Integrand size = 25, antiderivative size = 130 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {10 d x \sqrt {d^2-e^2 x^2}}{e}+\frac {20 \left (d^2-e^2 x^2\right )^{3/2}}{3 e^2}+\frac {8 \left (d^2-e^2 x^2\right )^{5/2}}{e^2 (d+e x)^2}+\frac {\left (d^2-e^2 x^2\right )^{7/2}}{e^2 (d+e x)^4}+\frac {10 d^3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2} \]
20/3*(-e^2*x^2+d^2)^(3/2)/e^2+8*(-e^2*x^2+d^2)^(5/2)/e^2/(e*x+d)^2+(-e^2*x ^2+d^2)^(7/2)/e^2/(e*x+d)^4+10*d^3*arctan(e*x/(-e^2*x^2+d^2)^(1/2))/e^2+10 *d*x*(-e^2*x^2+d^2)^(1/2)/e
Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {\sqrt {d^2-e^2 x^2} \left (47 d^3+17 d^2 e x-5 d e^2 x^2+e^3 x^3\right )}{3 e^2 (d+e x)}+\frac {10 d^3 \sqrt {-e^2} \log \left (-\sqrt {-e^2} x+\sqrt {d^2-e^2 x^2}\right )}{e^3} \]
(Sqrt[d^2 - e^2*x^2]*(47*d^3 + 17*d^2*e*x - 5*d*e^2*x^2 + e^3*x^3))/(3*e^2 *(d + e*x)) + (10*d^3*Sqrt[-e^2]*Log[-(Sqrt[-e^2]*x) + Sqrt[d^2 - e^2*x^2] ])/e^3
Time = 0.38 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {563, 2346, 25, 2346, 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.
| \(\displaystyle \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 563 |
| \(\displaystyle \frac {\int \frac {8 d^3-7 e x d^2+4 e^2 x^2 d-e^3 x^3}{\sqrt {d^2-e^2 x^2}}dx}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}-\frac {\int -\frac {12 d x^2 e^4-23 d^2 x e^3+24 d^3 e^2}{\sqrt {d^2-e^2 x^2}}dx}{3 e^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {12 d x^2 e^4-23 d^2 x e^3+24 d^3 e^2}{\sqrt {d^2-e^2 x^2}}dx}{3 e^2}+\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 2346 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {2 d^2 e^4 (30 d-23 e x)}{\sqrt {d^2-e^2 x^2}}dx}{2 e^2}-6 d e^2 x \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {d^2 e^2 \int \frac {30 d-23 e x}{\sqrt {d^2-e^2 x^2}}dx-6 d e^2 x \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 455 |
| \(\displaystyle \frac {\frac {d^2 e^2 \left (30 d \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx+\frac {23 \sqrt {d^2-e^2 x^2}}{e}\right )-6 d e^2 x \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {d^2 e^2 \left (30 d \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}+\frac {23 \sqrt {d^2-e^2 x^2}}{e}\right )-6 d e^2 x \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {d^2 e^2 \left (\frac {30 d \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e}+\frac {23 \sqrt {d^2-e^2 x^2}}{e}\right )-6 d e^2 x \sqrt {d^2-e^2 x^2}}{3 e^2}+\frac {1}{3} e x^2 \sqrt {d^2-e^2 x^2}}{e}+\frac {8 d^3 \sqrt {d^2-e^2 x^2}}{e^2 (d+e x)}\) |
(8*d^3*Sqrt[d^2 - e^2*x^2])/(e^2*(d + e*x)) + ((e*x^2*Sqrt[d^2 - e^2*x^2]) /3 + (-6*d*e^2*x*Sqrt[d^2 - e^2*x^2] + d^2*e^2*((23*Sqrt[d^2 - e^2*x^2])/e + (30*d*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e))/(3*e^2))/e
3.3.2.3.1 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] :> Simp[(-(-c)^(m - n - 2))*d^(2*n - m + 3)*(Sqrt[a + b*x^2]/(2^(n + 1)* b^(n + 2)*(c + d*x))), x] - Simp[d^(2*n - m + 2)/b^(n + 1) Int[(1/Sqrt[a + b*x^2])*ExpandToSum[(2^(-n - 1)*(-c)^(m - n - 1) - d^m*x^m*(-c + d*x)^(-n - 1))/(c + d*x), x], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2 , 0] && IGtQ[m, 0] && ILtQ[n, 0] && EqQ[n + p, -3/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], e = Coeff[Pq, x, Expon[Pq, x]]}, Simp[e*x^(q - 1)*((a + b*x^2)^(p + 1)/(b*( q + 2*p + 1))), x] + Simp[1/(b*(q + 2*p + 1)) Int[(a + b*x^2)^p*ExpandToS um[b*(q + 2*p + 1)*Pq - a*e*(q - 1)*x^(q - 2) - b*e*(q + 2*p + 1)*x^q, x], x], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && !LeQ[p, -1]
Time = 0.46 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92
| method | result | size |
| risch | \(\frac {\left (e^{2} x^{2}-6 d e x +23 d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{3 e^{2}}+\frac {10 d^{3} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e \sqrt {e^{2}}}+\frac {8 d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{e^{3} \left (x +\frac {d}{e}\right )}\) | \(119\) |
| default | \(\frac {\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{d e \left (x +\frac {d}{e}\right )^{3}}+\frac {4 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{d}}{e^{4}}-\frac {d \left (-\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{d e \left (x +\frac {d}{e}\right )^{4}}-\frac {3 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{d e \left (x +\frac {d}{e}\right )^{3}}+\frac {4 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {7}{2}}}{3 d e \left (x +\frac {d}{e}\right )^{2}}+\frac {5 e \left (\frac {\left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {5}{2}}}{5}+d e \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \left (-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )\right )^{\frac {3}{2}}}{8 e^{2}}+\frac {3 d^{2} \left (-\frac {\left (-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e \right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}{4 e^{2}}+\frac {d^{2} \arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{2 \sqrt {e^{2}}}\right )}{4}\right )\right )}{3 d}\right )}{d}\right )}{d}\right )}{e^{5}}\) | \(644\) |
1/3*(e^2*x^2-6*d*e*x+23*d^2)/e^2*(-e^2*x^2+d^2)^(1/2)+10*d^3/e/(e^2)^(1/2) *arctan((e^2)^(1/2)*x/(-e^2*x^2+d^2)^(1/2))+8*d^3/e^3/(x+d/e)*(-(x+d/e)^2* e^2+2*d*e*(x+d/e))^(1/2)
Time = 0.29 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.85 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {47 \, d^{3} e x + 47 \, d^{4} - 60 \, {\left (d^{3} e x + d^{4}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (e^{3} x^{3} - 5 \, d e^{2} x^{2} + 17 \, d^{2} e x + 47 \, d^{3}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (e^{3} x + d e^{2}\right )}} \]
1/3*(47*d^3*e*x + 47*d^4 - 60*(d^3*e*x + d^4)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (e^3*x^3 - 5*d*e^2*x^2 + 17*d^2*e*x + 47*d^3)*sqrt(-e^2*x^ 2 + d^2))/(e^3*x + d*e^2)
\[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {x \left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {5}{2}}}{\left (d + e x\right )^{4}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.81 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=-\frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} d}{2 \, {\left (e^{5} x^{3} + 3 \, d e^{4} x^{2} + 3 \, d^{2} e^{3} x + d^{3} e^{2}\right )}} - \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d^{2}}{2 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \frac {15 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{3}}{e^{3} x + d e^{2}} + \frac {10 \, d^{3} \arcsin \left (\frac {e x}{d}\right )}{e^{2}} + \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}} + \frac {5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} d}{6 \, {\left (e^{3} x + d e^{2}\right )}} + \frac {5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2}}{2 \, e^{2}} \]
-1/2*(-e^2*x^2 + d^2)^(5/2)*d/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e ^2) - 5/2*(-e^2*x^2 + d^2)^(3/2)*d^2/(e^4*x^2 + 2*d*e^3*x + d^2*e^2) + 15* sqrt(-e^2*x^2 + d^2)*d^3/(e^3*x + d*e^2) + 10*d^3*arcsin(e*x/d)/e^2 + 1/3* (-e^2*x^2 + d^2)^(5/2)/(e^4*x^2 + 2*d*e^3*x + d^2*e^2) + 5/6*(-e^2*x^2 + d ^2)^(3/2)*d/(e^3*x + d*e^2) + 5/2*sqrt(-e^2*x^2 + d^2)*d^2/e^2
Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\frac {10 \, d^{3} \arcsin \left (\frac {e x}{d}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}{e {\left | e \right |}} + \frac {1}{3} \, \sqrt {-e^{2} x^{2} + d^{2}} {\left ({\left (x - \frac {6 \, d}{e}\right )} x + \frac {23 \, d^{2}}{e^{2}}\right )} - \frac {16 \, d^{3}}{e {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} + 1\right )} {\left | e \right |}} \]
10*d^3*arcsin(e*x/d)*sgn(d)*sgn(e)/(e*abs(e)) + 1/3*sqrt(-e^2*x^2 + d^2)*( (x - 6*d/e)*x + 23*d^2/e^2) - 16*d^3/(e*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e ))/(e^2*x) + 1)*abs(e))
Timed out. \[ \int \frac {x \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^4} \, dx=\int \frac {x\,{\left (d^2-e^2\,x^2\right )}^{5/2}}{{\left (d+e\,x\right )}^4} \,d x \]