Integrand size = 25, antiderivative size = 147 \[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {x^5 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {x^3 (5 d+6 e x)}{15 e^4 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {x (5 d+8 e x)}{5 e^6 \sqrt {d^2-e^2 x^2}}+\frac {16 \sqrt {d^2-e^2 x^2}}{5 e^7}-\frac {d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^7} \]
1/5*x^5*(e*x+d)/e^2/(-e^2*x^2+d^2)^(5/2)-1/15*x^3*(6*e*x+5*d)/e^4/(-e^2*x^ 2+d^2)^(3/2)-d*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^7+1/5*x*(8*e*x+5*d)/e^6/ (-e^2*x^2+d^2)^(1/2)+16/5*(-e^2*x^2+d^2)^(1/2)/e^7
Time = 0.44 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.86 \[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (48 d^5-33 d^4 e x-87 d^3 e^2 x^2+52 d^2 e^3 x^3+38 d e^4 x^4-15 e^5 x^5\right )}{(d-e x)^3 (d+e x)^2}+30 d \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{15 e^7} \]
((Sqrt[d^2 - e^2*x^2]*(48*d^5 - 33*d^4*e*x - 87*d^3*e^2*x^2 + 52*d^2*e^3*x ^3 + 38*d*e^4*x^4 - 15*e^5*x^5))/((d - e*x)^3*(d + e*x)^2) + 30*d*ArcTan[( e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(15*e^7)
Time = 0.45 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.14, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {529, 2345, 2345, 27, 455, 224, 216}
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.
| \(\displaystyle \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {\frac {d^6}{e^6}+\frac {5 x d^5}{e^5}+\frac {5 x^2 d^4}{e^4}+\frac {5 x^3 d^3}{e^3}+\frac {5 x^4 d^2}{e^2}+\frac {5 x^5 d}{e}}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^4 (15 d+11 e x)}{3 e^7 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {\frac {8 d^6}{e^6}+\frac {30 x d^5}{e^5}+\frac {15 x^2 d^4}{e^4}+\frac {15 x^3 d^3}{e^3}}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^4 (15 d+11 e x)}{3 e^7 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^4 (45 d+23 e x)}{e^7 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^5 (d+e x)}{e^6 \sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^4 (15 d+11 e x)}{3 e^7 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^4 (45 d+23 e x)}{e^7 \sqrt {d^2-e^2 x^2}}-\frac {15 d^3 \int \frac {d+e x}{\sqrt {d^2-e^2 x^2}}dx}{e^6}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^4 (15 d+11 e x)}{3 e^7 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^4 (45 d+23 e x)}{e^7 \sqrt {d^2-e^2 x^2}}-\frac {15 d^3 \left (d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )}{e^6}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^4 (15 d+11 e x)}{3 e^7 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^4 (45 d+23 e x)}{e^7 \sqrt {d^2-e^2 x^2}}-\frac {15 d^3 \left (d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )}{e^6}}{3 d^2}}{5 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^5 (d+e x)}{5 e^7 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {d^4 (15 d+11 e x)}{3 e^7 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {d^4 (45 d+23 e x)}{e^7 \sqrt {d^2-e^2 x^2}}-\frac {15 d^3 \left (\frac {d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )}{e^6}}{3 d^2}}{5 d}\) |
(d^5*(d + e*x))/(5*e^7*(d^2 - e^2*x^2)^(5/2)) - ((d^4*(15*d + 11*e*x))/(3* e^7*(d^2 - e^2*x^2)^(3/2)) - ((d^4*(45*d + 23*e*x))/(e^7*Sqrt[d^2 - e^2*x^ 2]) - (15*d^3*(-(Sqrt[d^2 - e^2*x^2]/e) + (d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x ^2]])/e))/e^6)/(3*d^2))/(5*d)
3.1.20.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && !LeQ[p, -1]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m, a*d + b*c*x, x], R = PolynomialRem ainder[x^m, a*d + b*c*x, x]}, Simp[(-c)*R*(c + d*x)^n*((a + b*x^2)^(p + 1)/ (2*a*d*(p + 1))), x] + Simp[c/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b* x^2)^(p + 1)*ExpandToSum[2*a*d*(p + 1)*Qx + R*(n + 2*p + 2), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 1] && LtQ[p, -1] && EqQ[b* c^2 + a*d^2, 0]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuot ient[Pq, a + b*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*g - b *f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) In t[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]
Time = 0.42 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.50
| method | result | size |
| default | \(e \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+d \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )\) | \(220\) |
| risch | \(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{7}}-\frac {d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{6} \sqrt {e^{2}}}-\frac {d^{2} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{24 e^{9} \left (x +\frac {d}{e}\right )^{2}}+\frac {25 d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{48 e^{8} \left (x +\frac {d}{e}\right )}-\frac {23 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{60 e^{9} \left (x -\frac {d}{e}\right )^{2}}-\frac {493 d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{240 e^{8} \left (x -\frac {d}{e}\right )}-\frac {d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{20 e^{10} \left (x -\frac {d}{e}\right )^{3}}\) | \(283\) |
e*(-x^6/e^2/(-e^2*x^2+d^2)^(5/2)+6*d^2/e^2*(x^4/e^2/(-e^2*x^2+d^2)^(5/2)-4 *d^2/e^2*(1/3*x^2/e^2/(-e^2*x^2+d^2)^(5/2)-2/15*d^2/e^4/(-e^2*x^2+d^2)^(5/ 2))))+d*(1/5*x^5/e^2/(-e^2*x^2+d^2)^(5/2)-1/e^2*(1/3*x^3/e^2/(-e^2*x^2+d^2 )^(3/2)-1/e^2*(x/e^2/(-e^2*x^2+d^2)^(1/2)-1/e^2/(e^2)^(1/2)*arctan((e^2)^( 1/2)*x/(-e^2*x^2+d^2)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.79 \[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {48 \, d e^{5} x^{5} - 48 \, d^{2} e^{4} x^{4} - 96 \, d^{3} e^{3} x^{3} + 96 \, d^{4} e^{2} x^{2} + 48 \, d^{5} e x - 48 \, d^{6} + 30 \, {\left (d e^{5} x^{5} - d^{2} e^{4} x^{4} - 2 \, d^{3} e^{3} x^{3} + 2 \, d^{4} e^{2} x^{2} + d^{5} e x - d^{6}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{5} x^{5} - 38 \, d e^{4} x^{4} - 52 \, d^{2} e^{3} x^{3} + 87 \, d^{3} e^{2} x^{2} + 33 \, d^{4} e x - 48 \, d^{5}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{12} x^{5} - d e^{11} x^{4} - 2 \, d^{2} e^{10} x^{3} + 2 \, d^{3} e^{9} x^{2} + d^{4} e^{8} x - d^{5} e^{7}\right )}} \]
1/15*(48*d*e^5*x^5 - 48*d^2*e^4*x^4 - 96*d^3*e^3*x^3 + 96*d^4*e^2*x^2 + 48 *d^5*e*x - 48*d^6 + 30*(d*e^5*x^5 - d^2*e^4*x^4 - 2*d^3*e^3*x^3 + 2*d^4*e^ 2*x^2 + d^5*e*x - d^6)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (15*e^5 *x^5 - 38*d*e^4*x^4 - 52*d^2*e^3*x^3 + 87*d^3*e^2*x^2 + 33*d^4*e*x - 48*d^ 5)*sqrt(-e^2*x^2 + d^2))/(e^12*x^5 - d*e^11*x^4 - 2*d^2*e^10*x^3 + 2*d^3*e ^9*x^2 + d^4*e^8*x - d^5*e^7)
Result contains complex when optimal does not.
Time = 10.51 (sec) , antiderivative size = 1821, normalized size of antiderivative = 12.39 \[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\text {Too large to display} \]
d*Piecewise((30*I*d**5*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e** 7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 15*pi*d**5*sqrt(-1 + e**2*x **2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt (-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 30*I* d**4*e*x/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt( -1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 60*I*d **3*e**2*x**2*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e* *11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*pi*d**3*e**2*x**2*sqrt(-1 + e**2* x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqr t(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 70*I *d**2*e**3*x**3/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x** 2*sqrt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) + 30*I*d*e**4*x**4*sqrt(-1 + e**2*x**2/d**2)*acosh(e*x/d)/(30*d**5*e**7*sqr t(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sqrt(-1 + e**2*x**2/d**2) + 30* d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 15*pi*d*e**4*x**4*sqrt(-1 + e**2 *x**2/d**2)/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2*sq rt(-1 + e**2*x**2/d**2) + 30*d*e**11*x**4*sqrt(-1 + e**2*x**2/d**2)) - 46* I*e**5*x**5/(30*d**5*e**7*sqrt(-1 + e**2*x**2/d**2) - 60*d**3*e**9*x**2...
Leaf count of result is larger than twice the leaf count of optimal. 290 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.97 \[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {1}{15} \, d x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {d x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{3 \, e^{2}} + \frac {6 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{3}} - \frac {8 \, d^{4} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{5}} + \frac {16 \, d^{6}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{7}} + \frac {4 \, d^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{6}} - \frac {7 \, d x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{6}} - \frac {d \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}} e^{6}} \]
1/15*d*x*(15*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 20*d^2*x^2/((-e^2*x^2 + d^ 2)^(5/2)*e^4) + 8*d^4/((-e^2*x^2 + d^2)^(5/2)*e^6)) - x^6/((-e^2*x^2 + d^2 )^(5/2)*e) - 1/3*d*x*(3*x^2/((-e^2*x^2 + d^2)^(3/2)*e^2) - 2*d^2/((-e^2*x^ 2 + d^2)^(3/2)*e^4))/e^2 + 6*d^2*x^4/((-e^2*x^2 + d^2)^(5/2)*e^3) - 8*d^4* x^2/((-e^2*x^2 + d^2)^(5/2)*e^5) + 16/5*d^6/((-e^2*x^2 + d^2)^(5/2)*e^7) + 4/15*d^3*x/((-e^2*x^2 + d^2)^(3/2)*e^6) - 7/15*d*x/(sqrt(-e^2*x^2 + d^2)* e^6) - d*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^6)
\[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )} x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^6 (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^6\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]