Integrand size = 24, antiderivative size = 104 \[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=-\frac {2 (b c+a d) \sqrt {c+d x^3}}{3 b^2 d^2}+\frac {2 \left (c+d x^3\right )^{3/2}}{9 b d^2}-\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{5/2} \sqrt {b c-a d}} \] Output:
-2/3*(a*d+b*c)*(d*x^3+c)^(1/2)/b^2/d^2+2/9*(d*x^3+c)^(3/2)/b/d^2-2/3*a^2*a rctanh(b^(1/2)*(d*x^3+c)^(1/2)/(-a*d+b*c)^(1/2))/b^(5/2)/(-a*d+b*c)^(1/2)
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.88 \[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {2 \sqrt {c+d x^3} \left (-2 b c-3 a d+b d x^3\right )}{9 b^2 d^2}+\frac {2 a^2 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {-b c+a d}}\right )}{3 b^{5/2} \sqrt {-b c+a d}} \] Input:
Integrate[x^8/((a + b*x^3)*Sqrt[c + d*x^3]),x]
Output:
(2*Sqrt[c + d*x^3]*(-2*b*c - 3*a*d + b*d*x^3))/(9*b^2*d^2) + (2*a^2*ArcTan [(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[-(b*c) + a*d]])/(3*b^(5/2)*Sqrt[-(b*c) + a *d])
Time = 0.43 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.
\(\displaystyle \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {1}{3} \int \frac {x^6}{\left (b x^3+a\right ) \sqrt {d x^3+c}}dx^3\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{3} \int \left (\frac {a^2}{b^2 \left (b x^3+a\right ) \sqrt {d x^3+c}}+\frac {\sqrt {d x^3+c}}{b d}+\frac {-b c-a d}{b^2 d \sqrt {d x^3+c}}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (-\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{b^{5/2} \sqrt {b c-a d}}-\frac {2 \sqrt {c+d x^3} (a d+b c)}{b^2 d^2}+\frac {2 \left (c+d x^3\right )^{3/2}}{3 b d^2}\right )\) |
Input:
Int[x^8/((a + b*x^3)*Sqrt[c + d*x^3]),x]
Output:
((-2*(b*c + a*d)*Sqrt[c + d*x^3])/(b^2*d^2) + (2*(c + d*x^3)^(3/2))/(3*b*d ^2) - (2*a^2*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^3])/Sqrt[b*c - a*d]])/(b^(5/2)* Sqrt[b*c - a*d]))/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 = 1.41 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.76
method | result | size |
risch | \(-\frac {2 \left (-b d \,x^{3}+3 a d +2 b c \right ) \sqrt {d \,x^{3}+c}}{9 d^{2} b^{2}}+\frac {2 a^{2} \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 b^{2} \sqrt {\left (a d -b c \right ) b}}\) | \(79\) |
pseudoelliptic | \(-\frac {2 \left (-\arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right ) a^{2} d^{2}+\sqrt {\left (a d -b c \right ) b}\, \left (\frac {\left (-d \,x^{3}+2 c \right ) b}{3}+a d \right ) \sqrt {d \,x^{3}+c}\right )}{3 \sqrt {\left (a d -b c \right ) b}\, b^{2} d^{2}}\) | \(93\) |
default | \(\frac {\frac {2 x^{3} \sqrt {d \,x^{3}+c}}{9 d}-\frac {4 c \sqrt {d \,x^{3}+c}}{9 d^{2}}}{b}+\frac {2 a^{2} \arctan \left (\frac {b \sqrt {d \,x^{3}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{3 b^{2} \sqrt {\left (a d -b c \right ) b}}-\frac {2 a \sqrt {d \,x^{3}+c}}{3 b^{2} d}\) | \(101\) |
elliptic | \(\frac {2 x^{3} \sqrt {d \,x^{3}+c}}{9 d b}+\frac {2 \left (-\frac {a}{b^{2}}-\frac {2 c}{3 d b}\right ) \sqrt {d \,x^{3}+c}}{3 d}-\frac {i a^{2} \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \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 {b \left (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 \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 ) \sqrt {d \,x^{3}+c}}\right )}{3 b^{2} d^{2}}\) | \(483\) |
Input:
int(x^8/(b*x^3+a)/(d*x^3+c)^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/9*(-b*d*x^3+3*a*d+2*b*c)*(d*x^3+c)^(1/2)/d^2/b^2+2/3*a^2/b^2/((a*d-b*c) *b)^(1/2)*arctan(b*(d*x^3+c)^(1/2)/((a*d-b*c)*b)^(1/2))
Time = 0.12 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.78 \[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\left [\frac {3 \, \sqrt {b^{2} c - a b d} a^{2} d^{2} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) - 2 \, {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} - {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{9 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}, \frac {2 \, {\left (3 \, \sqrt {-b^{2} c + a b d} a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) - {\left (2 \, b^{3} c^{2} + a b^{2} c d - 3 \, a^{2} b d^{2} - {\left (b^{3} c d - a b^{2} d^{2}\right )} x^{3}\right )} \sqrt {d x^{3} + c}\right )}}{9 \, {\left (b^{4} c d^{2} - a b^{3} d^{3}\right )}}\right ] \] Input:
integrate(x^8/(b*x^3+a)/(d*x^3+c)^(1/2),x, algorithm="fricas")
Output:
[1/9*(3*sqrt(b^2*c - a*b*d)*a^2*d^2*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*b^3*c^2 + a*b^2*c*d - 3* a^2*b*d^2 - (b^3*c*d - a*b^2*d^2)*x^3)*sqrt(d*x^3 + c))/(b^4*c*d^2 - a*b^3 *d^3), 2/9*(3*sqrt(-b^2*c + a*b*d)*a^2*d^2*arctan(sqrt(d*x^3 + c)*sqrt(-b^ 2*c + a*b*d)/(b*d*x^3 + b*c)) - (2*b^3*c^2 + a*b^2*c*d - 3*a^2*b*d^2 - (b^ 3*c*d - a*b^2*d^2)*x^3)*sqrt(d*x^3 + c))/(b^4*c*d^2 - a*b^3*d^3)]
\[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\int \frac {x^{8}}{\left (a + b x^{3}\right ) \sqrt {c + d x^{3}}}\, dx \] Input:
integrate(x**8/(b*x**3+a)/(d*x**3+c)**(1/2),x)
Output:
Integral(x**8/((a + b*x**3)*sqrt(c + d*x**3)), x)
Exception generated. \[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^8/(b*x^3+a)/(d*x^3+c)^(1/2),x, algorithm="maxima")
Output:
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 = 0.12 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98 \[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {2 \, {\left (\frac {3 \, a^{2} d^{2} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{2}} + \frac {{\left (d x^{3} + c\right )}^{\frac {3}{2}} b^{2} - 3 \, \sqrt {d x^{3} + c} b^{2} c - 3 \, \sqrt {d x^{3} + c} a b d}{b^{3}}\right )}}{9 \, d^{2}} \] Input:
integrate(x^8/(b*x^3+a)/(d*x^3+c)^(1/2),x, algorithm="giac")
Output:
2/9*(3*a^2*d^2*arctan(sqrt(d*x^3 + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^2) + ((d*x^3 + c)^(3/2)*b^2 - 3*sqrt(d*x^3 + c)*b^2*c - 3*sqrt (d*x^3 + c)*a*b*d)/b^3)/d^2
Time = 4.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.16 \[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {2\,x^3\,\sqrt {d\,x^3+c}}{9\,b\,d}-\frac {\left (\frac {2\,a}{b^2}+\frac {4\,c}{3\,b\,d}\right )\,\sqrt {d\,x^3+c}}{3\,d}+\frac {a^2\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,1{}\mathrm {i}}{3\,b^{5/2}\,\sqrt {a\,d-b\,c}} \] Input:
int(x^8/((a + b*x^3)*(c + d*x^3)^(1/2)),x)
Output:
(2*x^3*(c + d*x^3)^(1/2))/(9*b*d) - (((2*a)/b^2 + (4*c)/(3*b*d))*(c + d*x^ 3)^(1/2))/(3*d) + (a^2*log((2*b*c - a*d + b^(1/2)*(c + d*x^3)^(1/2)*(a*d - b*c)^(1/2)*2i + b*d*x^3)/(a + b*x^3))*1i)/(3*b^(5/2)*(a*d - b*c)^(1/2))
\[ \int \frac {x^8}{\left (a+b x^3\right ) \sqrt {c+d x^3}} \, dx=\frac {-4 \sqrt {d \,x^{3}+c}\, c +2 \sqrt {d \,x^{3}+c}\, d \,x^{3}-9 \left (\int \frac {\sqrt {d \,x^{3}+c}\, x^{5}}{b d \,x^{6}+a d \,x^{3}+b c \,x^{3}+a c}d x \right ) a \,d^{2}}{9 b \,d^{2}} \] Input:
int(x^8/(b*x^3+a)/(d*x^3+c)^(1/2),x)
Output:
( - 4*sqrt(c + d*x**3)*c + 2*sqrt(c + d*x**3)*d*x**3 - 9*int((sqrt(c + d*x **3)*x**5)/(a*c + a*d*x**3 + b*c*x**3 + b*d*x**6),x)*a*d**2)/(9*b*d**2)