Integrand size = 23, antiderivative size = 188 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=-\frac {c \sqrt {c-d x} \sqrt {c+d x}}{6 x^6}+\frac {d^2 \sqrt {c-d x} \sqrt {c+d x}}{24 c x^4}+\frac {d^4 \sqrt {c-d x} \sqrt {c+d x}}{16 c^3 x^2}-\frac {d (c-d x)^{3/2} (c+d x)^{3/2}}{5 c^2 x^5}-\frac {2 d^3 (c-d x)^{3/2} (c+d x)^{3/2}}{15 c^4 x^3}+\frac {d^6 \text {arctanh}\left (\frac {\sqrt {c-d x} \sqrt {c+d x}}{c}\right )}{16 c^4} \] Output:
-1/6*c*(-d*x+c)^(1/2)*(d*x+c)^(1/2)/x^6+1/24*d^2*(-d*x+c)^(1/2)*(d*x+c)^(1 /2)/c/x^4+1/16*d^4*(-d*x+c)^(1/2)*(d*x+c)^(1/2)/c^3/x^2-1/5*d*(-d*x+c)^(3/ 2)*(d*x+c)^(3/2)/c^2/x^5-2/15*d^3*(-d*x+c)^(3/2)*(d*x+c)^(3/2)/c^4/x^3+1/1 6*d^6*arctanh((-d*x+c)^(1/2)*(d*x+c)^(1/2)/c)/c^4
Time = 0.14 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=\frac {\frac {\sqrt {c-d x} \left (-40 c^6-88 c^5 d x-38 c^4 d^2 x^2+26 c^3 d^3 x^3+31 c^2 d^4 x^4+47 c d^5 x^5+32 d^6 x^6\right )}{x^6 \sqrt {c+d x}}+30 d^6 \text {arctanh}\left (\frac {\sqrt {c-d x}}{\sqrt {c+d x}}\right )}{240 c^4} \] Input:
Integrate[(Sqrt[c - d*x]*(c + d*x)^(3/2))/x^7,x]
Output:
((Sqrt[c - d*x]*(-40*c^6 - 88*c^5*d*x - 38*c^4*d^2*x^2 + 26*c^3*d^3*x^3 + 31*c^2*d^4*x^4 + 47*c*d^5*x^5 + 32*d^6*x^6))/(x^6*Sqrt[c + d*x]) + 30*d^6* ArcTanh[Sqrt[c - d*x]/Sqrt[c + d*x]])/(240*c^4)
Time = 0.32 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.26, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.739, Rules used = {108, 27, 166, 25, 27, 168, 27, 168, 25, 27, 168, 25, 27, 168, 27, 103, 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.
| \(\displaystyle \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{6} \int \frac {d (c-2 d x) \sqrt {c+d x}}{x^6 \sqrt {c-d x}}dx-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} d \int \frac {(c-2 d x) \sqrt {c+d x}}{x^6 \sqrt {c-d x}}dx-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle \frac {1}{6} d \left (\frac {\int -\frac {c d (5 c+6 d x)}{x^5 \sqrt {c-d x} \sqrt {c+d x}}dx}{5 c}-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {\int \frac {c d (5 c+6 d x)}{x^5 \sqrt {c-d x} \sqrt {c+d x}}dx}{5 c}-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \int \frac {5 c+6 d x}{x^5 \sqrt {c-d x} \sqrt {c+d x}}dx-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (-\frac {\int -\frac {3 c d (8 c+5 d x)}{x^4 \sqrt {c-d x} \sqrt {c+d x}}dx}{4 c^2}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \int \frac {8 c+5 d x}{x^4 \sqrt {c-d x} \sqrt {c+d x}}dx}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (-\frac {\int -\frac {c d (15 c+16 d x)}{x^3 \sqrt {c-d x} \sqrt {c+d x}}dx}{3 c^2}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {\int \frac {c d (15 c+16 d x)}{x^3 \sqrt {c-d x} \sqrt {c+d x}}dx}{3 c^2}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \int \frac {15 c+16 d x}{x^3 \sqrt {c-d x} \sqrt {c+d x}}dx}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (-\frac {\int -\frac {c d (32 c+15 d x)}{x^2 \sqrt {c-d x} \sqrt {c+d x}}dx}{2 c^2}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (\frac {\int \frac {c d (32 c+15 d x)}{x^2 \sqrt {c-d x} \sqrt {c+d x}}dx}{2 c^2}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (\frac {d \int \frac {32 c+15 d x}{x^2 \sqrt {c-d x} \sqrt {c+d x}}dx}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 168 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (\frac {d \left (-\frac {\int -\frac {15 c^2 d}{x \sqrt {c-d x} \sqrt {c+d x}}dx}{c^2}-\frac {32 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (\frac {d \left (15 d \int \frac {1}{x \sqrt {c-d x} \sqrt {c+d x}}dx-\frac {32 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 103 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (\frac {d \left (-15 d^2 \int \frac {1}{c^2 d-d (c-d x) (c+d x)}d\left (\sqrt {c-d x} \sqrt {c+d x}\right )-\frac {32 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{6} d \left (-\frac {1}{5} d \left (\frac {3 d \left (\frac {d \left (\frac {d \left (-\frac {15 d \text {arctanh}\left (\frac {\sqrt {c-d x} \sqrt {c+d x}}{c}\right )}{c}-\frac {32 \sqrt {c-d x} \sqrt {c+d x}}{c x}\right )}{2 c}-\frac {15 \sqrt {c-d x} \sqrt {c+d x}}{2 c x^2}\right )}{3 c}-\frac {8 \sqrt {c-d x} \sqrt {c+d x}}{3 c x^3}\right )}{4 c}-\frac {5 \sqrt {c-d x} \sqrt {c+d x}}{4 c x^4}\right )-\frac {\sqrt {c-d x} \sqrt {c+d x}}{5 x^5}\right )-\frac {\sqrt {c-d x} (c+d x)^{3/2}}{6 x^6}\) |
Input:
Int[(Sqrt[c - d*x]*(c + d*x)^(3/2))/x^7,x]
Output:
-1/6*(Sqrt[c - d*x]*(c + d*x)^(3/2))/x^6 + (d*(-1/5*(Sqrt[c - d*x]*Sqrt[c + d*x])/x^5 - (d*((-5*Sqrt[c - d*x]*Sqrt[c + d*x])/(4*c*x^4) + (3*d*((-8*S qrt[c - d*x]*Sqrt[c + d*x])/(3*c*x^3) + (d*((-15*Sqrt[c - d*x]*Sqrt[c + d* x])/(2*c*x^2) + (d*((-32*Sqrt[c - d*x]*Sqrt[c + d*x])/(c*x) - (15*d*ArcTan h[(Sqrt[c - d*x]*Sqrt[c + d*x])/c])/c))/(2*c)))/(3*c)))/(4*c)))/5))/6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
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]
Time = 0.24 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80
| method | result | size |
| risch | \(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-32 d^{5} x^{5}-15 c \,d^{4} x^{4}-16 c^{2} d^{3} x^{3}-10 c^{3} d^{2} x^{2}+48 c^{4} d x +40 c^{5}\right )}{240 x^{6} c^{4}}+\frac {d^{6} \ln \left (\frac {2 c^{2}+2 \sqrt {c^{2}}\, \sqrt {-d^{2} x^{2}+c^{2}}}{x}\right ) \sqrt {\left (-x d +c \right ) \left (x d +c \right )}}{16 c^{3} \sqrt {c^{2}}\, \sqrt {-x d +c}\, \sqrt {x d +c}}\) | \(151\) |
| default | \(-\frac {\sqrt {-x d +c}\, \sqrt {x d +c}\, \left (-15 \ln \left (\frac {2 c \left (\sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right )+c \right )}{x}\right ) d^{6} x^{6}-32 \sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right ) d^{5} x^{5}-15 \sqrt {-d^{2} x^{2}+c^{2}}\, \operatorname {csgn}\left (c \right ) c \,d^{4} x^{4}-16 \,\operatorname {csgn}\left (c \right ) c^{2} d^{3} x^{3} \sqrt {-d^{2} x^{2}+c^{2}}-10 \,\operatorname {csgn}\left (c \right ) c^{3} d^{2} x^{2} \sqrt {-d^{2} x^{2}+c^{2}}+48 \,\operatorname {csgn}\left (c \right ) c^{4} d x \sqrt {-d^{2} x^{2}+c^{2}}+40 \,\operatorname {csgn}\left (c \right ) c^{5} \sqrt {-d^{2} x^{2}+c^{2}}\right ) \operatorname {csgn}\left (c \right )}{240 c^{4} \sqrt {-d^{2} x^{2}+c^{2}}\, x^{6}}\) | \(222\) |
Input:
int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^7,x,method=_RETURNVERBOSE)
Output:
-1/240*(-d*x+c)^(1/2)*(d*x+c)^(1/2)*(-32*d^5*x^5-15*c*d^4*x^4-16*c^2*d^3*x ^3-10*c^3*d^2*x^2+48*c^4*d*x+40*c^5)/x^6/c^4+1/16*d^6/c^3/(c^2)^(1/2)*ln(( 2*c^2+2*(c^2)^(1/2)*(-d^2*x^2+c^2)^(1/2))/x)*((-d*x+c)*(d*x+c))^(1/2)/(-d* x+c)^(1/2)/(d*x+c)^(1/2)
Time = 0.09 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=-\frac {15 \, d^{6} x^{6} \log \left (\frac {\sqrt {d x + c} \sqrt {-d x + c} - c}{x}\right ) - {\left (32 \, d^{5} x^{5} + 15 \, c d^{4} x^{4} + 16 \, c^{2} d^{3} x^{3} + 10 \, c^{3} d^{2} x^{2} - 48 \, c^{4} d x - 40 \, c^{5}\right )} \sqrt {d x + c} \sqrt {-d x + c}}{240 \, c^{4} x^{6}} \] Input:
integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^7,x, algorithm="fricas")
Output:
-1/240*(15*d^6*x^6*log((sqrt(d*x + c)*sqrt(-d*x + c) - c)/x) - (32*d^5*x^5 + 15*c*d^4*x^4 + 16*c^2*d^3*x^3 + 10*c^3*d^2*x^2 - 48*c^4*d*x - 40*c^5)*s qrt(d*x + c)*sqrt(-d*x + c))/(c^4*x^6)
\[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=\int \frac {\sqrt {c - d x} \left (c + d x\right )^{\frac {3}{2}}}{x^{7}}\, dx \] Input:
integrate((-d*x+c)**(1/2)*(d*x+c)**(3/2)/x**7,x)
Output:
Integral(sqrt(c - d*x)*(c + d*x)**(3/2)/x**7, x)
Time = 0.11 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=\frac {d^{6} \log \left (\frac {2 \, c^{2}}{{\left | x \right |}} + \frac {2 \, \sqrt {-d^{2} x^{2} + c^{2}} c}{{\left | x \right |}}\right )}{16 \, c^{4}} - \frac {\sqrt {-d^{2} x^{2} + c^{2}} d^{6}}{16 \, c^{5}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{4}}{16 \, c^{5} x^{2}} - \frac {2 \, {\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{3}}{15 \, c^{4} x^{3}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d^{2}}{8 \, c^{3} x^{4}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}} d}{5 \, c^{2} x^{5}} - \frac {{\left (-d^{2} x^{2} + c^{2}\right )}^{\frac {3}{2}}}{6 \, c x^{6}} \] Input:
integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^7,x, algorithm="maxima")
Output:
1/16*d^6*log(2*c^2/abs(x) + 2*sqrt(-d^2*x^2 + c^2)*c/abs(x))/c^4 - 1/16*sq rt(-d^2*x^2 + c^2)*d^6/c^5 - 1/16*(-d^2*x^2 + c^2)^(3/2)*d^4/(c^5*x^2) - 2 /15*(-d^2*x^2 + c^2)^(3/2)*d^3/(c^4*x^3) - 1/8*(-d^2*x^2 + c^2)^(3/2)*d^2/ (c^3*x^4) - 1/5*(-d^2*x^2 + c^2)^(3/2)*d/(c^2*x^5) - 1/6*(-d^2*x^2 + c^2)^ (3/2)/(c*x^6)
Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (152) = 304\).
Time = 0.46 (sec) , antiderivative size = 584, normalized size of antiderivative = 3.11 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=\frac {\frac {15 \, d^{7} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} + 2 \right |}\right )}{c^{4}} - \frac {15 \, d^{7} \log \left ({\left | -\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} + \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}} - 2 \right |}\right )}{c^{4}} + \frac {4 \, {\left (15 \, d^{7} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{11} - 340 \, d^{7} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{9} + 9312 \, d^{7} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{7} + 12672 \, d^{7} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{5} + 142080 \, d^{7} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{3} + 15360 \, d^{7} {\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}\right )}}{{\left ({\left (\frac {\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}{\sqrt {d x + c}} - \frac {\sqrt {d x + c}}{\sqrt {2} \sqrt {c} - \sqrt {-d x + c}}\right )}^{2} - 4\right )}^{6} c^{4}}}{240 \, d} \] Input:
integrate((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^7,x, algorithm="giac")
Output:
1/240*(15*d^7*log(abs(-(sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) + sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)) + 2))/c^4 - 15*d^7*log(ab s(-(sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) + sqrt(d*x + c)/(sqrt( 2)*sqrt(c) - sqrt(-d*x + c)) - 2))/c^4 + 4*(15*d^7*((sqrt(2)*sqrt(c) - sqr t(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)))^11 - 340*d^7*((sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt (d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)))^9 + 9312*d^7*((sqrt(2)*sqrt( c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt (-d*x + c)))^7 + 12672*d^7*((sqrt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c)))^5 + 142080*d^7*((sq rt(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqr t(c) - sqrt(-d*x + c)))^3 + 15360*d^7*((sqrt(2)*sqrt(c) - sqrt(-d*x + c))/ sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt(c) - sqrt(-d*x + c))))/((((sqr t(2)*sqrt(c) - sqrt(-d*x + c))/sqrt(d*x + c) - sqrt(d*x + c)/(sqrt(2)*sqrt (c) - sqrt(-d*x + c)))^2 - 4)^6*c^4))/d
Timed out. \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=\int \frac {{\left (c+d\,x\right )}^{3/2}\,\sqrt {c-d\,x}}{x^7} \,d x \] Input:
int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^7,x)
Output:
int(((c + d*x)^(3/2)*(c - d*x)^(1/2))/x^7, x)
Time = 0.20 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.46 \[ \int \frac {\sqrt {c-d x} (c+d x)^{3/2}}{x^7} \, dx=\frac {-40 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{5}-48 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{4} d x +10 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{3} d^{2} x^{2}+16 \sqrt {d x +c}\, \sqrt {-d x +c}\, c^{2} d^{3} x^{3}+15 \sqrt {d x +c}\, \sqrt {-d x +c}\, c \,d^{4} x^{4}+32 \sqrt {d x +c}\, \sqrt {-d x +c}\, d^{5} x^{5}+15 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{6} x^{6}-15 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{6} x^{6}+15 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )-1\right ) d^{6} x^{6}-15 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-d x +c}}{\sqrt {c}\, \sqrt {2}}\right )}{2}\right )+1\right ) d^{6} x^{6}}{240 c^{4} x^{6}} \] Input:
int((-d*x+c)^(1/2)*(d*x+c)^(3/2)/x^7,x)
Output:
( - 40*sqrt(c + d*x)*sqrt(c - d*x)*c**5 - 48*sqrt(c + d*x)*sqrt(c - d*x)*c **4*d*x + 10*sqrt(c + d*x)*sqrt(c - d*x)*c**3*d**2*x**2 + 16*sqrt(c + d*x) *sqrt(c - d*x)*c**2*d**3*x**3 + 15*sqrt(c + d*x)*sqrt(c - d*x)*c*d**4*x**4 + 32*sqrt(c + d*x)*sqrt(c - d*x)*d**5*x**5 + 15*log( - sqrt(2) + tan(asin (sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) - 1)*d**6*x**6 - 15*log( - sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) + 1)*d**6*x**6 + 15*log(sqrt( 2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) - 1)*d**6*x**6 - 15*log( sqrt(2) + tan(asin(sqrt(c - d*x)/(sqrt(c)*sqrt(2)))/2) + 1)*d**6*x**6)/(24 0*c**4*x**6)