Integrand size = 27, antiderivative size = 100 \[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {34 c \sqrt {c+d x^3}}{3 d^2}+\frac {24 c^2 \sqrt {c+d x^3}}{d^2 \left (8 c-d x^3\right )}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 d^2}-\frac {42 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2} \] Output:
34/3*c*(d*x^3+c)^(1/2)/d^2+24*c^2*(d*x^3+c)^(1/2)/d^2/(-d*x^3+8*c)+2/9*(d* x^3+c)^(3/2)/d^2-42*c^(3/2)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^2
Time = 0.12 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.81 \[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {2 \sqrt {c+d x^3} \left (-524 c^2+44 c d x^3+d^2 x^6\right )}{9 d^2 \left (-8 c+d x^3\right )}-\frac {42 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^2} \] Input:
Integrate[(x^5*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]
Output:
(2*Sqrt[c + d*x^3]*(-524*c^2 + 44*c*d*x^3 + d^2*x^6))/(9*d^2*(-8*c + d*x^3 )) - (42*c^(3/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^2
Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {948, 87, 60, 60, 73, 219}
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.
| \(\displaystyle \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {x^3 \left (d x^3+c\right )^{3/2}}{\left (8 c-d x^3\right )^2}dx^3\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 \left (c+d x^3\right )^{5/2}}{9 d^2 \left (8 c-d x^3\right )}-\frac {7 \int \frac {\left (d x^3+c\right )^{3/2}}{8 c-d x^3}dx^3}{3 d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 \left (c+d x^3\right )^{5/2}}{9 d^2 \left (8 c-d x^3\right )}-\frac {7 \left (9 c \int \frac {\sqrt {d x^3+c}}{8 c-d x^3}dx^3-\frac {2 \left (c+d x^3\right )^{3/2}}{3 d}\right )}{3 d}\right )\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 \left (c+d x^3\right )^{5/2}}{9 d^2 \left (8 c-d x^3\right )}-\frac {7 \left (9 c \left (9 c \int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3-\frac {2 \sqrt {c+d x^3}}{d}\right )-\frac {2 \left (c+d x^3\right )^{3/2}}{3 d}\right )}{3 d}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 \left (c+d x^3\right )^{5/2}}{9 d^2 \left (8 c-d x^3\right )}-\frac {7 \left (9 c \left (\frac {18 c \int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}}{d}-\frac {2 \sqrt {c+d x^3}}{d}\right )-\frac {2 \left (c+d x^3\right )^{3/2}}{3 d}\right )}{3 d}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{3} \left (\frac {8 \left (c+d x^3\right )^{5/2}}{9 d^2 \left (8 c-d x^3\right )}-\frac {7 \left (9 c \left (\frac {6 \sqrt {c} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d}-\frac {2 \sqrt {c+d x^3}}{d}\right )-\frac {2 \left (c+d x^3\right )^{3/2}}{3 d}\right )}{3 d}\right )\) |
Input:
Int[(x^5*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x]
Output:
((8*(c + d*x^3)^(5/2))/(9*d^2*(8*c - d*x^3)) - (7*((-2*(c + d*x^3)^(3/2))/ (3*d) + 9*c*((-2*Sqrt[c + d*x^3])/d + (6*Sqrt[c]*ArcTanh[Sqrt[c + d*x^3]/( 3*Sqrt[c])])/d)))/(3*d))/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, 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_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Time = 1.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.79
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 \left (d \,x^{3}+c \right )^{\frac {3}{2}}}{9}+\frac {34 c \sqrt {d \,x^{3}+c}}{3}+6 c^{2} \left (\frac {4 \sqrt {d \,x^{3}+c}}{-d \,x^{3}+8 c}-\frac {7 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{\sqrt {c}}\right )}{d^{2}}\) | \(79\) |
| risch | \(\frac {2 \left (d \,x^{3}+52 c \right ) \sqrt {d \,x^{3}+c}}{9 d^{2}}+\frac {9 c^{2} \left (-\frac {50 \,\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{9 \sqrt {c}\, d}+\frac {8 c \left (-\frac {\sqrt {d \,x^{3}+c}}{c \left (d \,x^{3}-8 c \right )}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{3 c^{\frac {3}{2}}}\right )}{3 d}\right )}{d}\) | \(109\) |
| default | \(-\frac {2 \left (81 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )-\left (d \,x^{3}+28 c \right ) \sqrt {d \,x^{3}+c}\right )}{9 d^{2}}+\frac {8 c \left (\frac {2 \sqrt {d \,x^{3}+c}}{3}-3 c \left (-\frac {\sqrt {d \,x^{3}+c}}{-d \,x^{3}+8 c}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{\sqrt {c}}\right )\right )}{d^{2}}\) | \(112\) |
| elliptic | \(\frac {24 c^{2} \sqrt {d \,x^{3}+c}}{d^{2} \left (-d \,x^{3}+8 c \right )}+\frac {2 x^{3} \sqrt {d \,x^{3}+c}}{9 d}+\frac {104 c \sqrt {d \,x^{3}+c}}{9 d^{2}}+\frac {7 i c \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-8 c \right )}{\sum }\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i d \left (2 x +\frac {-i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {d \left (x -\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{-3 \left (-c \,d^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i d \left (2 x +\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}+\left (-c \,d^{2}\right )^{\frac {1}{3}}}{d}\right )}{2 \left (-c \,d^{2}\right )^{\frac {1}{3}}}}\, \left (i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, d -i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} d^{2}-\left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha d -\left (-c \,d^{2}\right )^{\frac {2}{3}}\right ) \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}-\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right ) \sqrt {3}\, d}{\left (-c \,d^{2}\right )^{\frac {1}{3}}}}}{3}, -\frac {2 i \left (-c \,d^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}\, d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}+i \sqrt {3}\, c d -3 \left (-c \,d^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 c d}{18 d c}, \sqrt {\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{d \left (-\frac {3 \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}+\frac {i \sqrt {3}\, \left (-c \,d^{2}\right )^{\frac {1}{3}}}{2 d}\right )}}\right )}{2 \sqrt {d \,x^{3}+c}}\right )}{d^{4}}\) | \(473\) |
Input:
int(x^5*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x,method=_RETURNVERBOSE)
Output:
2*(1/9*(d*x^3+c)^(3/2)+17/3*c*(d*x^3+c)^(1/2)+3*c^2*(4*(d*x^3+c)^(1/2)/(-d *x^3+8*c)-7*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/c^(1/2)))/d^2
Time = 0.08 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.89 \[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\left [\frac {189 \, {\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt {c} \log \left (\frac {d x^{3} - 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) + 2 \, {\left (d^{2} x^{6} + 44 \, c d x^{3} - 524 \, c^{2}\right )} \sqrt {d x^{3} + c}}{9 \, {\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}, \frac {2 \, {\left (189 \, {\left (c d x^{3} - 8 \, c^{2}\right )} \sqrt {-c} \arctan \left (\frac {3 \, \sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + {\left (d^{2} x^{6} + 44 \, c d x^{3} - 524 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{9 \, {\left (d^{3} x^{3} - 8 \, c d^{2}\right )}}\right ] \] Input:
integrate(x^5*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="fricas")
Output:
[1/9*(189*(c*d*x^3 - 8*c^2)*sqrt(c)*log((d*x^3 - 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) + 2*(d^2*x^6 + 44*c*d*x^3 - 524*c^2)*sqrt(d*x^3 + c))/(d^3*x^3 - 8*c*d^2), 2/9*(189*(c*d*x^3 - 8*c^2)*sqrt(-c)*arctan(3*sqrt (-c)/sqrt(d*x^3 + c)) + (d^2*x^6 + 44*c*d*x^3 - 524*c^2)*sqrt(d*x^3 + c))/ (d^3*x^3 - 8*c*d^2)]
\[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\int \frac {x^{5} \left (c + d x^{3}\right )^{\frac {3}{2}}}{\left (- 8 c + d x^{3}\right )^{2}}\, dx \] Input:
integrate(x**5*(d*x**3+c)**(3/2)/(-d*x**3+8*c)**2,x)
Output:
Integral(x**5*(c + d*x**3)**(3/2)/(-8*c + d*x**3)**2, x)
Time = 0.11 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {189 \, c^{\frac {3}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 2 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} + 102 \, \sqrt {d x^{3} + c} c - \frac {216 \, \sqrt {d x^{3} + c} c^{2}}{d x^{3} - 8 \, c}}{9 \, d^{2}} \] Input:
integrate(x^5*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="maxima")
Output:
1/9*(189*c^(3/2)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3*sq rt(c))) + 2*(d*x^3 + c)^(3/2) + 102*sqrt(d*x^3 + c)*c - 216*sqrt(d*x^3 + c )*c^2/(d*x^3 - 8*c))/d^2
Time = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.93 \[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {42 \, c^{2} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{2}} - \frac {24 \, \sqrt {d x^{3} + c} c^{2}}{{\left (d x^{3} - 8 \, c\right )} d^{2}} + \frac {2 \, {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} d^{4} + 51 \, \sqrt {d x^{3} + c} c d^{4}\right )}}{9 \, d^{6}} \] Input:
integrate(x^5*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x, algorithm="giac")
Output:
42*c^2*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^2) - 24*sqrt(d*x^3 + c)*c^2/((d*x^3 - 8*c)*d^2) + 2/9*((d*x^3 + c)^(3/2)*d^4 + 51*sqrt(d*x^3 + c)*c*d^4)/d^6
Time = 2.65 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07 \[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {104\,c\,\sqrt {d\,x^3+c}}{9\,d^2}+\frac {21\,c^{3/2}\,\ln \left (\frac {10\,c+d\,x^3-6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^2}+\frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,d}+\frac {24\,c^2\,\sqrt {d\,x^3+c}}{d^2\,\left (8\,c-d\,x^3\right )} \] Input:
int((x^5*(c + d*x^3)^(3/2))/(8*c - d*x^3)^2,x)
Output:
(104*c*(c + d*x^3)^(1/2))/(9*d^2) + (21*c^(3/2)*log((10*c + d*x^3 - 6*c^(1 /2)*(c + d*x^3)^(1/2))/(8*c - d*x^3)))/d^2 + (2*x^3*(c + d*x^3)^(1/2))/(9* d) + (24*c^2*(c + d*x^3)^(1/2))/(d^2*(8*c - d*x^3))
\[ \int \frac {x^5 \left (c+d x^3\right )^{3/2}}{\left (8 c-d x^3\right )^2} \, dx=\frac {704 \sqrt {d \,x^{3}+c}\, c^{2}-440 \sqrt {d \,x^{3}+c}\, c d \,x^{3}-10 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}+122472 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{3} d^{2}-15309 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{d^{3} x^{9}-15 c \,d^{2} x^{6}+48 c^{2} d \,x^{3}+64 c^{3}}d x \right ) c^{2} d^{3} x^{3}}{45 d^{2} \left (-d \,x^{3}+8 c \right )} \] Input:
int(x^5*(d*x^3+c)^(3/2)/(-d*x^3+8*c)^2,x)
Output:
(704*sqrt(c + d*x**3)*c**2 - 440*sqrt(c + d*x**3)*c*d*x**3 - 10*sqrt(c + d *x**3)*d**2*x**6 + 122472*int((sqrt(c + d*x**3)*x**5)/(64*c**3 + 48*c**2*d *x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c**3*d**2 - 15309*int((sqrt(c + d*x **3)*x**5)/(64*c**3 + 48*c**2*d*x**3 - 15*c*d**2*x**6 + d**3*x**9),x)*c**2 *d**3*x**3)/(45*d**2*(8*c - d*x**3))