Integrand size = 27, antiderivative size = 55 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=-\frac {2}{27 c d \sqrt {c+d x^3}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{81 c^{3/2} d} \]
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.95 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {2 \left (-\frac {3 \sqrt {c}}{\sqrt {c+d x^3}}+\text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )\right )}{81 c^{3/2} d} \]
Time = 0.21 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {946, 61, 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.
\(\displaystyle \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 946 |
\(\displaystyle \frac {1}{3} \int \frac {1}{\left (8 c-d x^3\right ) \left (d x^3+c\right )^{3/2}}dx^3\) |
\(\Big \downarrow \) 61 |
\(\displaystyle \frac {1}{3} \left (\frac {\int \frac {1}{\left (8 c-d x^3\right ) \sqrt {d x^3+c}}dx^3}{9 c}-\frac {2}{9 c d \sqrt {c+d x^3}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{3} \left (\frac {2 \int \frac {1}{9 c-x^6}d\sqrt {d x^3+c}}{9 c d}-\frac {2}{9 c d \sqrt {c+d x^3}}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {c+d x^3}}{3 \sqrt {c}}\right )}{27 c^{3/2} d}-\frac {2}{9 c d \sqrt {c+d x^3}}\right )\) |
3.4.28.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[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_)^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[(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] && EqQ[m - n + 1, 0]
Time = 4.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87
method | result | size |
default | \(-\frac {2 \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right ) \sqrt {d \,x^{3}+c}}{3}+\sqrt {c}\right )}{27 \sqrt {d \,x^{3}+c}\, c^{\frac {3}{2}} d}\) | \(48\) |
pseudoelliptic | \(-\frac {2 \left (-\frac {\operatorname {arctanh}\left (\frac {\sqrt {d \,x^{3}+c}}{3 \sqrt {c}}\right ) \sqrt {d \,x^{3}+c}}{3}+\sqrt {c}\right )}{27 \sqrt {d \,x^{3}+c}\, c^{\frac {3}{2}} d}\) | \(48\) |
elliptic | \(-\frac {2}{27 d c \sqrt {\left (x^{3}+\frac {c}{d}\right ) d}}-\frac {i \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 ) \Pi \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}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d -i \left (-c \,d^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +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 )}{243 d^{3} c^{2}}\) | \(435\) |
-2/27/(d*x^3+c)^(1/2)/c^(3/2)*(-1/3*arctanh(1/3*(d*x^3+c)^(1/2)/c^(1/2))*( d*x^3+c)^(1/2)+c^(1/2))/d
Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 2.67 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\left [\frac {{\left (d x^{3} + c\right )} \sqrt {c} \log \left (\frac {d x^{3} + 6 \, \sqrt {d x^{3} + c} \sqrt {c} + 10 \, c}{d x^{3} - 8 \, c}\right ) - 6 \, \sqrt {d x^{3} + c} c}{81 \, {\left (c^{2} d^{2} x^{3} + c^{3} d\right )}}, -\frac {2 \, {\left ({\left (d x^{3} + c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-c}}{3 \, c}\right ) + 3 \, \sqrt {d x^{3} + c} c\right )}}{81 \, {\left (c^{2} d^{2} x^{3} + c^{3} d\right )}}\right ] \]
[1/81*((d*x^3 + c)*sqrt(c)*log((d*x^3 + 6*sqrt(d*x^3 + c)*sqrt(c) + 10*c)/ (d*x^3 - 8*c)) - 6*sqrt(d*x^3 + c)*c)/(c^2*d^2*x^3 + c^3*d), -2/81*((d*x^3 + c)*sqrt(-c)*arctan(1/3*sqrt(d*x^3 + c)*sqrt(-c)/c) + 3*sqrt(d*x^3 + c)* c)/(c^2*d^2*x^3 + c^3*d)]
Time = 6.51 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {1}{27 c \sqrt {c + d x^{3}}} - \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{3}}}{3 \sqrt {- c}} \right )}}{81 c \sqrt {- c}}\right )}{d} & \text {for}\: d \neq 0 \\\frac {x^{3}}{24 c^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(-1/(27*c*sqrt(c + d*x**3)) - atan(sqrt(c + d*x**3)/(3*sqrt(- c)))/(81*c*sqrt(-c)))/d, Ne(d, 0)), (x**3/(24*c**(5/2)), True))
Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=-\frac {\frac {\log \left (\frac {\sqrt {d x^{3} + c} - 3 \, \sqrt {c}}{\sqrt {d x^{3} + c} + 3 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {6}{\sqrt {d x^{3} + c} c}}{81 \, d} \]
-1/81*(log((sqrt(d*x^3 + c) - 3*sqrt(c))/(sqrt(d*x^3 + c) + 3*sqrt(c)))/c^ (3/2) + 6/(sqrt(d*x^3 + c)*c))/d
Time = 0.45 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {d x^{3} + c}}{3 \, \sqrt {-c}}\right )}{81 \, \sqrt {-c} c d} - \frac {2}{27 \, \sqrt {d x^{3} + c} c d} \]
Time = 7.89 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\left (8 c-d x^3\right ) \left (c+d x^3\right )^{3/2}} \, dx=\frac {\ln \left (\frac {10\,c+d\,x^3+6\,\sqrt {c}\,\sqrt {d\,x^3+c}}{8\,c-d\,x^3}\right )}{81\,c^{3/2}\,d}-\frac {2}{27\,c\,d\,\sqrt {d\,x^3+c}} \]