Integrand size = 24, antiderivative size = 67 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9} \] Output:
-1/11*(-e^2*x^2+d^2)^(9/2)/d/e/(e*x+d)^10-1/99*(-e^2*x^2+d^2)^(9/2)/d^2/e/ (e*x+d)^9
Time = 0.56 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.72 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=-\frac {(d-e x)^4 (10 d+e x) \sqrt {d^2-e^2 x^2}}{99 d^2 e (d+e x)^6} \] Input:
Integrate[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]
Output:
-1/99*((d - e*x)^4*(10*d + e*x)*Sqrt[d^2 - e^2*x^2])/(d^2*e*(d + e*x)^6)
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {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.
\(\displaystyle \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx\) |
\(\Big \downarrow \) 461 |
\(\displaystyle \frac {\int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^9}dx}{11 d}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}\) |
\(\Big \downarrow \) 460 |
\(\displaystyle -\frac {\left (d^2-e^2 x^2\right )^{9/2}}{99 d^2 e (d+e x)^9}-\frac {\left (d^2-e^2 x^2\right )^{9/2}}{11 d e (d+e x)^{10}}\) |
Input:
Int[(d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x]
Output:
-1/11*(d^2 - e^2*x^2)^(9/2)/(d*e*(d + e*x)^10) - (d^2 - e^2*x^2)^(9/2)/(99 *d^2*e*(d + e*x)^9)
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 = 2.70 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64
method | result | size |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +10 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{99 \left (e x +d \right )^{9} d^{2} e}\) | \(43\) |
orering | \(-\frac {\left (-e x +d \right ) \left (e x +10 d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}{99 \left (e x +d \right )^{9} d^{2} e}\) | \(43\) |
trager | \(-\frac {\left (e^{5} x^{5}+6 d \,e^{4} x^{4}-34 d^{2} e^{3} x^{3}+56 d^{3} e^{2} x^{2}-39 d^{4} e x +10 d^{5}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{99 d^{2} \left (e x +d \right )^{6} e}\) | \(81\) |
default | \(\frac {-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{11 d e \left (x +\frac {d}{e}\right )^{10}}-\frac {\left (-e^{2} \left (x +\frac {d}{e}\right )^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {9}{2}}}{99 d^{2} \left (x +\frac {d}{e}\right )^{9}}}{e^{10}}\) | \(93\) |
Input:
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x,method=_RETURNVERBOSE)
Output:
-1/99*(-e*x+d)*(e*x+10*d)*(-e^2*x^2+d^2)^(7/2)/(e*x+d)^9/d^2/e
Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (59) = 118\).
Time = 0.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 3.01 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=-\frac {10 \, e^{6} x^{6} + 60 \, d e^{5} x^{5} + 150 \, d^{2} e^{4} x^{4} + 200 \, d^{3} e^{3} x^{3} + 150 \, d^{4} e^{2} x^{2} + 60 \, d^{5} e x + 10 \, d^{6} + {\left (e^{5} x^{5} + 6 \, d e^{4} x^{4} - 34 \, d^{2} e^{3} x^{3} + 56 \, d^{3} e^{2} x^{2} - 39 \, d^{4} e x + 10 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d^{2} e^{7} x^{6} + 6 \, d^{3} e^{6} x^{5} + 15 \, d^{4} e^{5} x^{4} + 20 \, d^{5} e^{4} x^{3} + 15 \, d^{6} e^{3} x^{2} + 6 \, d^{7} e^{2} x + d^{8} e\right )}} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="fricas")
Output:
-1/99*(10*e^6*x^6 + 60*d*e^5*x^5 + 150*d^2*e^4*x^4 + 200*d^3*e^3*x^3 + 150 *d^4*e^2*x^2 + 60*d^5*e*x + 10*d^6 + (e^5*x^5 + 6*d*e^4*x^4 - 34*d^2*e^3*x ^3 + 56*d^3*e^2*x^2 - 39*d^4*e*x + 10*d^5)*sqrt(-e^2*x^2 + d^2))/(d^2*e^7* x^6 + 6*d^3*e^6*x^5 + 15*d^4*e^5*x^4 + 20*d^5*e^4*x^3 + 15*d^6*e^3*x^2 + 6 *d^7*e^2*x + d^8*e)
Timed out. \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\text {Timed out} \] Input:
integrate((-e**2*x**2+d**2)**(7/2)/(e*x+d)**10,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (59) = 118\).
Time = 0.04 (sec) , antiderivative size = 659, normalized size of antiderivative = 9.84 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}{2 \, {\left (e^{10} x^{9} + 9 \, d e^{9} x^{8} + 36 \, d^{2} e^{8} x^{7} + 84 \, d^{3} e^{7} x^{6} + 126 \, d^{4} e^{6} x^{5} + 126 \, d^{5} e^{5} x^{4} + 84 \, d^{6} e^{4} x^{3} + 36 \, d^{7} e^{3} x^{2} + 9 \, d^{8} e^{2} x + d^{9} e\right )}} + \frac {7 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{6 \, {\left (e^{9} x^{8} + 8 \, d e^{8} x^{7} + 28 \, d^{2} e^{7} x^{6} + 56 \, d^{3} e^{6} x^{5} + 70 \, d^{4} e^{5} x^{4} + 56 \, d^{5} e^{4} x^{3} + 28 \, d^{6} e^{3} x^{2} + 8 \, d^{7} e^{2} x + d^{8} e\right )}} - \frac {35 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{24 \, {\left (e^{8} x^{7} + 7 \, d e^{7} x^{6} + 21 \, d^{2} e^{6} x^{5} + 35 \, d^{3} e^{5} x^{4} + 35 \, d^{4} e^{4} x^{3} + 21 \, d^{5} e^{3} x^{2} + 7 \, d^{6} e^{2} x + d^{7} e\right )}} + \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{44 \, {\left (e^{7} x^{6} + 6 \, d e^{6} x^{5} + 15 \, d^{2} e^{5} x^{4} + 20 \, d^{3} e^{4} x^{3} + 15 \, d^{4} e^{3} x^{2} + 6 \, d^{5} e^{2} x + d^{6} e\right )}} - \frac {35 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{792 \, {\left (e^{6} x^{5} + 5 \, d e^{5} x^{4} + 10 \, d^{2} e^{4} x^{3} + 10 \, d^{3} e^{3} x^{2} + 5 \, d^{4} e^{2} x + d^{5} e\right )}} - \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d}{198 \, {\left (e^{5} x^{4} + 4 \, d e^{4} x^{3} + 6 \, d^{2} e^{3} x^{2} + 4 \, d^{3} e^{2} x + d^{4} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{66 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d e^{3} x^{2} + 2 \, d^{2} e^{2} x + d^{3} e\right )}} - \frac {\sqrt {-e^{2} x^{2} + d^{2}}}{99 \, {\left (d^{2} e^{2} x + d^{3} e\right )}} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="maxima")
Output:
-1/2*(-e^2*x^2 + d^2)^(7/2)/(e^10*x^9 + 9*d*e^9*x^8 + 36*d^2*e^8*x^7 + 84* d^3*e^7*x^6 + 126*d^4*e^6*x^5 + 126*d^5*e^5*x^4 + 84*d^6*e^4*x^3 + 36*d^7* e^3*x^2 + 9*d^8*e^2*x + d^9*e) + 7/6*(-e^2*x^2 + d^2)^(5/2)*d/(e^9*x^8 + 8 *d*e^8*x^7 + 28*d^2*e^7*x^6 + 56*d^3*e^6*x^5 + 70*d^4*e^5*x^4 + 56*d^5*e^4 *x^3 + 28*d^6*e^3*x^2 + 8*d^7*e^2*x + d^8*e) - 35/24*(-e^2*x^2 + d^2)^(3/2 )*d^2/(e^8*x^7 + 7*d*e^7*x^6 + 21*d^2*e^6*x^5 + 35*d^3*e^5*x^4 + 35*d^4*e^ 4*x^3 + 21*d^5*e^3*x^2 + 7*d^6*e^2*x + d^7*e) + 35/44*sqrt(-e^2*x^2 + d^2) *d^3/(e^7*x^6 + 6*d*e^6*x^5 + 15*d^2*e^5*x^4 + 20*d^3*e^4*x^3 + 15*d^4*e^3 *x^2 + 6*d^5*e^2*x + d^6*e) - 35/792*sqrt(-e^2*x^2 + d^2)*d^2/(e^6*x^5 + 5 *d*e^5*x^4 + 10*d^2*e^4*x^3 + 10*d^3*e^3*x^2 + 5*d^4*e^2*x + d^5*e) - 5/19 8*sqrt(-e^2*x^2 + d^2)*d/(e^5*x^4 + 4*d*e^4*x^3 + 6*d^2*e^3*x^2 + 4*d^3*e^ 2*x + d^4*e) - 1/66*sqrt(-e^2*x^2 + d^2)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^ 2*x + d^3*e) - 1/99*sqrt(-e^2*x^2 + d^2)/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - 1/99*sqrt(-e^2*x^2 + d^2)/(d^2*e^2*x + d^3*e)
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (59) = 118\).
Time = 0.15 (sec) , antiderivative size = 351, normalized size of antiderivative = 5.24 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {2 \, {\left (\frac {11 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} + \frac {451 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {396 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} + \frac {2376 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{4}}{e^{8} x^{4}} + \frac {1386 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{5}}{e^{10} x^{5}} + \frac {3234 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{6}}{e^{12} x^{6}} + \frac {924 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{7}}{e^{14} x^{7}} + \frac {1254 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{8}}{e^{16} x^{8}} + \frac {99 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{9}}{e^{18} x^{9}} + \frac {99 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{10}}{e^{20} x^{10}} + 10\right )}}{99 \, d^{2} {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )}^{11} {\left | e \right |}} \] Input:
integrate((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x, algorithm="giac")
Output:
2/99*(11*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) + 451*(d*e + sqrt(-e^ 2*x^2 + d^2)*abs(e))^2/(e^4*x^2) + 396*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e)) ^3/(e^6*x^3) + 2376*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^4/(e^8*x^4) + 1386 *(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^5/(e^10*x^5) + 3234*(d*e + sqrt(-e^2* x^2 + d^2)*abs(e))^6/(e^12*x^6) + 924*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^ 7/(e^14*x^7) + 1254*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^8/(e^16*x^8) + 99* (d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^9/(e^18*x^9) + 99*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^10/(e^20*x^10) + 10)/(d^2*((d*e + sqrt(-e^2*x^2 + d^2)*abs (e))/(e^2*x) + 1)^11*abs(e))
Time = 8.14 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.54 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {16\,\sqrt {d^2-e^2\,x^2}}{33\,e\,{\left (d+e\,x\right )}^3}-\frac {184\,d\,\sqrt {d^2-e^2\,x^2}}{99\,e\,{\left (d+e\,x\right )}^4}-\frac {\sqrt {d^2-e^2\,x^2}}{99\,d\,e\,{\left (d+e\,x\right )}^2}-\frac {\sqrt {d^2-e^2\,x^2}}{99\,d^2\,e\,\left (d+e\,x\right )}+\frac {272\,d^2\,\sqrt {d^2-e^2\,x^2}}{99\,e\,{\left (d+e\,x\right )}^5}-\frac {16\,d^3\,\sqrt {d^2-e^2\,x^2}}{11\,e\,{\left (d+e\,x\right )}^6} \] Input:
int((d^2 - e^2*x^2)^(7/2)/(d + e*x)^10,x)
Output:
(16*(d^2 - e^2*x^2)^(1/2))/(33*e*(d + e*x)^3) - (184*d*(d^2 - e^2*x^2)^(1/ 2))/(99*e*(d + e*x)^4) - (d^2 - e^2*x^2)^(1/2)/(99*d*e*(d + e*x)^2) - (d^2 - e^2*x^2)^(1/2)/(99*d^2*e*(d + e*x)) + (272*d^2*(d^2 - e^2*x^2)^(1/2))/( 99*e*(d + e*x)^5) - (16*d^3*(d^2 - e^2*x^2)^(1/2))/(11*e*(d + e*x)^6)
Time = 0.24 (sec) , antiderivative size = 390, normalized size of antiderivative = 5.82 \[ \int \frac {\left (d^2-e^2 x^2\right )^{7/2}}{(d+e x)^{10}} \, dx=\frac {18 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}+\sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e x +136 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{2} x^{2}+46 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{3} x^{3}+46 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{4} x^{4}+9 \sqrt {-e^{2} x^{2}+d^{2}}\, e^{5} x^{5}-18 d^{6}+d^{5} e x -215 d^{4} e^{2} x^{2}-70 d^{3} e^{3} x^{3}-160 d^{2} e^{4} x^{4}-43 d \,e^{5} x^{5}-7 e^{6} x^{6}}{99 d^{2} e \left (\sqrt {-e^{2} x^{2}+d^{2}}\, d^{5}+5 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{4} e x +10 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{3} e^{2} x^{2}+10 \sqrt {-e^{2} x^{2}+d^{2}}\, d^{2} e^{3} x^{3}+5 \sqrt {-e^{2} x^{2}+d^{2}}\, d \,e^{4} x^{4}+\sqrt {-e^{2} x^{2}+d^{2}}\, e^{5} x^{5}-d^{6}-6 d^{5} e x -15 d^{4} e^{2} x^{2}-20 d^{3} e^{3} x^{3}-15 d^{2} e^{4} x^{4}-6 d \,e^{5} x^{5}-e^{6} x^{6}\right )} \] Input:
int((-e^2*x^2+d^2)^(7/2)/(e*x+d)^10,x)
Output:
(18*sqrt(d**2 - e**2*x**2)*d**5 + sqrt(d**2 - e**2*x**2)*d**4*e*x + 136*sq rt(d**2 - e**2*x**2)*d**3*e**2*x**2 + 46*sqrt(d**2 - e**2*x**2)*d**2*e**3* x**3 + 46*sqrt(d**2 - e**2*x**2)*d*e**4*x**4 + 9*sqrt(d**2 - e**2*x**2)*e* *5*x**5 - 18*d**6 + d**5*e*x - 215*d**4*e**2*x**2 - 70*d**3*e**3*x**3 - 16 0*d**2*e**4*x**4 - 43*d*e**5*x**5 - 7*e**6*x**6)/(99*d**2*e*(sqrt(d**2 - e **2*x**2)*d**5 + 5*sqrt(d**2 - e**2*x**2)*d**4*e*x + 10*sqrt(d**2 - e**2*x **2)*d**3*e**2*x**2 + 10*sqrt(d**2 - e**2*x**2)*d**2*e**3*x**3 + 5*sqrt(d* *2 - e**2*x**2)*d*e**4*x**4 + sqrt(d**2 - e**2*x**2)*e**5*x**5 - d**6 - 6* d**5*e*x - 15*d**4*e**2*x**2 - 20*d**3*e**3*x**3 - 15*d**2*e**4*x**4 - 6*d *e**5*x**5 - e**6*x**6))