Integrand size = 24, antiderivative size = 133 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{21 a^2 b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^4}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{315 a^4 b (a+b x)^3} \] Output:
-1/9*(-b^2*x^2+a^2)^(3/2)/a/b/(b*x+a)^6-1/21*(-b^2*x^2+a^2)^(3/2)/a^2/b/(b *x+a)^5-2/105*(-b^2*x^2+a^2)^(3/2)/a^3/b/(b*x+a)^4-2/315*(-b^2*x^2+a^2)^(3 /2)/a^4/b/(b*x+a)^3
Time = 0.48 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=\frac {\sqrt {a^2-b^2 x^2} \left (-58 a^4+25 a^3 b x+21 a^2 b^2 x^2+10 a b^3 x^3+2 b^4 x^4\right )}{315 a^4 b (a+b x)^5} \] Input:
Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^6,x]
Output:
(Sqrt[a^2 - b^2*x^2]*(-58*a^4 + 25*a^3*b*x + 21*a^2*b^2*x^2 + 10*a*b^3*x^3 + 2*b^4*x^4))/(315*a^4*b*(a + b*x)^5)
Time = 0.38 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {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.
\(\displaystyle \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5}dx}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^4}dx}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\frac {2 \left (\frac {\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3}dx}{5 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4}\right )}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {\frac {2 \left (-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{15 a^2 b (a+b x)^3}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4}\right )}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}}{3 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{9 a b (a+b x)^6}\) |
Input:
Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^6,x]
Output:
-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)/(15*a^2*b*(a + b*x)^3)))/(7*a))/(3*a)
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 = 0.37 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.50
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (2 b^{3} x^{3}+12 a \,b^{2} x^{2}+33 a^{2} b x +58 a^{3}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{315 \left (b x +a \right )^{5} a^{4} b}\) | \(66\) |
orering | \(-\frac {\left (-b x +a \right ) \left (2 b^{3} x^{3}+12 a \,b^{2} x^{2}+33 a^{2} b x +58 a^{3}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{315 \left (b x +a \right )^{5} a^{4} b}\) | \(66\) |
trager | \(-\frac {\left (-2 b^{4} x^{4}-10 a \,b^{3} x^{3}-21 a^{2} b^{2} x^{2}-25 a^{3} b x +58 a^{4}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{315 a^{4} \left (b x +a \right )^{5} b}\) | \(71\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{9 a b \left (x +\frac {a}{b}\right )^{6}}+\frac {b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{5}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{5 a b \left (x +\frac {a}{b}\right )^{4}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{15 a^{2} \left (x +\frac {a}{b}\right )^{3}}\right )}{7 a}\right )}{3 a}}{b^{6}}\) | \(197\) |
Input:
int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^6,x,method=_RETURNVERBOSE)
Output:
-1/315*(-b*x+a)*(2*b^3*x^3+12*a*b^2*x^2+33*a^2*b*x+58*a^3)*(-b^2*x^2+a^2)^ (1/2)/(b*x+a)^5/a^4/b
Time = 0.09 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.29 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=-\frac {58 \, b^{5} x^{5} + 290 \, a b^{4} x^{4} + 580 \, a^{2} b^{3} x^{3} + 580 \, a^{3} b^{2} x^{2} + 290 \, a^{4} b x + 58 \, a^{5} - {\left (2 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 21 \, a^{2} b^{2} x^{2} + 25 \, a^{3} b x - 58 \, a^{4}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{4} b^{6} x^{5} + 5 \, a^{5} b^{5} x^{4} + 10 \, a^{6} b^{4} x^{3} + 10 \, a^{7} b^{3} x^{2} + 5 \, a^{8} b^{2} x + a^{9} b\right )}} \] Input:
integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^6,x, algorithm="fricas")
Output:
-1/315*(58*b^5*x^5 + 290*a*b^4*x^4 + 580*a^2*b^3*x^3 + 580*a^3*b^2*x^2 + 2 90*a^4*b*x + 58*a^5 - (2*b^4*x^4 + 10*a*b^3*x^3 + 21*a^2*b^2*x^2 + 25*a^3* b*x - 58*a^4)*sqrt(-b^2*x^2 + a^2))/(a^4*b^6*x^5 + 5*a^5*b^5*x^4 + 10*a^6* b^4*x^3 + 10*a^7*b^3*x^2 + 5*a^8*b^2*x + a^9*b)
\[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=\int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{6}}\, dx \] Input:
integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**6,x)
Output:
Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**6, x)
Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (117) = 234\).
Time = 0.04 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.98 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=-\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{9 \, {\left (b^{6} x^{5} + 5 \, a b^{5} x^{4} + 10 \, a^{2} b^{4} x^{3} + 10 \, a^{3} b^{3} x^{2} + 5 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{63 \, {\left (a b^{5} x^{4} + 4 \, a^{2} b^{4} x^{3} + 6 \, a^{3} b^{3} x^{2} + 4 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{2} b^{4} x^{3} + 3 \, a^{3} b^{3} x^{2} + 3 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{3} b^{3} x^{2} + 2 \, a^{4} b^{2} x + a^{5} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{315 \, {\left (a^{4} b^{2} x + a^{5} b\right )}} \] Input:
integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^6,x, algorithm="maxima")
Output:
-2/9*sqrt(-b^2*x^2 + a^2)/(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)/(a*b^5*x^4 + 4 *a^2*b^4*x^3 + 6*a^3*b^3*x^2 + 4*a^4*b^2*x + a^5*b) + 1/105*sqrt(-b^2*x^2 + a^2)/(a^2*b^4*x^3 + 3*a^3*b^3*x^2 + 3*a^4*b^2*x + a^5*b) + 2/315*sqrt(-b ^2*x^2 + a^2)/(a^3*b^3*x^2 + 2*a^4*b^2*x + a^5*b) + 2/315*sqrt(-b^2*x^2 + a^2)/(a^4*b^2*x + a^5*b)
Leaf count of result is larger than twice the leaf count of optimal. 289 vs. \(2 (117) = 234\).
Time = 0.14 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.17 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=\frac {2 \, {\left (\frac {207 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}}{b^{2} x} + \frac {1143 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{2}}{b^{4} x^{2}} + \frac {2247 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{3}}{b^{6} x^{3}} + \frac {3843 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{4}}{b^{8} x^{4}} + \frac {3465 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{5}}{b^{10} x^{5}} + \frac {2625 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{6}}{b^{12} x^{6}} + \frac {945 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{7}}{b^{14} x^{7}} + \frac {315 \, {\left (a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}\right )}^{8}}{b^{16} x^{8}} + 58\right )}}{315 \, a^{4} {\left (\frac {a b + \sqrt {-b^{2} x^{2} + a^{2}} {\left | b \right |}}{b^{2} x} + 1\right )}^{9} {\left | b \right |}} \] Input:
integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^6,x, algorithm="giac")
Output:
2/315*(207*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1143*(a*b + sqrt( -b^2*x^2 + a^2)*abs(b))^2/(b^4*x^2) + 2247*(a*b + sqrt(-b^2*x^2 + a^2)*abs (b))^3/(b^6*x^3) + 3843*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^4/(b^8*x^4) + 3465*(a*b + sqrt(-b^2*x^2 + a^2)*abs(b))^5/(b^10*x^5) + 2625*(a*b + sqrt(- b^2*x^2 + a^2)*abs(b))^6/(b^12*x^6) + 945*(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) + 58)/(a^4*((a*b + sqrt(-b^2*x^2 + a^2)*abs(b))/(b^2*x) + 1)^9*abs(b))
Time = 6.78 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=\frac {\sqrt {a^2-b^2\,x^2}}{63\,a\,b\,{\left (a+b\,x\right )}^4}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{9\,b\,{\left (a+b\,x\right )}^5}+\frac {\sqrt {a^2-b^2\,x^2}}{105\,a^2\,b\,{\left (a+b\,x\right )}^3}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^3\,b\,{\left (a+b\,x\right )}^2}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{315\,a^4\,b\,\left (a+b\,x\right )} \] Input:
int((a^2 - b^2*x^2)^(1/2)/(a + b*x)^6,x)
Output:
(a^2 - b^2*x^2)^(1/2)/(63*a*b*(a + b*x)^4) - (2*(a^2 - b^2*x^2)^(1/2))/(9* b*(a + b*x)^5) + (a^2 - b^2*x^2)^(1/2)/(105*a^2*b*(a + b*x)^3) + (2*(a^2 - b^2*x^2)^(1/2))/(315*a^3*b*(a + b*x)^2) + (2*(a^2 - b^2*x^2)^(1/2))/(315* a^4*b*(a + b*x))
Time = 0.22 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.42 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^6} \, dx=\frac {70 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{4}+23 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{3} b x +51 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{2} b^{2} x^{2}+38 \sqrt {-b^{2} x^{2}+a^{2}}\, a \,b^{3} x^{3}+10 \sqrt {-b^{2} x^{2}+a^{2}}\, b^{4} x^{4}-70 a^{5}+23 a^{4} b x -124 a^{3} b^{2} x^{2}-131 a^{2} b^{3} x^{3}-68 a \,b^{4} x^{4}-14 b^{5} x^{5}}{315 a^{4} b \left (\sqrt {-b^{2} x^{2}+a^{2}}\, a^{4}+4 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{3} b x +6 \sqrt {-b^{2} x^{2}+a^{2}}\, a^{2} b^{2} x^{2}+4 \sqrt {-b^{2} x^{2}+a^{2}}\, a \,b^{3} x^{3}+\sqrt {-b^{2} x^{2}+a^{2}}\, b^{4} x^{4}-a^{5}-5 a^{4} b x -10 a^{3} b^{2} x^{2}-10 a^{2} b^{3} x^{3}-5 a \,b^{4} x^{4}-b^{5} x^{5}\right )} \] Input:
int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^6,x)
Output:
(70*sqrt(a**2 - b**2*x**2)*a**4 + 23*sqrt(a**2 - b**2*x**2)*a**3*b*x + 51* sqrt(a**2 - b**2*x**2)*a**2*b**2*x**2 + 38*sqrt(a**2 - b**2*x**2)*a*b**3*x **3 + 10*sqrt(a**2 - b**2*x**2)*b**4*x**4 - 70*a**5 + 23*a**4*b*x - 124*a* *3*b**2*x**2 - 131*a**2*b**3*x**3 - 68*a*b**4*x**4 - 14*b**5*x**5)/(315*a* *4*b*(sqrt(a**2 - b**2*x**2)*a**4 + 4*sqrt(a**2 - b**2*x**2)*a**3*b*x + 6* sqrt(a**2 - b**2*x**2)*a**2*b**2*x**2 + 4*sqrt(a**2 - b**2*x**2)*a*b**3*x* *3 + sqrt(a**2 - b**2*x**2)*b**4*x**4 - a**5 - 5*a**4*b*x - 10*a**3*b**2*x **2 - 10*a**2*b**3*x**3 - 5*a*b**4*x**4 - b**5*x**5))