Integrand size = 24, antiderivative size = 123 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2 \sqrt {c+d x^3}}{3 b^2 d}-\frac {a^2 \sqrt {c+d x^3}}{3 b^2 (b c-a d) \left (a+b x^3\right )}+\frac {a (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{5/2} (b c-a d)^{3/2}} \]
1/3*a*(-3*a*d+4*b*c)*arctanh(b^(1/2)*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))/b^( 5/2)/(-a*d+b*c)^(3/2)+2/3*(d*x^3+c)^(1/2)/b^2/d-1/3*a^2*(d*x^3+c)^(1/2)/b^ 2/(-a*d+b*c)/(b*x^3+a)
Time = 0.62 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.06 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {\frac {\sqrt {b} \sqrt {c+d x^3} \left (-3 a^2 d+2 b^2 c x^3+2 a b \left (c-d x^3\right )\right )}{d (b c-a d) \left (a+b x^3\right )}+\frac {a (4 b c-3 a d) \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{3/2}}}{3 b^{5/2}} \]
((Sqrt[b]*Sqrt[c + d*x^3]*(-3*a^2*d + 2*b^2*c*x^3 + 2*a*b*(c - d*x^3)))/(d *(b*c - a*d)*(a + b*x^3)) + (a*(4*b*c - 3*a*d)*ArcTan[(Sqrt[b]*Sqrt[c + d* x^3])/Sqrt[-(b*c) + a*d]])/(-(b*c) + a*d)^(3/2))/(3*b^(5/2))
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {948, 100, 27, 90, 73, 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.
| \(\displaystyle \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx\) |
| \(\Big \downarrow \) 948 |
| \(\displaystyle \frac {1}{3} \int \frac {x^6}{\left (b x^3+a\right )^2 \sqrt {d x^3+c}}dx^3\) |
| \(\Big \downarrow \) 100 |
| \(\displaystyle \frac {1}{3} \left (\frac {\int -\frac {a (2 b c-a d)-2 b (b c-a d) x^3}{2 \left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{b^2 (b c-a d)}-\frac {a^2 \sqrt {c+d x^3}}{b^2 \left (a+b x^3\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\int \frac {a (2 b c-a d)-2 b (b c-a d) x^3}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3}{2 b^2 (b c-a d)}-\frac {a^2 \sqrt {c+d x^3}}{b^2 \left (a+b x^3\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \frac {1}{3} \left (-\frac {a (4 b c-3 a d) \int \frac {1}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3-\frac {4 \sqrt {c+d x^3} (b c-a d)}{d}}{2 b^2 (b c-a d)}-\frac {a^2 \sqrt {c+d x^3}}{b^2 \left (a+b x^3\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\frac {2 a (4 b c-3 a d) \int \frac {1}{\frac {b x^6}{d}+a-\frac {b c}{d}}d\sqrt {d x^3+c}}{d}-\frac {4 \sqrt {c+d x^3} (b c-a d)}{d}}{2 b^2 (b c-a d)}-\frac {a^2 \sqrt {c+d x^3}}{b^2 \left (a+b x^3\right ) (b c-a d)}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{3} \left (-\frac {a^2 \sqrt {c+d x^3}}{b^2 \left (a+b x^3\right ) (b c-a d)}-\frac {-\frac {2 a (4 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \sqrt {b c-a d}}-\frac {4 \sqrt {c+d x^3} (b c-a d)}{d}}{2 b^2 (b c-a d)}\right )\) |
(-((a^2*Sqrt[c + d*x^3])/(b^2*(b*c - a*d)*(a + b*x^3))) - ((-4*(b*c - a*d) *Sqrt[c + d*x^3])/d - (2*a*(4*b*c - 3*a*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3 ])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/(2*b^2*(b*c - a*d)))/3
3.5.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[(b*c - a*d)^2*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d^2*(d *e - c*f)*(n + 1))), x] - Simp[1/(d^2*(d*e - c*f)*(n + 1)) Int[(c + d*x)^ (n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*( p + 1)) - 2*a*b*d*(d*e*(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x , x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] || !SumSimplerQ[p, 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]
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 = 4.60 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.07
| method | result | size |
| pseudoelliptic | \(\frac {-d \left (b \,x^{3}+a \right ) a \left (a d -\frac {4 b c}{3}\right ) \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )+\sqrt {d \,x^{3}+c}\, \left (-\frac {2 b^{2} c \,x^{3}}{3}-\frac {2 a \left (-d \,x^{3}+c \right ) b}{3}+a^{2} d \right ) \sqrt {\left (a d -b c \right ) b}}{\sqrt {\left (a d -b c \right ) b}\, d \,b^{2} \left (a d -b c \right ) \left (b \,x^{3}+a \right )}\) | \(132\) |
| risch | \(\frac {2 \sqrt {d \,x^{3}+c}}{3 b^{2} d}-\frac {a \left (\frac {4 \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 \sqrt {\left (a d -b c \right ) b}}-\frac {a \left (d \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) \left (b \,x^{3}+a \right )+\sqrt {d \,x^{3}+c}\, \sqrt {\left (a d -b c \right ) b}\right )}{3 \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right ) \left (b \,x^{3}+a \right )}\right )}{b^{2}}\) | \(154\) |
| default | \(\frac {2 \sqrt {d \,x^{3}+c}}{3 b^{2} d}+\frac {a^{2} \left (d \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) \left (b \,x^{3}+a \right )+\sqrt {d \,x^{3}+c}\, \sqrt {\left (a d -b c \right ) b}\right )}{3 b^{2} \sqrt {\left (a d -b c \right ) b}\, \left (a d -b c \right ) \left (b \,x^{3}+a \right )}-\frac {4 a \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(156\) |
| elliptic | \(\frac {a^{2} \sqrt {d \,x^{3}+c}}{3 \left (a d -b c \right ) b^{2} \left (b \,x^{3}+a \right )}+\frac {2 \sqrt {d \,x^{3}+c}}{3 b^{2} d}+\frac {i a \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (3 a d -4 b c \right ) \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 {b \left (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 \right )}{2 d \left (a d -b c \right )}, \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 \left (a d -b c \right )^{2} \sqrt {d \,x^{3}+c}}\right )}{6 b^{2} d^{2}}\) | \(493\) |
1/((a*d-b*c)*b)^(1/2)*(-d*(b*x^3+a)*a*(a*d-4/3*b*c)*arctan(b*(d*x^3+c)^(1/ 2)/((a*d-b*c)*b)^(1/2))+(d*x^3+c)^(1/2)*(-2/3*b^2*c*x^3-2/3*a*(-d*x^3+c)*b +a^2*d)*((a*d-b*c)*b)^(1/2))/d/b^2/(a*d-b*c)/(b*x^3+a)
Leaf count of result is larger than twice the leaf count of optimal. 230 vs. \(2 (103) = 206\).
Time = 0.29 (sec) , antiderivative size = 475, normalized size of antiderivative = 3.86 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\left [\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{3} + 2 \, b c - a d + 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) + 2 \, {\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} + {\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{3}\right )}}, -\frac {{\left (4 \, a^{2} b c d - 3 \, a^{3} d^{2} + {\left (4 \, a b^{2} c d - 3 \, a^{2} b d^{2}\right )} x^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) - {\left (2 \, a b^{3} c^{2} - 5 \, a^{2} b^{2} c d + 3 \, a^{3} b d^{2} + 2 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a b^{5} c^{2} d - 2 \, a^{2} b^{4} c d^{2} + a^{3} b^{3} d^{3} + {\left (b^{6} c^{2} d - 2 \, a b^{5} c d^{2} + a^{2} b^{4} d^{3}\right )} x^{3}\right )}}\right ] \]
[1/6*((4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^3)*sqrt(b^2 *c - a*b*d)*log((b*d*x^3 + 2*b*c - a*d + 2*sqrt(d*x^3 + c)*sqrt(b^2*c - a* b*d))/(b*x^3 + a)) + 2*(2*a*b^3*c^2 - 5*a^2*b^2*c*d + 3*a^3*b*d^2 + 2*(b^4 *c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^3)*sqrt(d*x^3 + c))/(a*b^5*c^2*d - 2*a ^2*b^4*c*d^2 + a^3*b^3*d^3 + (b^6*c^2*d - 2*a*b^5*c*d^2 + a^2*b^4*d^3)*x^3 ), -1/3*((4*a^2*b*c*d - 3*a^3*d^2 + (4*a*b^2*c*d - 3*a^2*b*d^2)*x^3)*sqrt( -b^2*c + a*b*d)*arctan(sqrt(d*x^3 + c)*sqrt(-b^2*c + a*b*d)/(b*d*x^3 + b*c )) - (2*a*b^3*c^2 - 5*a^2*b^2*c*d + 3*a^3*b*d^2 + 2*(b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*x^3)*sqrt(d*x^3 + c))/(a*b^5*c^2*d - 2*a^2*b^4*c*d^2 + a^3 *b^3*d^3 + (b^6*c^2*d - 2*a*b^5*c*d^2 + a^2*b^4*d^3)*x^3)]
\[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\int \frac {x^{8}}{\left (a + b x^{3}\right )^{2} \sqrt {c + d x^{3}}}\, dx \]
Exception generated. \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m ore detail
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.09 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=-\frac {\sqrt {d x^{3} + c} a^{2} d}{3 \, {\left (b^{3} c - a b^{2} d\right )} {\left ({\left (d x^{3} + c\right )} b - b c + a d\right )}} - \frac {{\left (4 \, a b c - 3 \, a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (b^{3} c - a b^{2} d\right )} \sqrt {-b^{2} c + a b d}} + \frac {2 \, \sqrt {d x^{3} + c}}{3 \, b^{2} d} \]
-1/3*sqrt(d*x^3 + c)*a^2*d/((b^3*c - a*b^2*d)*((d*x^3 + c)*b - b*c + a*d)) - 1/3*(4*a*b*c - 3*a^2*d)*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/ ((b^3*c - a*b^2*d)*sqrt(-b^2*c + a*b*d)) + 2/3*sqrt(d*x^3 + c)/(b^2*d)
Time = 11.61 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.30 \[ \int \frac {x^8}{\left (a+b x^3\right )^2 \sqrt {c+d x^3}} \, dx=\frac {2\,\sqrt {d\,x^3+c}\,\left (2\,b^2\,c-2\,a\,b\,d\right )}{3\,d\,\left (2\,b^4\,c-2\,a\,b^3\,d\right )}-\frac {2\,a^2\,\sqrt {d\,x^3+c}}{3\,b\,\left (b\,x^3+a\right )\,\left (2\,b^2\,c-2\,a\,b\,d\right )}+\frac {a\,\ln \left (\frac {a\,d-2\,b\,c-b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (3\,a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{3/2}} \]