Integrand size = 27, antiderivative size = 89 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {x^2 (d-e x)}{3 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {2 d-3 e x}{3 e^4 \sqrt {d^2-e^2 x^2}}-\frac {\arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^4} \]
1/3*x^2*(-e*x+d)/e^2/(-e^2*x^2+d^2)^(3/2)-arctan(e*x/(-e^2*x^2+d^2)^(1/2)) /e^4+1/3*(3*e*x-2*d)/e^4/(-e^2*x^2+d^2)^(1/2)
Time = 0.27 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.03 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {d^2-e^2 x^2} \left (-2 d^2+d e x+4 e^2 x^2\right )}{(d-e x) (d+e x)^2}+6 \arctan \left (\frac {e x}{\sqrt {d^2}-\sqrt {d^2-e^2 x^2}}\right )}{3 e^4} \]
((Sqrt[d^2 - e^2*x^2]*(-2*d^2 + d*e*x + 4*e^2*x^2))/((d - e*x)*(d + e*x)^2 ) + 6*ArcTan[(e*x)/(Sqrt[d^2] - Sqrt[d^2 - e^2*x^2])])/(3*e^4)
Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {568, 530, 27, 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.
\(\displaystyle \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 568 |
\(\displaystyle \frac {x^2}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\int \frac {x (2 d-3 e x)}{\left (d^2-e^2 x^2\right )^{3/2}}dx}{3 e^2}\) |
\(\Big \downarrow \) 530 |
\(\displaystyle \frac {x^2}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\frac {2 d-3 e x}{e^2 \sqrt {d^2-e^2 x^2}}-\frac {\int -\frac {3 d^2}{e \sqrt {d^2-e^2 x^2}}dx}{d^2}}{3 e^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {x^2}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\frac {3 \int \frac {1}{\sqrt {d^2-e^2 x^2}}dx}{e}+\frac {2 d-3 e x}{e^2 \sqrt {d^2-e^2 x^2}}}{3 e^2}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {x^2}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\frac {3 \int \frac {1}{\frac {e^2 x^2}{d^2-e^2 x^2}+1}d\frac {x}{\sqrt {d^2-e^2 x^2}}}{e}+\frac {2 d-3 e x}{e^2 \sqrt {d^2-e^2 x^2}}}{3 e^2}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {x^2}{3 e^2 (d+e x) \sqrt {d^2-e^2 x^2}}-\frac {\frac {3 \arctan \left (\frac {e x}{\sqrt {d^2-e^2 x^2}}\right )}{e^2}+\frac {2 d-3 e x}{e^2 \sqrt {d^2-e^2 x^2}}}{3 e^2}\) |
x^2/(3*e^2*(d + e*x)*Sqrt[d^2 - e^2*x^2]) - ((2*d - 3*e*x)/(e^2*Sqrt[d^2 - e^2*x^2]) + (3*ArcTan[(e*x)/Sqrt[d^2 - e^2*x^2]])/e^2)/(3*e^2)
3.2.29.3.1 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[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Co eff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Po lynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x )*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*Qx + e*(2*p + 3), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && EqQ[n, 1] && IntegerQ[2*p]
Int[((x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] : > Simp[x^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*p*(c + d*x))), x] + Simp[1/(2*d^ 2*p) Int[x^(m - 2)*(a + b*x^2)^p*(c*(m - 1) - d*m*x), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + a*d^2, 0] && IGtQ[m, 1] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(79)=158\).
Time = 0.38 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.29
method | result | size |
default | \(\frac {\frac {x}{e^{2} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {\arctan \left (\frac {\sqrt {e^{2}}\, x}{\sqrt {-e^{2} x^{2}+d^{2}}}\right )}{e^{2} \sqrt {e^{2}}}}{e}+\frac {x}{\sqrt {-e^{2} x^{2}+d^{2}}\, e^{3}}-\frac {d}{e^{4} \sqrt {-e^{2} x^{2}+d^{2}}}-\frac {d^{3} \left (-\frac {1}{3 d e \left (x +\frac {d}{e}\right ) \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}-\frac {-2 \left (x +\frac {d}{e}\right ) e^{2}+2 d e}{3 e \,d^{3} \sqrt {-\left (x +\frac {d}{e}\right )^{2} e^{2}+2 d e \left (x +\frac {d}{e}\right )}}\right )}{e^{4}}\) | \(204\) |
1/e*(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)))+1/(-e^2*x^2+d^2)^(1/2)/e^3*x-d/e^4/(-e^2*x^2+d^2)^(1/2 )-d^3/e^4*(-1/3/d/e/(x+d/e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2)-1/3/e/d^3 *(-2*(x+d/e)*e^2+2*d*e)/(-(x+d/e)^2*e^2+2*d*e*(x+d/e))^(1/2))
Time = 0.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.76 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=-\frac {2 \, e^{3} x^{3} + 2 \, d e^{2} x^{2} - 2 \, d^{2} e x - 2 \, d^{3} - 6 \, {\left (e^{3} x^{3} + d e^{2} x^{2} - d^{2} e x - d^{3}\right )} \arctan \left (-\frac {d - \sqrt {-e^{2} x^{2} + d^{2}}}{e x}\right ) + {\left (4 \, e^{2} x^{2} + d e x - 2 \, d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{3 \, {\left (e^{7} x^{3} + d e^{6} x^{2} - d^{2} e^{5} x - d^{3} e^{4}\right )}} \]
-1/3*(2*e^3*x^3 + 2*d*e^2*x^2 - 2*d^2*e*x - 2*d^3 - 6*(e^3*x^3 + d*e^2*x^2 - d^2*e*x - d^3)*arctan(-(d - sqrt(-e^2*x^2 + d^2))/(e*x)) + (4*e^2*x^2 + d*e*x - 2*d^2)*sqrt(-e^2*x^2 + d^2))/(e^7*x^3 + d*e^6*x^2 - d^2*e^5*x - d ^3*e^4)
\[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {x^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {3}{2}} \left (d + e x\right )}\, dx \]
Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.11 \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\frac {d^{2}}{3 \, {\left (\sqrt {-e^{2} x^{2} + d^{2}} e^{5} x + \sqrt {-e^{2} x^{2} + d^{2}} d e^{4}\right )}} + \frac {4 \, x}{3 \, \sqrt {-e^{2} x^{2} + d^{2}} e^{3}} - \frac {\arcsin \left (\frac {e x}{d}\right )}{e^{4}} - \frac {d}{\sqrt {-e^{2} x^{2} + d^{2}} e^{4}} \]
1/3*d^2/(sqrt(-e^2*x^2 + d^2)*e^5*x + sqrt(-e^2*x^2 + d^2)*d*e^4) + 4/3*x/ (sqrt(-e^2*x^2 + d^2)*e^3) - arcsin(e*x/d)/e^4 - d/(sqrt(-e^2*x^2 + d^2)*e ^4)
\[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int { \frac {x^{3}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {x^3}{(d+e x) \left (d^2-e^2 x^2\right )^{3/2}} \, dx=\int \frac {x^3}{{\left (d^2-e^2\,x^2\right )}^{3/2}\,\left (d+e\,x\right )} \,d x \]