Integrand size = 27, antiderivative size = 109 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {384 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{15 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{21 d^3}+\frac {1152 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3} \] Output:
-384*c^3*(d*x^3+c)^(1/2)/d^3-128/9*c^2*(d*x^3+c)^(3/2)/d^3-14/15*c*(d*x^3+ c)^(5/2)/d^3-2/21*(d*x^3+c)^(7/2)/d^3+1152*c^(7/2)*arctanh(1/3*(d*x^3+c)^( 1/2)/c^(1/2))/d^3
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.75 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {2 \sqrt {c+d x^3} \left (62882 c^3+2579 c^2 d x^3+192 c d^2 x^6+15 d^3 x^9\right )}{315 d^3}+\frac {1152 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3} \] Input:
Integrate[(x^8*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
(-2*Sqrt[c + d*x^3]*(62882*c^3 + 2579*c^2*d*x^3 + 192*c*d^2*x^6 + 15*d^3*x ^9))/(315*d^3) + (1152*c^(7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3
Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {948, 99, 2009}
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.
| \(\displaystyle \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {x^6 \left (d x^3+c\right )^{3/2}}{8 c-d x^3}dx^3\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {1}{3} \int \left (-\frac {\left (d x^3+c\right )^{5/2}}{d^2}+\frac {64 c^2 \left (d x^3+c\right )^{3/2}}{d^2 \left (8 c-d x^3\right )}-\frac {7 c \left (d x^3+c\right )^{3/2}}{d^2}\right )dx^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} \left (\frac {3456 c^{7/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {1152 c^3 \sqrt {c+d x^3}}{d^3}-\frac {128 c^2 \left (c+d x^3\right )^{3/2}}{3 d^3}-\frac {14 c \left (c+d x^3\right )^{5/2}}{5 d^3}-\frac {2 \left (c+d x^3\right )^{7/2}}{7 d^3}\right )\) |
Input:
Int[(x^8*(c + d*x^3)^(3/2))/(8*c - d*x^3),x]
Output:
((-1152*c^3*Sqrt[c + d*x^3])/d^3 - (128*c^2*(c + d*x^3)^(3/2))/(3*d^3) - ( 14*c*(c + d*x^3)^(5/2))/(5*d^3) - (2*(c + d*x^3)^(7/2))/(7*d^3) + (3456*c^ (7/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3)/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 1.16 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
| method | result | size |
| pseudoelliptic | \(\frac {\frac {2 \left (-15 d^{3} x^{9}-192 c \,d^{2} x^{6}-2579 c^{2} d \,x^{3}-62882 c^{3}\right ) \sqrt {d \,x^{3}+c}}{315}+1152 c^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{d^{3}}\) | \(69\) |
| risch | \(-\frac {2 \left (15 d^{3} x^{9}+192 c \,d^{2} x^{6}+2579 c^{2} d \,x^{3}+62882 c^{3}\right ) \sqrt {d \,x^{3}+c}}{315 d^{3}}+\frac {1152 c^{\frac {7}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{d^{3}}\) | \(71\) |
| default | \(-\frac {\frac {2 d \,x^{9} \sqrt {d \,x^{3}+c}}{21}+\frac {16 c \,x^{6} \sqrt {d \,x^{3}+c}}{105}+\frac {2 c^{2} x^{3} \sqrt {d \,x^{3}+c}}{105 d}-\frac {4 c^{3} \sqrt {d \,x^{3}+c}}{105 d^{2}}}{d}-\frac {16 c \left (d \,x^{3}+c \right )^{\frac {5}{2}}}{15 d^{3}}+\frac {128 c^{2} \left (81 c^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )-\left (d \,x^{3}+28 c \right ) \sqrt {d \,x^{3}+c}\right )}{9 d^{3}}\) | \(139\) |
| elliptic | \(-\frac {2 x^{9} \sqrt {d \,x^{3}+c}}{21}-\frac {128 c \,x^{6} \sqrt {d \,x^{3}+c}}{105 d}-\frac {5158 c^{2} x^{3} \sqrt {d \,x^{3}+c}}{315 d^{2}}-\frac {125764 c^{3} \sqrt {d \,x^{3}+c}}{315 d^{3}}-\frac {192 i c^{3} \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^{5}}\) | \(483\) |
Input:
int(x^8*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
2/315*((-15*d^3*x^9-192*c*d^2*x^6-2579*c^2*d*x^3-62882*c^3)*(d*x^3+c)^(1/2 )+181440*c^(7/2)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2)))/d^3
Time = 0.11 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.52 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\left [\frac {2 \, {\left (90720 \, c^{\frac {7}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{315 \, d^{3}}, -\frac {2 \, {\left (181440 \, \sqrt {-c} c^{3} \arctan \left (\frac {3 \, \sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + {\left (15 \, d^{3} x^{9} + 192 \, c d^{2} x^{6} + 2579 \, c^{2} d x^{3} + 62882 \, c^{3}\right )} \sqrt {d x^{3} + c}\right )}}{315 \, d^{3}}\right ] \] Input:
integrate(x^8*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="fricas")
Output:
[2/315*(90720*c^(7/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^ 3 - 8*c)) - (15*d^3*x^9 + 192*c*d^2*x^6 + 2579*c^2*d*x^3 + 62882*c^3)*sqrt (d*x^3 + c))/d^3, -2/315*(181440*sqrt(-c)*c^3*arctan(3*sqrt(-c)/sqrt(d*x^3 + c)) + (15*d^3*x^9 + 192*c*d^2*x^6 + 2579*c^2*d*x^3 + 62882*c^3)*sqrt(d* x^3 + c))/d^3]
Time = 30.91 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.12 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\begin {cases} \frac {2 \left (- \frac {576 c^{4} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{d^{2} \sqrt {- c}} - \frac {192 c^{3} \sqrt {c + d x^{3}}}{d^{2}} - \frac {64 c^{2} \left (c + d x^{3}\right )^{\frac {3}{2}}}{9 d^{2}} - \frac {7 c \left (c + d x^{3}\right )^{\frac {5}{2}}}{15 d^{2}} - \frac {\left (c + d x^{3}\right )^{\frac {7}{2}}}{21 d^{2}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {\sqrt {c} x^{9}}{72} & \text {otherwise} \end {cases} \] Input:
integrate(x**8*(d*x**3+c)**(3/2)/(-d*x**3+8*c),x)
Output:
Piecewise((2*(-576*c**4*atan(sqrt(c + d*x**3)/(3*sqrt(-c)))/(d**2*sqrt(-c) ) - 192*c**3*sqrt(c + d*x**3)/d**2 - 64*c**2*(c + d*x**3)**(3/2)/(9*d**2) - 7*c*(c + d*x**3)**(5/2)/(15*d**2) - (c + d*x**3)**(7/2)/(21*d**2))/d, Ne (d, 0)), (sqrt(c)*x**9/72, True))
Time = 0.11 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.88 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {2 \, {\left (90720 \, c^{\frac {7}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 15 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} + 147 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c + 2240 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{2} + 60480 \, \sqrt {d x^{3} + c} c^{3}\right )}}{315 \, d^{3}} \] Input:
integrate(x^8*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-2/315*(90720*c^(7/2)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3*sqrt(c))) + 15*(d*x^3 + c)^(7/2) + 147*(d*x^3 + c)^(5/2)*c + 2240*(d*x^ 3 + c)^(3/2)*c^2 + 60480*sqrt(d*x^3 + c)*c^3)/d^3
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=-\frac {1152 \, c^{4} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{3}} - \frac {2 \, {\left (15 \, {\left (d x^{3} + c\right )}^{\frac {7}{2}} d^{18} + 147 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} c d^{18} + 2240 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c^{2} d^{18} + 60480 \, \sqrt {d x^{3} + c} c^{3} d^{18}\right )}}{315 \, d^{21}} \] Input:
integrate(x^8*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x, algorithm="giac")
Output:
-1152*c^4*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 2/315*(15* (d*x^3 + c)^(7/2)*d^18 + 147*(d*x^3 + c)^(5/2)*c*d^18 + 2240*(d*x^3 + c)^( 3/2)*c^2*d^18 + 60480*sqrt(d*x^3 + c)*c^3*d^18)/d^21
Time = 1.67 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.06 \[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {576\,c^{7/2}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^3}-\frac {2\,x^9\,\sqrt {d\,x^3+c}}{21}-\frac {125764\,c^3\,\sqrt {d\,x^3+c}}{315\,d^3}-\frac {128\,c\,x^6\,\sqrt {d\,x^3+c}}{105\,d}-\frac {5158\,c^2\,x^3\,\sqrt {d\,x^3+c}}{315\,d^2} \] Input:
int((x^8*(c + d*x^3)^(3/2))/(8*c - d*x^3),x)
Output:
(576*c^(7/2)*log((10*c + d*x^3 + 6*c^(1/2)*(c + d*x^3)^(1/2))/(8*c - d*x^3 )))/d^3 - (2*x^9*(c + d*x^3)^(1/2))/21 - (125764*c^3*(c + d*x^3)^(1/2))/(3 15*d^3) - (128*c*x^6*(c + d*x^3)^(1/2))/(105*d) - (5158*c^2*x^3*(c + d*x^3 )^(1/2))/(315*d^2)
\[ \int \frac {x^8 \left (c+d x^3\right )^{3/2}}{8 c-d x^3} \, dx=\frac {\frac {10316 \sqrt {d \,x^{3}+c}\, c^{3}}{315}-\frac {5158 \sqrt {d \,x^{3}+c}\, c^{2} d \,x^{3}}{315}-\frac {128 \sqrt {d \,x^{3}+c}\, c \,d^{2} x^{6}}{105}-\frac {2 \sqrt {d \,x^{3}+c}\, d^{3} x^{9}}{21}+648 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{3} d^{2}}{d^{3}} \] Input:
int(x^8*(d*x^3+c)^(3/2)/(-d*x^3+8*c),x)
Output:
(2*(5158*sqrt(c + d*x**3)*c**3 - 2579*sqrt(c + d*x**3)*c**2*d*x**3 - 192*s qrt(c + d*x**3)*c*d**2*x**6 - 15*sqrt(c + d*x**3)*d**3*x**9 + 102060*int(( sqrt(c + d*x**3)*x**5)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)*c**3*d**2))/(3 15*d**3)