Integrand size = 24, antiderivative size = 100 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{35 a^2 b (a+b x)^4}-\frac {2 \left (a^2-b^2 x^2\right )^{3/2}}{105 a^3 b (a+b x)^3} \] Output:
-1/7*(-b^2*x^2+a^2)^(3/2)/a/b/(b*x+a)^5-2/35*(-b^2*x^2+a^2)^(3/2)/a^2/b/(b *x+a)^4-2/105*(-b^2*x^2+a^2)^(3/2)/a^3/b/(b*x+a)^3
Time = 0.45 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=\frac {\sqrt {a^2-b^2 x^2} \left (-23 a^3+13 a^2 b x+8 a b^2 x^2+2 b^3 x^3\right )}{105 a^3 b (a+b x)^4} \] Input:
Integrate[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]
Output:
(Sqrt[a^2 - b^2*x^2]*(-23*a^3 + 13*a^2*b*x + 8*a*b^2*x^2 + 2*b^3*x^3))/(10 5*a^3*b*(a + b*x)^4)
Time = 0.34 (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.
\(\displaystyle \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {2 \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^4}dx}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {2 \left (\frac {\int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^3}dx}{5 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4}\right )}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {2 \left (-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{15 a^2 b (a+b x)^3}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{5 a b (a+b x)^4}\right )}{7 a}-\frac {\left (a^2-b^2 x^2\right )^{3/2}}{7 a b (a+b x)^5}\) |
Input:
Int[Sqrt[a^2 - b^2*x^2]/(a + b*x)^5,x]
Output:
-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)
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 = 0.33 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.55
method | result | size |
gosper | \(-\frac {\left (-b x +a \right ) \left (2 b^{2} x^{2}+10 a b x +23 a^{2}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{105 \left (b x +a \right )^{4} a^{3} b}\) | \(55\) |
orering | \(-\frac {\left (-b x +a \right ) \left (2 b^{2} x^{2}+10 a b x +23 a^{2}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{105 \left (b x +a \right )^{4} a^{3} b}\) | \(55\) |
trager | \(-\frac {\left (-2 b^{3} x^{3}-8 a \,b^{2} x^{2}-13 a^{2} b x +23 a^{3}\right ) \sqrt {-b^{2} x^{2}+a^{2}}}{105 a^{3} \left (b x +a \right )^{4} b}\) | \(60\) |
default | \(\frac {-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{7 a b \left (x +\frac {a}{b}\right )^{5}}+\frac {2 b \left (-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{5 a b \left (x +\frac {a}{b}\right )^{4}}-\frac {\left (-b^{2} \left (x +\frac {a}{b}\right )^{2}+2 a b \left (x +\frac {a}{b}\right )\right )^{\frac {3}{2}}}{15 a^{2} \left (x +\frac {a}{b}\right )^{3}}\right )}{7 a}}{b^{5}}\) | \(145\) |
Input:
int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x,method=_RETURNVERBOSE)
Output:
-1/105*(-b*x+a)*(2*b^2*x^2+10*a*b*x+23*a^2)*(-b^2*x^2+a^2)^(1/2)/(b*x+a)^4 /a^3/b
Time = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=-\frac {23 \, b^{4} x^{4} + 92 \, a b^{3} x^{3} + 138 \, a^{2} b^{2} x^{2} + 92 \, a^{3} b x + 23 \, a^{4} - {\left (2 \, b^{3} x^{3} + 8 \, a b^{2} x^{2} + 13 \, a^{2} b x - 23 \, a^{3}\right )} \sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{3} b^{5} x^{4} + 4 \, a^{4} b^{4} x^{3} + 6 \, a^{5} b^{3} x^{2} + 4 \, a^{6} b^{2} x + a^{7} b\right )}} \] Input:
integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x, algorithm="fricas")
Output:
-1/105*(23*b^4*x^4 + 92*a*b^3*x^3 + 138*a^2*b^2*x^2 + 92*a^3*b*x + 23*a^4 - (2*b^3*x^3 + 8*a*b^2*x^2 + 13*a^2*b*x - 23*a^3)*sqrt(-b^2*x^2 + a^2))/(a ^3*b^5*x^4 + 4*a^4*b^4*x^3 + 6*a^5*b^3*x^2 + 4*a^6*b^2*x + a^7*b)
\[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=\int \frac {\sqrt {- \left (- a + b x\right ) \left (a + b x\right )}}{\left (a + b x\right )^{5}}\, dx \] Input:
integrate((-b**2*x**2+a**2)**(1/2)/(b*x+a)**5,x)
Output:
Integral(sqrt(-(-a + b*x)*(a + b*x))/(a + b*x)**5, x)
Leaf count of result is larger than twice the leaf count of optimal. 188 vs. \(2 (88) = 176\).
Time = 0.04 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.88 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=-\frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{7 \, {\left (b^{5} x^{4} + 4 \, a b^{4} x^{3} + 6 \, a^{2} b^{3} x^{2} + 4 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {\sqrt {-b^{2} x^{2} + a^{2}}}{35 \, {\left (a b^{4} x^{3} + 3 \, a^{2} b^{3} x^{2} + 3 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{2} b^{3} x^{2} + 2 \, a^{3} b^{2} x + a^{4} b\right )}} + \frac {2 \, \sqrt {-b^{2} x^{2} + a^{2}}}{105 \, {\left (a^{3} b^{2} x + a^{4} b\right )}} \] Input:
integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x, algorithm="maxima")
Output:
-2/7*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) + 1/35*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) + 2/105*sqrt(-b^2*x^2 + a^2)/(a^2*b^3*x^2 + 2*a^3*b^2*x + a^4*b) + 2/105*sqrt(-b^2*x^2 + a^2)/(a^3*b^2*x + a^4*b)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.78 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=-\frac {1}{420} \, {\left (\frac {\frac {3 \, {\left (5 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {7}{2}} + 21 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {5}{2}} + 35 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {3}{2}} + 35 \, \sqrt {\frac {2 \, a}{b x + a} - 1}\right )} \mathrm {sgn}\left (\frac {1}{b x + a}\right ) \mathrm {sgn}\left (b\right )}{a^{2} b^{2}} - \frac {7 \, {\left (3 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {5}{2}} + 10 \, {\left (\frac {2 \, a}{b x + a} - 1\right )}^{\frac {3}{2}} + 15 \, \sqrt {\frac {2 \, a}{b x + a} - 1}\right )} \mathrm {sgn}\left (\frac {1}{b x + a}\right ) \mathrm {sgn}\left (b\right )}{a^{2} b^{2}}}{a} + \frac {8 i \, \mathrm {sgn}\left (\frac {1}{b x + a}\right ) \mathrm {sgn}\left (b\right )}{a^{3} b^{2}}\right )} {\left | b \right |} \] Input:
integrate((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x, algorithm="giac")
Output:
-1/420*((3*(5*(2*a/(b*x + a) - 1)^(7/2) + 21*(2*a/(b*x + a) - 1)^(5/2) + 3 5*(2*a/(b*x + a) - 1)^(3/2) + 35*sqrt(2*a/(b*x + a) - 1))*sgn(1/(b*x + a)) *sgn(b)/(a^2*b^2) - 7*(3*(2*a/(b*x + a) - 1)^(5/2) + 10*(2*a/(b*x + a) - 1 )^(3/2) + 15*sqrt(2*a/(b*x + a) - 1))*sgn(1/(b*x + a))*sgn(b)/(a^2*b^2))/a + 8*I*sgn(1/(b*x + a))*sgn(b)/(a^3*b^2))*abs(b)
Time = 6.22 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=\frac {\sqrt {a^2-b^2\,x^2}}{35\,a\,b\,{\left (a+b\,x\right )}^3}-\frac {2\,\sqrt {a^2-b^2\,x^2}}{7\,b\,{\left (a+b\,x\right )}^4}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{105\,a^2\,b\,{\left (a+b\,x\right )}^2}+\frac {2\,\sqrt {a^2-b^2\,x^2}}{105\,a^3\,b\,\left (a+b\,x\right )} \] Input:
int((a^2 - b^2*x^2)^(1/2)/(a + b*x)^5,x)
Output:
(a^2 - b^2*x^2)^(1/2)/(35*a*b*(a + b*x)^3) - (2*(a^2 - b^2*x^2)^(1/2))/(7* b*(a + b*x)^4) + (2*(a^2 - b^2*x^2)^(1/2))/(105*a^2*b*(a + b*x)^2) + (2*(a ^2 - b^2*x^2)^(1/2))/(105*a^3*b*(a + b*x))
\[ \int \frac {\sqrt {a^2-b^2 x^2}}{(a+b x)^5} \, dx=\int \frac {\sqrt {-b^{2} x^{2}+a^{2}}}{\left (b x +a \right )^{5}}d x \] Input:
int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x)
Output:
int((-b^2*x^2+a^2)^(1/2)/(b*x+a)^5,x)