Integrand size = 27, antiderivative size = 218 \[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {d^2 (7 c-9 d x)}{7 c^4 \left (c^2-d^2 x^2\right )^{5/2}}+\frac {4 d^2}{7 c^2 (c+d x) \left (c^2-d^2 x^2\right )^{5/2}}+\frac {d^2 (14 c-19 d x)}{7 c^6 \left (c^2-d^2 x^2\right )^{3/2}}+\frac {d^2 (49 c-59 d x)}{7 c^8 \sqrt {c^2-d^2 x^2}}-\frac {\sqrt {c^2-d^2 x^2}}{2 c^7 x^2}+\frac {3 d \sqrt {c^2-d^2 x^2}}{c^8 x}-\frac {15 d^2 \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{c}\right )}{2 c^8} \] Output:
1/7*d^2*(-9*d*x+7*c)/c^4/(-d^2*x^2+c^2)^(5/2)+4/7*d^2/c^2/(d*x+c)/(-d^2*x^ 2+c^2)^(5/2)+1/7*d^2*(-19*d*x+14*c)/c^6/(-d^2*x^2+c^2)^(3/2)+1/7*d^2*(-59* d*x+49*c)/c^8/(-d^2*x^2+c^2)^(1/2)-1/2*(-d^2*x^2+c^2)^(1/2)/c^7/x^2+3*d*(- d^2*x^2+c^2)^(1/2)/c^8/x-15/2*d^2*arctanh((-d^2*x^2+c^2)^(1/2)/c)/c^8
Time = 0.65 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {\frac {c \sqrt {c^2-d^2 x^2} \left (7 c^6-21 c^5 d x-260 c^4 d^2 x^2-360 c^3 d^3 x^3+85 c^2 d^4 x^4+375 c d^5 x^5+160 d^6 x^6\right )}{x^2 (-c+d x) (c+d x)^4}-105 \sqrt {c^2} d^2 \log (x)+105 \sqrt {c^2} d^2 \log \left (\sqrt {c^2}-\sqrt {c^2-d^2 x^2}\right )}{14 c^9} \] Input:
Integrate[1/(x^3*(c + d*x)^3*(c^2 - d^2*x^2)^(3/2)),x]
Output:
((c*Sqrt[c^2 - d^2*x^2]*(7*c^6 - 21*c^5*d*x - 260*c^4*d^2*x^2 - 360*c^3*d^ 3*x^3 + 85*c^2*d^4*x^4 + 375*c*d^5*x^5 + 160*d^6*x^6))/(x^2*(-c + d*x)*(c + d*x)^4) - 105*Sqrt[c^2]*d^2*Log[x] + 105*Sqrt[c^2]*d^2*Log[Sqrt[c^2] - S qrt[c^2 - d^2*x^2]])/(14*c^9)
Time = 1.45 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {570, 532, 25, 2336, 27, 2336, 27, 2336, 27, 2338, 27, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 570 |
| \(\displaystyle \int \frac {(c-d x)^3}{x^3 \left (c^2-d^2 x^2\right )^{9/2}}dx\) |
| \(\Big \downarrow \) 532 |
| \(\displaystyle \frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}-\frac {\int -\frac {7 c^3-21 d x c^2+28 d^2 x^2 c-24 d^3 x^3}{x^3 \left (c^2-d^2 x^2\right )^{7/2}}dx}{7 c^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {7 c^3-21 d x c^2+28 d^2 x^2 c-24 d^3 x^3}{x^3 \left (c^2-d^2 x^2\right )^{7/2}}dx}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}-\frac {\int -\frac {5 \left (7 c^3-21 d x c^2+35 d^2 x^2 c-36 d^3 x^3\right )}{x^3 \left (c^2-d^2 x^2\right )^{5/2}}dx}{5 c^2}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {7 c^3-21 d x c^2+35 d^2 x^2 c-36 d^3 x^3}{x^3 \left (c^2-d^2 x^2\right )^{5/2}}dx}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}-\frac {\int -\frac {3 \left (7 c^3-21 d x c^2+42 d^2 x^2 c-38 d^3 x^3\right )}{x^3 \left (c^2-d^2 x^2\right )^{3/2}}dx}{3 c^2}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {7 c^3-21 d x c^2+42 d^2 x^2 c-38 d^3 x^3}{x^3 \left (c^2-d^2 x^2\right )^{3/2}}dx}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2336 |
| \(\displaystyle \frac {\frac {\frac {\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}-\frac {\int -\frac {7 \left (c^3-3 d x c^2+7 d^2 x^2 c\right )}{x^3 \sqrt {c^2-d^2 x^2}}dx}{c^2}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \int \frac {c^3-3 d x c^2+7 d^2 x^2 c}{x^3 \sqrt {c^2-d^2 x^2}}dx}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \left (-\frac {\int \frac {3 c^3 d (2 c-5 d x)}{x^2 \sqrt {c^2-d^2 x^2}}dx}{2 c^2}-\frac {c \sqrt {c^2-d^2 x^2}}{2 x^2}\right )}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \left (-\frac {3}{2} c d \int \frac {2 c-5 d x}{x^2 \sqrt {c^2-d^2 x^2}}dx-\frac {c \sqrt {c^2-d^2 x^2}}{2 x^2}\right )}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \left (-\frac {3}{2} c d \left (-5 d \int \frac {1}{x \sqrt {c^2-d^2 x^2}}dx-\frac {2 \sqrt {c^2-d^2 x^2}}{c x}\right )-\frac {c \sqrt {c^2-d^2 x^2}}{2 x^2}\right )}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \left (-\frac {3}{2} c d \left (-\frac {5}{2} d \int \frac {1}{x^2 \sqrt {c^2-d^2 x^2}}dx^2-\frac {2 \sqrt {c^2-d^2 x^2}}{c x}\right )-\frac {c \sqrt {c^2-d^2 x^2}}{2 x^2}\right )}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \left (-\frac {3}{2} c d \left (\frac {5 \int \frac {1}{\frac {c^2}{d^2}-\frac {x^4}{d^2}}d\sqrt {c^2-d^2 x^2}}{d}-\frac {2 \sqrt {c^2-d^2 x^2}}{c x}\right )-\frac {c \sqrt {c^2-d^2 x^2}}{2 x^2}\right )}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\frac {7 \left (-\frac {3}{2} c d \left (\frac {5 d \text {arctanh}\left (\frac {\sqrt {c^2-d^2 x^2}}{c}\right )}{c}-\frac {2 \sqrt {c^2-d^2 x^2}}{c x}\right )-\frac {c \sqrt {c^2-d^2 x^2}}{2 x^2}\right )}{c^2}+\frac {d^2 (49 c-59 d x)}{c^2 \sqrt {c^2-d^2 x^2}}}{c^2}+\frac {d^2 (14 c-19 d x)}{c^2 \left (c^2-d^2 x^2\right )^{3/2}}}{c^2}+\frac {d^2 (7 c-9 d x)}{c^2 \left (c^2-d^2 x^2\right )^{5/2}}}{7 c^2}+\frac {4 d^2 (c-d x)}{7 c^2 \left (c^2-d^2 x^2\right )^{7/2}}\) |
Input:
Int[1/(x^3*(c + d*x)^3*(c^2 - d^2*x^2)^(3/2)),x]
Output:
(4*d^2*(c - d*x))/(7*c^2*(c^2 - d^2*x^2)^(7/2)) + ((d^2*(7*c - 9*d*x))/(c^ 2*(c^2 - d^2*x^2)^(5/2)) + ((d^2*(14*c - 19*d*x))/(c^2*(c^2 - d^2*x^2)^(3/ 2)) + ((d^2*(49*c - 59*d*x))/(c^2*Sqrt[c^2 - d^2*x^2]) + (7*(-1/2*(c*Sqrt[ c^2 - d^2*x^2])/x^2 - (3*c*d*((-2*Sqrt[c^2 - d^2*x^2])/(c*x) + (5*d*ArcTan h[Sqrt[c^2 - d^2*x^2]/c])/c))/2))/c^2)/c^2)/c^2)/(7*c^2)
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[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[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {Q = PolynomialQuotient[(c*x)^m*Pq, a + b*x^2, x], f = Coeff[PolynomialRema inder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[(c*x) ^m*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)) Int[(c*x)^m*(a + b*x^2)^(p + 1)*Ex pandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x], x]] /; F reeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Time = 0.33 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.36
| method | result | size |
| risch | \(-\frac {\sqrt {-d^{2} x^{2}+c^{2}}\, \left (-6 d x +c \right )}{2 c^{8} x^{2}}-\frac {15 d^{2} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{2 c^{7} \sqrt {c^{2}}}-\frac {d \sqrt {-d^{2} \left (x -\frac {c}{d}\right )^{2}-2 c d \left (x -\frac {c}{d}\right )}}{16 c^{8} \left (x -\frac {c}{d}\right )}+\frac {951 d \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{112 c^{8} \left (x +\frac {c}{d}\right )}+\frac {87 \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{56 c^{7} \left (x +\frac {c}{d}\right )^{2}}+\frac {11 \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{28 c^{6} d \left (x +\frac {c}{d}\right )^{3}}+\frac {\sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}{14 c^{5} d^{2} \left (x +\frac {c}{d}\right )^{4}}\) | \(296\) |
| default | \(\frac {-\frac {1}{2 c^{2} x^{2} \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {3 d^{2} \left (\frac {1}{c^{2} \sqrt {-d^{2} x^{2}+c^{2}}}-\frac {\ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{c^{2} \sqrt {c^{2}}}\right )}{2 c^{2}}}{c^{3}}+\frac {6 d^{2} \left (\frac {1}{c^{2} \sqrt {-d^{2} x^{2}+c^{2}}}-\frac {\ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right )}{c^{2} \sqrt {c^{2}}}\right )}{c^{5}}-\frac {3 d \left (-\frac {1}{c^{2} x \sqrt {-d^{2} x^{2}+c^{2}}}+\frac {2 d^{2} x}{c^{4} \sqrt {-d^{2} x^{2}+c^{2}}}\right )}{c^{4}}-\frac {6 d^{2} \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{c^{5}}-\frac {3 d \left (-\frac {1}{5 c d \left (x +\frac {c}{d}\right )^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {3 d \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{c^{4}}-\frac {-\frac {1}{7 c d \left (x +\frac {c}{d}\right )^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {4 d \left (-\frac {1}{5 c d \left (x +\frac {c}{d}\right )^{2} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}+\frac {3 d \left (-\frac {1}{3 c d \left (x +\frac {c}{d}\right ) \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}-\frac {-2 d^{2} \left (x +\frac {c}{d}\right )+2 c d}{3 d \,c^{3} \sqrt {-d^{2} \left (x +\frac {c}{d}\right )^{2}+2 c d \left (x +\frac {c}{d}\right )}}\right )}{5 c}\right )}{7 c}}{c^{3}}\) | \(691\) |
Input:
int(1/x^3/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(-d^2*x^2+c^2)^(1/2)*(-6*d*x+c)/c^8/x^2-15/2/c^7*d^2/(c^2)^(1/2)*ln(( 2*c^2+2*(c^2)^(1/2)*(-d^2*x^2+c^2)^(1/2))/x)-1/16/c^8*d/(x-c/d)*(-d^2*(x-c /d)^2-2*c*d*(x-c/d))^(1/2)+951/112/c^8*d/(x+c/d)*(-d^2*(x+c/d)^2+2*c*d*(x+ c/d))^(1/2)+87/56/c^7/(x+c/d)^2*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)+11/28 /c^6/d/(x+c/d)^3*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)+1/14/c^5/d^2/(x+c/d) ^4*(-d^2*(x+c/d)^2+2*c*d*(x+c/d))^(1/2)
Time = 0.17 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.34 \[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {148 \, d^{7} x^{7} + 444 \, c d^{6} x^{6} + 296 \, c^{2} d^{5} x^{5} - 296 \, c^{3} d^{4} x^{4} - 444 \, c^{4} d^{3} x^{3} - 148 \, c^{5} d^{2} x^{2} + 105 \, {\left (d^{7} x^{7} + 3 \, c d^{6} x^{6} + 2 \, c^{2} d^{5} x^{5} - 2 \, c^{3} d^{4} x^{4} - 3 \, c^{4} d^{3} x^{3} - c^{5} d^{2} x^{2}\right )} \log \left (-\frac {c - \sqrt {-d^{2} x^{2} + c^{2}}}{x}\right ) + {\left (160 \, d^{6} x^{6} + 375 \, c d^{5} x^{5} + 85 \, c^{2} d^{4} x^{4} - 360 \, c^{3} d^{3} x^{3} - 260 \, c^{4} d^{2} x^{2} - 21 \, c^{5} d x + 7 \, c^{6}\right )} \sqrt {-d^{2} x^{2} + c^{2}}}{14 \, {\left (c^{8} d^{5} x^{7} + 3 \, c^{9} d^{4} x^{6} + 2 \, c^{10} d^{3} x^{5} - 2 \, c^{11} d^{2} x^{4} - 3 \, c^{12} d x^{3} - c^{13} x^{2}\right )}} \] Input:
integrate(1/x^3/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x, algorithm="fricas")
Output:
1/14*(148*d^7*x^7 + 444*c*d^6*x^6 + 296*c^2*d^5*x^5 - 296*c^3*d^4*x^4 - 44 4*c^4*d^3*x^3 - 148*c^5*d^2*x^2 + 105*(d^7*x^7 + 3*c*d^6*x^6 + 2*c^2*d^5*x ^5 - 2*c^3*d^4*x^4 - 3*c^4*d^3*x^3 - c^5*d^2*x^2)*log(-(c - sqrt(-d^2*x^2 + c^2))/x) + (160*d^6*x^6 + 375*c*d^5*x^5 + 85*c^2*d^4*x^4 - 360*c^3*d^3*x ^3 - 260*c^4*d^2*x^2 - 21*c^5*d*x + 7*c^6)*sqrt(-d^2*x^2 + c^2))/(c^8*d^5* x^7 + 3*c^9*d^4*x^6 + 2*c^10*d^3*x^5 - 2*c^11*d^2*x^4 - 3*c^12*d*x^3 - c^1 3*x^2)
\[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (- \left (- c + d x\right ) \left (c + d x\right )\right )^{\frac {3}{2}} \left (c + d x\right )^{3}}\, dx \] Input:
integrate(1/x**3/(d*x+c)**3/(-d**2*x**2+c**2)**(3/2),x)
Output:
Integral(1/(x**3*(-(-c + d*x)*(c + d*x))**(3/2)*(c + d*x)**3), x)
\[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{3} x^{3}} \,d x } \] Input:
integrate(1/x^3/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^3*x^3), x)
\[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} {\left (d x + c\right )}^{3} x^{3}} \,d x } \] Input:
integrate(1/x^3/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x, algorithm="giac")
Output:
integrate(1/((-d^2*x^2 + c^2)^(3/2)*(d*x + c)^3*x^3), x)
Timed out. \[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (c^2-d^2\,x^2\right )}^{3/2}\,{\left (c+d\,x\right )}^3} \,d x \] Input:
int(1/(x^3*(c^2 - d^2*x^2)^(3/2)*(c + d*x)^3),x)
Output:
int(1/(x^3*(c^2 - d^2*x^2)^(3/2)*(c + d*x)^3), x)
Time = 0.21 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.56 \[ \int \frac {1}{x^3 (c+d x)^3 \left (c^2-d^2 x^2\right )^{3/2}} \, dx=\frac {420 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {d x}{c}\right )}{2}\right )\right ) c^{3} d^{2} x^{2}+1260 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {d x}{c}\right )}{2}\right )\right ) c^{2} d^{3} x^{3}+1260 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {d x}{c}\right )}{2}\right )\right ) c \,d^{4} x^{4}+420 \sqrt {-d^{2} x^{2}+c^{2}}\, \mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {d x}{c}\right )}{2}\right )\right ) d^{5} x^{5}+515 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{3} d^{2} x^{2}+1545 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{2} d^{3} x^{3}+1545 \sqrt {-d^{2} x^{2}+c^{2}}\, c \,d^{4} x^{4}+515 \sqrt {-d^{2} x^{2}+c^{2}}\, d^{5} x^{5}-28 c^{6}+84 c^{5} d x +1040 c^{4} d^{2} x^{2}+1440 c^{3} d^{3} x^{3}-340 c^{2} d^{4} x^{4}-1500 c \,d^{5} x^{5}-640 d^{6} x^{6}}{56 \sqrt {-d^{2} x^{2}+c^{2}}\, c^{8} x^{2} \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right )} \] Input:
int(1/x^3/(d*x+c)^3/(-d^2*x^2+c^2)^(3/2),x)
Output:
(420*sqrt(c**2 - d**2*x**2)*log(tan(asin((d*x)/c)/2))*c**3*d**2*x**2 + 126 0*sqrt(c**2 - d**2*x**2)*log(tan(asin((d*x)/c)/2))*c**2*d**3*x**3 + 1260*s qrt(c**2 - d**2*x**2)*log(tan(asin((d*x)/c)/2))*c*d**4*x**4 + 420*sqrt(c** 2 - d**2*x**2)*log(tan(asin((d*x)/c)/2))*d**5*x**5 + 515*sqrt(c**2 - d**2* x**2)*c**3*d**2*x**2 + 1545*sqrt(c**2 - d**2*x**2)*c**2*d**3*x**3 + 1545*s qrt(c**2 - d**2*x**2)*c*d**4*x**4 + 515*sqrt(c**2 - d**2*x**2)*d**5*x**5 - 28*c**6 + 84*c**5*d*x + 1040*c**4*d**2*x**2 + 1440*c**3*d**3*x**3 - 340*c **2*d**4*x**4 - 1500*c*d**5*x**5 - 640*d**6*x**6)/(56*sqrt(c**2 - d**2*x** 2)*c**8*x**2*(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3))