Integrand size = 25, antiderivative size = 86 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {2 (d+e x)}{5 e^2 \left (d^2-e^2 x^2\right )^{3/2}}-\frac {x}{5 d^2 e \sqrt {d^2-e^2 x^2}} \]
1/5*(e*x+d)^3/e^2/(-e^2*x^2+d^2)^(5/2)-2/5*(e*x+d)/e^2/(-e^2*x^2+d^2)^(3/2 )-1/5*x/d^2/e/(-e^2*x^2+d^2)^(1/2)
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.58 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\sqrt {d^2-e^2 x^2} \left (d^2-3 d e x+e^2 x^2\right )}{5 d^2 e^2 (d-e x)^3} \]
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {531, 27, 457, 208}
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.
\(\displaystyle \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 531 |
\(\displaystyle \frac {\int -\frac {3 d^2 e (d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 d^2 e^2}+\frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 \int \frac {(d+e x)^2}{\left (d^2-e^2 x^2\right )^{5/2}}dx}{5 e}\) |
\(\Big \downarrow \) 457 |
\(\displaystyle \frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 \left (\frac {1}{3} \int \frac {1}{\left (d^2-e^2 x^2\right )^{3/2}}dx+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 e}\) |
\(\Big \downarrow \) 208 |
\(\displaystyle \frac {(d+e x)^3}{5 e^2 \left (d^2-e^2 x^2\right )^{5/2}}-\frac {3 \left (\frac {x}{3 d^2 \sqrt {d^2-e^2 x^2}}+\frac {2 (d+e x)}{3 e \left (d^2-e^2 x^2\right )^{3/2}}\right )}{5 e}\) |
(d + e*x)^3/(5*e^2*(d^2 - e^2*x^2)^(5/2)) - (3*((2*(d + e*x))/(3*e*(d^2 - e^2*x^2)^(3/2)) + x/(3*d^2*Sqrt[d^2 - e^2*x^2])))/(5*e)
3.1.87.3.1 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)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 1))) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ b*c^2 + a*d^2, 0] && LtQ[p, -1]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symb ol] :> With[{Qx = PolynomialQuotient[x^m, a + b*x^2, x], e = Coeff[Polynomi alRemainder[x^m, a + b*x^2, x], x, 0], f = Coeff[PolynomialRemainder[x^m, a + b*x^2, x], x, 1]}, Simp[(c + d*x)^n*(a*f - b*e*x)*((a + b*x^2)^(p + 1)/( 2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b *x^2)^(p + 1)*ExpandToSum[2*a*b*(p + 1)*(c + d*x)*Qx - a*d*f*n + b*c*e*(2*p + 3) + b*d*e*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGt Q[n, 0] && IGtQ[m, 0] && LtQ[p, -1] && GtQ[n, 1] && IntegerQ[2*p]
Time = 0.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.55
method | result | size |
trager | \(-\frac {\left (e^{2} x^{2}-3 d e x +d^{2}\right ) \sqrt {-e^{2} x^{2}+d^{2}}}{5 d^{2} \left (-e x +d \right )^{3} e^{2}}\) | \(47\) |
gosper | \(-\frac {\left (-e x +d \right ) \left (e x +d \right )^{4} \left (e^{2} x^{2}-3 d e x +d^{2}\right )}{5 d^{2} e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\) | \(52\) |
default | \(e^{3} \left (\frac {x^{3}}{2 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {3 d^{2} \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )}{2 e^{2}}\right )+\frac {d^{3}}{5 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+3 d \,e^{2} \left (\frac {x^{2}}{3 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {2 d^{2}}{15 e^{4} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}\right )+3 d^{2} e \left (\frac {x}{4 e^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}-\frac {d^{2} \left (\frac {x}{5 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}+\frac {\frac {4 x}{15 d^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {3}{2}}}+\frac {8 x}{15 d^{4} \sqrt {-e^{2} x^{2}+d^{2}}}}{d^{2}}\right )}{4 e^{2}}\right )\) | \(308\) |
Time = 0.33 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {e^{3} x^{3} - 3 \, d e^{2} x^{2} + 3 \, d^{2} e x - d^{3} - {\left (e^{2} x^{2} - 3 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}}}{5 \, {\left (d^{2} e^{5} x^{3} - 3 \, d^{3} e^{4} x^{2} + 3 \, d^{4} e^{3} x - d^{5} e^{2}\right )}} \]
-1/5*(e^3*x^3 - 3*d*e^2*x^2 + 3*d^2*e*x - d^3 - (e^2*x^2 - 3*d*e*x + d^2)* sqrt(-e^2*x^2 + d^2))/(d^2*e^5*x^3 - 3*d^3*e^4*x^2 + 3*d^4*e^3*x - d^5*e^2 )
\[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\int \frac {x \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Time = 0.20 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.49 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {e x^{3}}{2 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {d x^{2}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}}} + \frac {3 \, d^{2} x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e} - \frac {d^{3}}{5 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} e^{2}} - \frac {x}{10 \, {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {3}{2}} e} - \frac {x}{5 \, \sqrt {-e^{2} x^{2} + d^{2}} d^{2} e} \]
1/2*e*x^3/(-e^2*x^2 + d^2)^(5/2) + d*x^2/(-e^2*x^2 + d^2)^(5/2) + 3/10*d^2 *x/((-e^2*x^2 + d^2)^(5/2)*e) - 1/5*d^3/((-e^2*x^2 + d^2)^(5/2)*e^2) - 1/1 0*x/((-e^2*x^2 + d^2)^(3/2)*e) - 1/5*x/(sqrt(-e^2*x^2 + d^2)*d^2*e)
Time = 0.31 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.59 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=\frac {2 \, {\left (\frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}}{e^{2} x} - \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{2}}{e^{4} x^{2}} + \frac {5 \, {\left (d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}\right )}^{3}}{e^{6} x^{3}} - 1\right )}}{5 \, d^{2} e {\left (\frac {d e + \sqrt {-e^{2} x^{2} + d^{2}} {\left | e \right |}}{e^{2} x} - 1\right )}^{5} {\left | e \right |}} \]
2/5*(5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 5*(d*e + sqrt(-e^2*x^ 2 + d^2)*abs(e))^2/(e^4*x^2) + 5*(d*e + sqrt(-e^2*x^2 + d^2)*abs(e))^3/(e^ 6*x^3) - 1)/(d^2*e*((d*e + sqrt(-e^2*x^2 + d^2)*abs(e))/(e^2*x) - 1)^5*abs (e))
Time = 11.73 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.53 \[ \int \frac {x (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx=-\frac {\sqrt {d^2-e^2\,x^2}\,\left (d^2-3\,d\,e\,x+e^2\,x^2\right )}{5\,d^2\,e^2\,{\left (d-e\,x\right )}^3} \]