (a + b x\right )^{7}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (88) = 176\).
Time = 0.20 (sec) , antiderivative size = 342, normalized size of antiderivative = 3.42 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=-\frac {{\left (-b^{2} x^{2} + a^{2}\right )}^{\frac {3}{2}}}{3 \, {\left (b^{7} x^{6} + 6 \, a b^{6} x^{5} + 15 \, a^{2} b^{5} x^{4} + 20 \, a^{3} b^{4} x^{3} + 15 \, a^{4} b^{3} x^{2} + 6 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}} a}{9 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{63 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a b^{4} x^{3} + 3 \, a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} - \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{2} x + a^{4} b\right )}} \]
-1/3*(-b^2*x^2 + a^2)^(3/2)/(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) + 2/9*sqrt(-b^2*x^2 + a ^2)*a/(b^6*x^5 + 5*a*b^5*x^4 + 10*a^2*b^4*x^3 + 10*a^3*b^3*x^2 + 5*a^4*b^2 *x + a^5*b) - 1/63*sqrt(-b^2*x^2 + a^2)/(b^5*x^4 + 4*a*b^4*x^3 + 6*a^2*b^3 *x^2 + 4*a^3*b^2*x + a^4*b) - 1/105*sqrt(-b^2*x^2 + a^2)/(a*b^4*x^3 + 3*a^ 2*b^3*x^2 + 3*a^3*b^2*x + a^4*b) - 2/315*sqrt(-b^2*x^2 + a^2)/(a^2*b^3*x^2 + 2*a^3*b^2*x + a^4*b) - 2/315*sqrt(-b^2*x^2 + a^2)/(a^3*b^2*x + a^4*b)
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (88) = 176\).
Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.89 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=\frac {2 \, {\left (\frac {108 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {1062 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {1638 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {3402 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {2520 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {2310 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {630 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {315 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 47\right )}}{315 \, a^{3} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{9} {\left | b \right |}} \]
2/315*(108*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1062*(a*b + sqrt( -b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 1638*(a*b + sqrt(-b^2*x^2 + a^2)*abs (b))^3/(b^6*x^3) + 3402*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 2520*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 2310*(a*b + sqrt(- b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6) + 630*(a*b + sqrt(-b^2*x^2 + a^2)*abs( b))^7/(b^14*x^7) + 315*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^8/(b^16*x^8) + 47)/(a^3*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^9*abs(b))
Time = 10.44 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.41 \[ \int \frac {\left (a^2-b^2 x^2\right )^{3/2}}{(a+b x)^7} \, dx=\frac {20\,\sqrt {a^2-b^2\,x^2}}{63\,b\,{\left (a+b\,x\right )}^4}-\frac {4\,a\,\sqrt {a^2-b^2\,x^2}}{9\,b\,{\left (a+b\,x\right )}^5}-\frac {\sqrt {a^2-b^2\,x^2}}{105\,a\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^2\,b\,{\left (a+b\,x\right )}^2}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^3\,b\,\left (a+b\,x\right )} \]