Integrand size = 27, antiderivative size = 143 \[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {22 d^3 (d+e x)}{15 e^6 \left (d^2-e^2 x^2\right )^{3/2}}+\frac {2 d (30 d+23 e x)}{15 e^6 \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {d^2-e^2 x^2}}{e^6}-\frac {2 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^6} \]
1/5*d^4*(e*x+d)^2/e^6/(-e^2*x^2+d^2)^(5/2)-22/15*d^3*(e*x+d)/e^6/(-e^2*x^2 +d^2)^(3/2)-2*d*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^6+2/15*d*(23*e*x+30*d)/ e^6/(-e^2*x^2+d^2)^(1/2)+(-e^2*x^2+d^2)^(1/2)/e^6
Time = 0.43 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.88 \[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (-56 d^4+82 d^3 e x+32 d^2 e^2 x^2-76 d e^3 x^3+15 e^4 x^4\right )}{15 e^6 (-d+e x)^3 (d+e x)}+\frac {2 d \left (-e^2\right )^{3/2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^9} \]
(Sqrt[d^2 - e^2*x^2]*(-56*d^4 + 82*d^3*e*x + 32*d^2*e^2*x^2 - 76*d*e^3*x^3 + 15*e^4*x^4))/(15*e^6*(-d + e*x)^3*(d + e*x)) + (2*d*(-e^2)^(3/2)*Log[-( Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2]])/e^9
Time = 0.50 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.18, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {529, 2166, 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.
| \(\displaystyle \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 529 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(d+e x) \left (\frac {2 d^5}{e^5}+\frac {5 x d^4}{e^4}+\frac {5 x^2 d^3}{e^3}+\frac {5 x^3 d^2}{e^2}+\frac {5 x^4 d}{e}\right )}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d}\) |
| \(\Big \downarrow \) 2166 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {22 d^4 (d+e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\int \frac {\frac {16 d^5}{e^5}+\frac {45 x d^4}{e^4}+\frac {30 x^2 d^3}{e^3}+\frac {15 x^3 d^2}{e^2}}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 2345 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {22 d^4 (d+e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {2 d^3 (30 d+23 e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {15 d^4 (2 d+e x)}{e^5 \sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {22 d^4 (d+e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {2 d^3 (30 d+23 e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 \int \frac {2 d+e x}{\sqrt {d^2-e^2 x^2}}dx}{e^5}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {22 d^4 (d+e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {2 d^3 (30 d+23 e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 \left (2 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )}{e^5}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {22 d^4 (d+e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {2 d^3 (30 d+23 e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 \left (2 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )}{e^5}}{3 d}}{5 d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {d^4 (d+e x)^2}{5 e^6 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\frac {22 d^4 (d+e x)}{3 e^6 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {\frac {2 d^3 (30 d+23 e x)}{e^6 \sqrt {d^2-e^2 x^2}}-\frac {15 d^2 \left (\frac {2 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}-\frac {\sqrt {d^2-e^2 x^2}}{e}\right )}{e^5}}{3 d}}{5 d}\) |
(d^4*(d + e*x)^2)/(5*e^6*(d^2 - e^2*x^2)^(5/2)) - ((22*d^4*(d + e*x))/(3*e ^6*(d^2 - e^2*x^2)^(3/2)) - ((2*d^3*(30*d + 23*e*x))/(e^6*Sqrt[d^2 - e^2*x ^2]) - (15*d^2*(-(Sqrt[d^2 - e^2*x^2]/e) + (2*d*ArcTan[(e*x)/Sqrt[d^2 - e^ 2*x^2]])/e))/e^5)/(3*d))/(5*d)
3.1.44.3.1 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_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : > With[{Qx = PolynomialQuotient[Pq, a*e + b*d*x, x], R = PolynomialRemainde r[Pq, a*e + b*d*x, x]}, Simp[(-d)*R*(d + e*x)^m*((a + b*x^2)^(p + 1)/(2*a*e *(p + 1))), x] + Simp[d/(2*a*(p + 1)) Int[(d + e*x)^(m - 1)*(a + b*x^2)^( p + 1)*ExpandToSum[2*a*e*(p + 1)*Qx + R*(m + 2*p + 2), x], x], x]] /; FreeQ [{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && ILtQ[p + 1/2, 0] && GtQ[m, 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 = 238, normalized size of antiderivative = 1.66
| method | result | size |
| risch | \(\frac {\sqrt {-e^{2} x^{2}+d^{2}}}{e^{6}}-\frac {2 d \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{5} \sqrt {e^{2}}}+\frac {d \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{8 e^{7} \left (x +\frac {d}{e}\right )}-\frac {41 d^{2} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{60 e^{8} \left (x -\frac {d}{e}\right )^{2}}-\frac {383 d \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{120 e^{7} \left (x -\frac {d}{e}\right )}-\frac {d^{3} \sqrt {-\left (x -\frac {d}{e}\right )^{2} e^{2}-2 d e \left (x -\frac {d}{e}\right )}}{10 e^{9} \left (x -\frac {d}{e}\right )^{3}}\) | \(238\) |
| default | \(e^{2} \left (-\frac {x^{6}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {6 d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )}{e^{2}}\right )+d^{2} \left (\frac {x^{4}}{e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {4 d^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )}{e^{2}}\right )+2 d e \left (\frac {x^{5}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {\frac {x^{3}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}-\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e^{2}}}{e^{2}}\right )\) | \(303\) |
(-e^2*x^2+d^2)^(1/2)/e^6-2*d/e^5/(e^2)^(1/2)*arctan((e^2)^(1/2)*x/(-e^2*x^ 2+d^2)^(1/2))+1/8*d/e^7/(x+d/e)*(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-41/60 *d^2/e^8/(x-d/e)^2*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-383/120*d/e^7/(x-d /e)*(-(x-d/e)^2*e^2-2*d*e*(x-d/e))^(1/2)-1/10*d^3/e^9/(x-d/e)^3*(-(x-d/e)^ 2*e^2-2*d*e*(x-d/e))^(1/2)
Time = 0.36 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.31 \[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {56 \, d e^{4} x^{4} - 112 \, d^{2} e^{3} x^{3} + 112 \, d^{4} e x - 56 \, d^{5} + 60 \, {\left (d e^{4} x^{4} - 2 \, d^{2} e^{3} x^{3} + 2 \, d^{4} e x - d^{5}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (15 \, e^{4} x^{4} - 76 \, d e^{3} x^{3} + 32 \, d^{2} e^{2} x^{2} + 82 \, d^{3} e x - 56 \, d^{4}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{15 \, {\left (e^{10} x^{4} - 2 \, d e^{9} x^{3} + 2 \, d^{3} e^{7} x - d^{4} e^{6}\right )}} \]
1/15*(56*d*e^4*x^4 - 112*d^2*e^3*x^3 + 112*d^4*e*x - 56*d^5 + 60*(d*e^4*x^ 4 - 2*d^2*e^3*x^3 + 2*d^4*e*x - d^5)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e *x)) + (15*e^4*x^4 - 76*d*e^3*x^3 + 32*d^2*e^2*x^2 + 82*d^3*e*x - 56*d^4)* sqrt(-e^2*x^2 + d^2))/(e^10*x^4 - 2*d*e^9*x^3 + 2*d^3*e^7*x - d^4*e^6)
\[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^{5} \left (d + e x\right )^{2}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (127) = 254\).
Time = 0.28 (sec) , antiderivative size = 288, normalized size of antiderivative = 2.01 \[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2}{15} \, d e x {\left (\frac {15 \, x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {20 \, d^{2} x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {8 \, d^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}}\right )} - \frac {x^{6}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} - \frac {2 \, d x {\left (\frac {3 \, x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{2}} - \frac {2 \, d^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{4}}\right )}}{3 \, e} + \frac {7 \, d^{2} x^{4}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {28 \, d^{4} x^{2}}{3 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{4}} + \frac {56 \, d^{6}}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{6}} + \frac {8 \, d^{3} x}{15 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e^{5}} - \frac {14 \, d x}{15 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{5}} - \frac {2 \, d \arcsin \left (\frac {e^{2} x}{d \sqrt {e^{2}}}\right )}{\sqrt {e^{2}} e^{5}} \]
2/15*d*e*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) - 2/3*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 + 7*d^2*x^4/((-e^2*x^2 + d^2)^(5/2)*e^2) - 28/3*d^4*x ^2/((-e^2*x^2 + d^2)^(5/2)*e^4) + 56/15*d^6/((-e^2*x^2 + d^2)^(5/2)*e^6) + 8/15*d^3*x/((-e^2*x^2 + d^2)^(3/2)*e^5) - 14/15*d*x/(sqrt(-e^2*x^2 + d^2) *e^5) - 2*d*arcsin(e^2*x/(d*sqrt(e^2)))/(sqrt(e^2)*e^5)
\[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int { \frac {{\left (e x + d\right )}^{2} x^{5}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^5 (d+e x)^2}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x^5\,{\left (d+e\,x\right )}^2}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]