Integrand size = 27, antiderivative size = 90 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=-\frac {128 c^2 \sqrt {c+d x^3}}{3 d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{9 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{15 d^3}+\frac {128 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3} \] Output:
-128/3*c^2*(d*x^3+c)^(1/2)/d^3-14/9*c*(d*x^3+c)^(3/2)/d^3-2/15*(d*x^3+c)^( 5/2)/d^3+128*c^(5/2)*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))/d^3
Time = 0.11 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.79 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=-\frac {2 \sqrt {c+d x^3} \left (998 c^2+41 c d x^3+3 d^2 x^6\right )}{45 d^3}+\frac {128 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3} \] Input:
Integrate[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3),x]
Output:
(-2*Sqrt[c + d*x^3]*(998*c^2 + 41*c*d*x^3 + 3*d^2*x^6))/(45*d^3) + (128*c^ (5/2)*ArcTanh[Sqrt[c + d*x^3]/(3*Sqrt[c])])/d^3
Time = 0.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02, 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.
\(\displaystyle \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {x^6 \sqrt {d x^3+c}}{8 c-d x^3}dx^3\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{3} \int \left (\frac {64 \sqrt {d x^3+c} c^2}{d^2 \left (8 c-d x^3\right )}-\frac {7 \sqrt {d x^3+c} c}{d^2}-\frac {\left (d x^3+c\right )^{3/2}}{d^2}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {384 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{d^3}-\frac {128 c^2 \sqrt {c+d x^3}}{d^3}-\frac {14 c \left (c+d x^3\right )^{3/2}}{3 d^3}-\frac {2 \left (c+d x^3\right )^{5/2}}{5 d^3}\right )\) |
Input:
Int[(x^8*Sqrt[c + d*x^3])/(8*c - d*x^3),x]
Output:
((-128*c^2*Sqrt[c + d*x^3])/d^3 - (14*c*(c + d*x^3)^(3/2))/(3*d^3) - (2*(c + d*x^3)^(5/2))/(5*d^3) + (384*c^(5/2)*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 = 1.24 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(\frac {\frac {2 \left (-3 d^{2} x^{6}-41 c d \,x^{3}-998 c^{2}\right ) \sqrt {d \,x^{3}+c}}{45}+128 c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{d^{3}}\) | \(58\) |
risch | \(-\frac {2 \left (3 d^{2} x^{6}+41 c d \,x^{3}+998 c^{2}\right ) \sqrt {d \,x^{3}+c}}{45 d^{3}}+\frac {128 c^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )}{d^{3}}\) | \(60\) |
default | \(-\frac {\frac {2 x^{6} \sqrt {d \,x^{3}+c}}{15}+\frac {2 c \,x^{3} \sqrt {d \,x^{3}+c}}{45 d}-\frac {4 c^{2} \sqrt {d \,x^{3}+c}}{45 d^{2}}}{d}-\frac {16 c \left (d \,x^{3}+c \right )^{\frac {3}{2}}}{9 d^{3}}+\frac {64 c^{2} \left (-2 \sqrt {d \,x^{3}+c}+6 \sqrt {c}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right )\right )}{3 d^{3}}\) | \(112\) |
elliptic | \(-\frac {2 x^{6} \sqrt {d \,x^{3}+c}}{15 d}-\frac {82 c \,x^{3} \sqrt {d \,x^{3}+c}}{45 d^{2}}-\frac {1996 c^{2} \sqrt {d \,x^{3}+c}}{45 d^{3}}-\frac {64 i c^{2} \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 )}{3 d^{5}}\) | \(466\) |
Input:
int(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x,method=_RETURNVERBOSE)
Output:
2/45*((-3*d^2*x^6-41*c*d*x^3-998*c^2)*(d*x^3+c)^(1/2)+2880*c^(5/2)*arctanh (1/3*(d*x^3+c)^(1/2)/c^(1/2)))/d^3
Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.60 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\left [\frac {2 \, {\left (1440 \, c^{\frac {5}{2}} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - {\left (3 \, d^{2} x^{6} + 41 \, c d x^{3} + 998 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{3}}, -\frac {2 \, {\left (2880 \, \sqrt {-c} c^{2} \arctan \left (\frac {3 \, \sqrt {-c}}{\sqrt {d x^{3} + c}}\right ) + {\left (3 \, d^{2} x^{6} + 41 \, c d x^{3} + 998 \, c^{2}\right )} \sqrt {d x^{3} + c}\right )}}{45 \, d^{3}}\right ] \] Input:
integrate(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x, algorithm="fricas")
Output:
[2/45*(1440*c^(5/2)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/(d*x^3 - 8*c)) - (3*d^2*x^6 + 41*c*d*x^3 + 998*c^2)*sqrt(d*x^3 + c))/d^3, -2/45*( 2880*sqrt(-c)*c^2*arctan(3*sqrt(-c)/sqrt(d*x^3 + c)) + (3*d^2*x^6 + 41*c*d *x^3 + 998*c^2)*sqrt(d*x^3 + c))/d^3]
Time = 8.83 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.02 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\begin {cases} \frac {2 \left (- \frac {64 c^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{\sqrt {- c}} - \frac {64 c^{2} \sqrt {c + d x^{3}}}{3} - \frac {7 c \left (c + d x^{3}\right )^{\frac {3}{2}}}{9} - \frac {\left (c + d x^{3}\right )^{\frac {5}{2}}}{15}\right )}{d^{3}} & \text {for}\: d \neq 0 \\\frac {x^{9}}{72 \sqrt {c}} & \text {otherwise} \end {cases} \] Input:
integrate(x**8*(d*x**3+c)**(1/2)/(-d*x**3+8*c),x)
Output:
Piecewise((2*(-64*c**3*atan(sqrt(c + d*x**3)/(3*sqrt(-c)))/sqrt(-c) - 64*c **2*sqrt(c + d*x**3)/3 - 7*c*(c + d*x**3)**(3/2)/9 - (c + d*x**3)**(5/2)/1 5)/d**3, Ne(d, 0)), (x**9/(72*sqrt(c)), True))
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.91 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=-\frac {2 \, {\left (1440 \, c^{\frac {5}{2}} \log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right ) + 3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} + 35 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c + 960 \, \sqrt {d x^{3} + c} c^{2}\right )}}{45 \, d^{3}} \] Input:
integrate(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x, algorithm="maxima")
Output:
-2/45*(1440*c^(5/2)*log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3 *sqrt(c))) + 3*(d*x^3 + c)^(5/2) + 35*(d*x^3 + c)^(3/2)*c + 960*sqrt(d*x^3 + c)*c^2)/d^3
Time = 0.12 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.92 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=-\frac {128 \, c^{3} \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{\sqrt {-c} d^{3}} - \frac {2 \, {\left (3 \, {\left (d x^{3} + c\right )}^{\frac {5}{2}} d^{12} + 35 \, {\left (d x^{3} + c\right )}^{\frac {3}{2}} c d^{12} + 960 \, \sqrt {d x^{3} + c} c^{2} d^{12}\right )}}{45 \, d^{15}} \] Input:
integrate(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x, algorithm="giac")
Output:
-128*c^3*arctan(1/3*sqrt(d*x^3 + c)/sqrt(-c))/(sqrt(-c)*d^3) - 2/45*(3*(d* x^3 + c)^(5/2)*d^12 + 35*(d*x^3 + c)^(3/2)*c*d^12 + 960*sqrt(d*x^3 + c)*c^ 2*d^12)/d^15
Time = 1.61 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.09 \[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\frac {64\,c^{5/2}\,\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{d^3}-\frac {1996\,c^2\,\sqrt {d\,x^3+c}}{45\,d^3}-\frac {2\,x^6\,\sqrt {d\,x^3+c}}{15\,d}-\frac {82\,c\,x^3\,\sqrt {d\,x^3+c}}{45\,d^2} \] Input:
int((x^8*(c + d*x^3)^(1/2))/(8*c - d*x^3),x)
Output:
(64*c^(5/2)*log((10*c + d*x^3 + 6*c^(1/2)*(c + d*x^3)^(1/2))/(8*c - d*x^3) ))/d^3 - (1996*c^2*(c + d*x^3)^(1/2))/(45*d^3) - (2*x^6*(c + d*x^3)^(1/2)) /(15*d) - (82*c*x^3*(c + d*x^3)^(1/2))/(45*d^2)
\[ \int \frac {x^8 \sqrt {c+d x^3}}{8 c-d x^3} \, dx=\frac {\frac {164 \sqrt {d \,x^{3}+c}\, c^{2}}{45}-\frac {82 \sqrt {d \,x^{3}+c}\, c d \,x^{3}}{45}-\frac {2 \sqrt {d \,x^{3}+c}\, d^{2} x^{6}}{15}+72 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{-d^{2} x^{6}+7 c d \,x^{3}+8 c^{2}}d x \right ) c^{2} d^{2}}{d^{3}} \] Input:
int(x^8*(d*x^3+c)^(1/2)/(-d*x^3+8*c),x)
Output:
(2*(82*sqrt(c + d*x**3)*c**2 - 41*sqrt(c + d*x**3)*c*d*x**3 - 3*sqrt(c + d *x**3)*d**2*x**6 + 1620*int((sqrt(c + d*x**3)*x**5)/(8*c**2 + 7*c*d*x**3 - d**2*x**6),x)*c**2*d**2))/(45*d**3)