Integrand size = 24, antiderivative size = 166 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}-\frac {4 \sqrt {d^2-e^2 x^2}}{63 d^2 e (d+e x)^4}-\frac {4 \sqrt {d^2-e^2 x^2}}{105 d^3 e (d+e x)^3}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^4 e (d+e x)^2}-\frac {8 \sqrt {d^2-e^2 x^2}}{315 d^5 e (d+e x)} \]
-1/9*(-e^2*x^2+d^2)^(1/2)/d/e/(e*x+d)^5-4/63*(-e^2*x^2+d^2)^(1/2)/d^2/e/(e *x+d)^4-4/105*(-e^2*x^2+d^2)^(1/2)/d^3/e/(e*x+d)^3-8/315*(-e^2*x^2+d^2)^(1 /2)/d^4/e/(e*x+d)^2-8/315*(-e^2*x^2+d^2)^(1/2)/d^5/e/(e*x+d)
Time = 0.59 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.45 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-83 d^4-100 d^3 e x-84 d^2 e^2 x^2-40 d e^3 x^3-8 e^4 x^4\right )}{315 d^5 e (d+e x)^5} \]
(Sqrt[d^2 - e^2*x^2]*(-83*d^4 - 100*d^3*e*x - 84*d^2*e^2*x^2 - 40*d*e^3*x^ 3 - 8*e^4*x^4))/(315*d^5*e*(d + e*x)^5)
Time = 0.27 (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.
\(\displaystyle \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \int \frac {1}{(d+e x)^4 \sqrt {d^2-e^2 x^2}}dx}{9 d}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \left (\frac {3 \int \frac {1}{(d+e x)^3 \sqrt {d^2-e^2 x^2}}dx}{7 d}-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}\right )}{9 d}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \int \frac {1}{(d+e x)^2 \sqrt {d^2-e^2 x^2}}dx}{5 d}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}\right )}{7 d}-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}\right )}{9 d}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {1}{(d+e x) \sqrt {d^2-e^2 x^2}}dx}{3 d}-\frac {\sqrt {d^2-e^2 x^2}}{3 d e (d+e x)^2}\right )}{5 d}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}\right )}{7 d}-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}\right )}{9 d}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle \frac {4 \left (\frac {3 \left (\frac {2 \left (-\frac {\sqrt {d^2-e^2 x^2}}{3 d^2 e (d+e x)}-\frac {\sqrt {d^2-e^2 x^2}}{3 d e (d+e x)^2}\right )}{5 d}-\frac {\sqrt {d^2-e^2 x^2}}{5 d e (d+e x)^3}\right )}{7 d}-\frac {\sqrt {d^2-e^2 x^2}}{7 d e (d+e x)^4}\right )}{9 d}-\frac {\sqrt {d^2-e^2 x^2}}{9 d e (d+e x)^5}\) |
-1/9*Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)^5) + (4*(-1/7*Sqrt[d^2 - e^2*x^2]/ (d*e*(d + e*x)^4) + (3*(-1/5*Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)^3) + (2*(- 1/3*Sqrt[d^2 - e^2*x^2]/(d*e*(d + e*x)^2) - Sqrt[d^2 - e^2*x^2]/(3*d^2*e*( d + e*x))))/(5*d)))/(7*d)))/(9*d)
3.9.35.3.1 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 = 2.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.43
method | result | size |
trager | \(-\frac {\left (8 e^{4} x^{4}+40 d \,e^{3} x^{3}+84 d^{2} e^{2} x^{2}+100 d^{3} e x +83 d^{4}\right ) \sqrt {-x^{2} e^{2}+d^{2}}}{315 d^{5} \left (e x +d \right )^{5} e}\) | \(71\) |
gosper | \(-\frac {\left (-e x +d \right ) \left (8 e^{4} x^{4}+40 d \,e^{3} x^{3}+84 d^{2} e^{2} x^{2}+100 d^{3} e x +83 d^{4}\right )}{315 \left (e x +d \right )^{4} d^{5} e \sqrt {-x^{2} e^{2}+d^{2}}}\) | \(77\) |
default | \(\frac {-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{9 d e \left (x +\frac {d}{e}\right )^{5}}+\frac {4 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{7 d e \left (x +\frac {d}{e}\right )^{4}}+\frac {3 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{5 d e \left (x +\frac {d}{e}\right )^{3}}+\frac {2 e \left (-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d e \left (x +\frac {d}{e}\right )^{2}}-\frac {\sqrt {-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )}}{3 d^{2} \left (x +\frac {d}{e}\right )}\right )}{5 d}\right )}{7 d}\right )}{9 d}}{e^{5}}\) | \(249\) |
-1/315*(8*e^4*x^4+40*d*e^3*x^3+84*d^2*e^2*x^2+100*d^3*e*x+83*d^4)/d^5/(e*x +d)^5/e*(-e^2*x^2+d^2)^(1/2)
Time = 0.41 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {83 \, e^{5} x^{5} + 415 \, d e^{4} x^{4} + 830 \, d^{2} e^{3} x^{3} + 830 \, d^{3} e^{2} x^{2} + 415 \, d^{4} e x + 83 \, d^{5} + {\left (8 \, e^{4} x^{4} + 40 \, d e^{3} x^{3} + 84 \, d^{2} e^{2} x^{2} + 100 \, d^{3} e x + 83 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{5} e^{6} x^{5} + 5 \, d^{6} e^{5} x^{4} + 10 \, d^{7} e^{4} x^{3} + 10 \, d^{8} e^{3} x^{2} + 5 \, d^{9} e^{2} x + d^{10} e\right )}} \]
-1/315*(83*e^5*x^5 + 415*d*e^4*x^4 + 830*d^2*e^3*x^3 + 830*d^3*e^2*x^2 + 4 15*d^4*e*x + 83*d^5 + (8*e^4*x^4 + 40*d*e^3*x^3 + 84*d^2*e^2*x^2 + 100*d^3 *e*x + 83*d^4)*sqrt(-e^2*x^2 + d^2))/(d^5*e^6*x^5 + 5*d^6*e^5*x^4 + 10*d^7 *e^4*x^3 + 10*d^8*e^3*x^2 + 5*d^9*e^2*x + d^10*e)
\[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=\int \frac {1}{\sqrt {- \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{5}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.62 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {-e^{2} x^{2} + d^{2}}}{9 \, {\left (d e^{6} x^{5} + 5 \, d^{2} e^{5} x^{4} + 10 \, d^{3} e^{4} x^{3} + 10 \, d^{4} e^{3} x^{2} + 5 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{63 \, {\left (d^{2} e^{5} x^{4} + 4 \, d^{3} e^{4} x^{3} + 6 \, d^{4} e^{3} x^{2} + 4 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {4 \, \sqrt {-e^{2} x^{2} + d^{2}}}{105 \, {\left (d^{3} e^{4} x^{3} + 3 \, d^{4} e^{3} x^{2} + 3 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{4} e^{3} x^{2} + 2 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {8 \, \sqrt {-e^{2} x^{2} + d^{2}}}{315 \, {\left (d^{5} e^{2} x + d^{6} e\right )}} \]
-1/9*sqrt(-e^2*x^2 + d^2)/(d*e^6*x^5 + 5*d^2*e^5*x^4 + 10*d^3*e^4*x^3 + 10 *d^4*e^3*x^2 + 5*d^5*e^2*x + d^6*e) - 4/63*sqrt(-e^2*x^2 + d^2)/(d^2*e^5*x ^4 + 4*d^3*e^4*x^3 + 6*d^4*e^3*x^2 + 4*d^5*e^2*x + d^6*e) - 4/105*sqrt(-e^ 2*x^2 + d^2)/(d^3*e^4*x^3 + 3*d^4*e^3*x^2 + 3*d^5*e^2*x + d^6*e) - 8/315*s qrt(-e^2*x^2 + d^2)/(d^4*e^3*x^2 + 2*d^5*e^2*x + d^6*e) - 8/315*sqrt(-e^2* x^2 + d^2)/(d^5*e^2*x + d^6*e)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {-\frac {128 i \, \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{d^{5}} + \frac {35 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {9}{2}} + 180 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {7}{2}} + 378 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {5}{2}} + 420 \, {\left (\frac {2 \, d}{e x + d} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {\frac {2 \, d}{e x + d} - 1}}{d^{5} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}}{5040 \, {\left | e \right |}} \]
-1/5040*(-128*I*sgn(1/(e*x + d))*sgn(e)/d^5 + (35*(2*d/(e*x + d) - 1)^(9/2 ) + 180*(2*d/(e*x + d) - 1)^(7/2) + 378*(2*d/(e*x + d) - 1)^(5/2) + 420*(2 *d/(e*x + d) - 1)^(3/2) + 315*sqrt(2*d/(e*x + d) - 1))/(d^5*sgn(1/(e*x + d ))*sgn(e)))/abs(e)
Time = 9.77 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(d+e x)^5 \sqrt {d^2-e^2 x^2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}}{9\,d\,e\,{\left (d+e\,x\right )}^5}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{63\,d^2\,e\,{\left (d+e\,x\right )}^4}-\frac {4\,\sqrt {d^2-e^2\,x^2}}{105\,d^3\,e\,{\left (d+e\,x\right )}^3}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{315\,d^4\,e\,{\left (d+e\,x\right )}^2}-\frac {8\,\sqrt {d^2-e^2\,x^2}}{315\,d^5\,e\,\left (d+e\,x\right )} \]