Integrand size = 22, antiderivative size = 91 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=-\frac {(b c-a d) x^3}{3 b^2 \sqrt {a+b x^6}}+\frac {d x^3 \sqrt {a+b x^6}}{6 b^2}+\frac {(2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} x^3}{\sqrt {a+b x^6}}\right )}{6 b^{5/2}} \] Output:
-1/3*(-a*d+b*c)*x^3/b^2/(b*x^6+a)^(1/2)+1/6*d*x^3*(b*x^6+a)^(1/2)/b^2+1/6* (-3*a*d+2*b*c)*arctanh(b^(1/2)*x^3/(b*x^6+a)^(1/2))/b^(5/2)
Time = 0.60 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.85 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {x^3 \left (-2 b c+3 a d+b d x^6\right )}{6 b^2 \sqrt {a+b x^6}}+\frac {(2 b c-3 a d) \log \left (\sqrt {b} x^3+\sqrt {a+b x^6}\right )}{6 b^{5/2}} \] Input:
Integrate[(x^8*(c + d*x^6))/(a + b*x^6)^(3/2),x]
Output:
(x^3*(-2*b*c + 3*a*d + b*d*x^6))/(6*b^2*Sqrt[a + b*x^6]) + ((2*b*c - 3*a*d )*Log[Sqrt[b]*x^3 + Sqrt[a + b*x^6]])/(6*b^(5/2))
Time = 0.37 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {959, 807, 252, 224, 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.
\(\displaystyle \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 959 |
\(\displaystyle \frac {(2 b c-3 a d) \int \frac {x^8}{\left (b x^6+a\right )^{3/2}}dx}{2 b}+\frac {d x^9}{6 b \sqrt {a+b x^6}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {(2 b c-3 a d) \int \frac {x^6}{\left (b x^6+a\right )^{3/2}}dx^3}{6 b}+\frac {d x^9}{6 b \sqrt {a+b x^6}}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {(2 b c-3 a d) \left (\frac {\int \frac {1}{\sqrt {b x^6+a}}dx^3}{b}-\frac {x^3}{b \sqrt {a+b x^6}}\right )}{6 b}+\frac {d x^9}{6 b \sqrt {a+b x^6}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {(2 b c-3 a d) \left (\frac {\int \frac {1}{1-b x^6}d\frac {x^3}{\sqrt {b x^6+a}}}{b}-\frac {x^3}{b \sqrt {a+b x^6}}\right )}{6 b}+\frac {d x^9}{6 b \sqrt {a+b x^6}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x^3}{\sqrt {a+b x^6}}\right )}{b^{3/2}}-\frac {x^3}{b \sqrt {a+b x^6}}\right ) (2 b c-3 a d)}{6 b}+\frac {d x^9}{6 b \sqrt {a+b x^6}}\) |
Input:
Int[(x^8*(c + d*x^6))/(a + b*x^6)^(3/2),x]
Output:
(d*x^9)/(6*b*Sqrt[a + b*x^6]) + ((2*b*c - 3*a*d)*(-(x^3/(b*Sqrt[a + b*x^6] )) + ArcTanh[(Sqrt[b]*x^3)/Sqrt[a + b*x^6]]/b^(3/2)))/(6*b)
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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p + 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p + 1) + 1)) Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
Time = 5.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81
method | result | size |
pseudoelliptic | \(-\frac {\sqrt {b \,x^{6}+a}\, \left (-c b +\frac {3 a d}{2}\right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{6}+a}}{x^{3} \sqrt {b}}\right )+\left (\left (-\frac {d \,x^{6}}{2}+c \right ) b -\frac {3 a d}{2}\right ) x^{3} \sqrt {b}}{3 \sqrt {b \,x^{6}+a}\, b^{\frac {5}{2}}}\) | \(74\) |
Input:
int(x^8*(d*x^6+c)/(b*x^6+a)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/3*((b*x^6+a)^(1/2)*(-c*b+3/2*a*d)*arctanh((b*x^6+a)^(1/2)/x^3/b^(1/2))+ ((-1/2*d*x^6+c)*b-3/2*a*d)*x^3*b^(1/2))/(b*x^6+a)^(1/2)/b^(5/2)
Time = 0.10 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.46 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\left [-\frac {{\left ({\left (2 \, b^{2} c - 3 \, a b d\right )} x^{6} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt {b} \log \left (-2 \, b x^{6} + 2 \, \sqrt {b x^{6} + a} \sqrt {b} x^{3} - a\right ) - 2 \, {\left (b^{2} d x^{9} - {\left (2 \, b^{2} c - 3 \, a b d\right )} x^{3}\right )} \sqrt {b x^{6} + a}}{12 \, {\left (b^{4} x^{6} + a b^{3}\right )}}, -\frac {{\left ({\left (2 \, b^{2} c - 3 \, a b d\right )} x^{6} + 2 \, a b c - 3 \, a^{2} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{3}}{\sqrt {b x^{6} + a}}\right ) - {\left (b^{2} d x^{9} - {\left (2 \, b^{2} c - 3 \, a b d\right )} x^{3}\right )} \sqrt {b x^{6} + a}}{6 \, {\left (b^{4} x^{6} + a b^{3}\right )}}\right ] \] Input:
integrate(x^8*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="fricas")
Output:
[-1/12*(((2*b^2*c - 3*a*b*d)*x^6 + 2*a*b*c - 3*a^2*d)*sqrt(b)*log(-2*b*x^6 + 2*sqrt(b*x^6 + a)*sqrt(b)*x^3 - a) - 2*(b^2*d*x^9 - (2*b^2*c - 3*a*b*d) *x^3)*sqrt(b*x^6 + a))/(b^4*x^6 + a*b^3), -1/6*(((2*b^2*c - 3*a*b*d)*x^6 + 2*a*b*c - 3*a^2*d)*sqrt(-b)*arctan(sqrt(-b)*x^3/sqrt(b*x^6 + a)) - (b^2*d *x^9 - (2*b^2*c - 3*a*b*d)*x^3)*sqrt(b*x^6 + a))/(b^4*x^6 + a*b^3)]
Time = 43.38 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=c \left (\frac {\operatorname {asinh}{\left (\frac {\sqrt {b} x^{3}}{\sqrt {a}} \right )}}{3 b^{\frac {3}{2}}} - \frac {x^{3}}{3 \sqrt {a} b \sqrt {1 + \frac {b x^{6}}{a}}}\right ) + d \left (\frac {\sqrt {a} x^{3}}{2 b^{2} \sqrt {1 + \frac {b x^{6}}{a}}} - \frac {a \operatorname {asinh}{\left (\frac {\sqrt {b} x^{3}}{\sqrt {a}} \right )}}{2 b^{\frac {5}{2}}} + \frac {x^{9}}{6 \sqrt {a} b \sqrt {1 + \frac {b x^{6}}{a}}}\right ) \] Input:
integrate(x**8*(d*x**6+c)/(b*x**6+a)**(3/2),x)
Output:
c*(asinh(sqrt(b)*x**3/sqrt(a))/(3*b**(3/2)) - x**3/(3*sqrt(a)*b*sqrt(1 + b *x**6/a))) + d*(sqrt(a)*x**3/(2*b**2*sqrt(1 + b*x**6/a)) - a*asinh(sqrt(b) *x**3/sqrt(a))/(2*b**(5/2)) + x**9/(6*sqrt(a)*b*sqrt(1 + b*x**6/a)))
Leaf count of result is larger than twice the leaf count of optimal. 172 vs. \(2 (75) = 150\).
Time = 0.11 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.89 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=-\frac {1}{6} \, {\left (\frac {2 \, x^{3}}{\sqrt {b x^{6} + a} b} + \frac {\log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{6} + a}}{x^{3}}}{\sqrt {b} + \frac {\sqrt {b x^{6} + a}}{x^{3}}}\right )}{b^{\frac {3}{2}}}\right )} c + \frac {1}{12} \, d {\left (\frac {2 \, {\left (2 \, a b - \frac {3 \, {\left (b x^{6} + a\right )} a}{x^{6}}\right )}}{\frac {\sqrt {b x^{6} + a} b^{3}}{x^{3}} - \frac {{\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{2}}{x^{9}}} + \frac {3 \, a \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{6} + a}}{x^{3}}}{\sqrt {b} + \frac {\sqrt {b x^{6} + a}}{x^{3}}}\right )}{b^{\frac {5}{2}}}\right )} \] Input:
integrate(x^8*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="maxima")
Output:
-1/6*(2*x^3/(sqrt(b*x^6 + a)*b) + log(-(sqrt(b) - sqrt(b*x^6 + a)/x^3)/(sq rt(b) + sqrt(b*x^6 + a)/x^3))/b^(3/2))*c + 1/12*d*(2*(2*a*b - 3*(b*x^6 + a )*a/x^6)/(sqrt(b*x^6 + a)*b^3/x^3 - (b*x^6 + a)^(3/2)*b^2/x^9) + 3*a*log(- (sqrt(b) - sqrt(b*x^6 + a)/x^3)/(sqrt(b) + sqrt(b*x^6 + a)/x^3))/b^(5/2))
Exception generated. \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(x^8*(d*x^6+c)/(b*x^6+a)^(3/2),x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: Recu rsive assumption sageVARa>=(-sageVARb/t_nostep^6) ignored2*(486*sageVARb^4 *sageVARd*1/5832/sageVARb^5*sageVARx*sageVARx*sageVARx*sageVARx*sageVARx*s ageVARx-(972*sageVA
Timed out. \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\int \frac {x^8\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{3/2}} \,d x \] Input:
int((x^8*(c + d*x^6))/(a + b*x^6)^(3/2),x)
Output:
int((x^8*(c + d*x^6))/(a + b*x^6)^(3/2), x)
Time = 0.29 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.96 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{3/2}} \, dx=\frac {6 \sqrt {b \,x^{6}+a}\, a b d \,x^{3}-4 \sqrt {b \,x^{6}+a}\, b^{2} c \,x^{3}+2 \sqrt {b \,x^{6}+a}\, b^{2} d \,x^{9}+3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a^{2} d -2 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a b c +3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a b d \,x^{6}-2 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) b^{2} c \,x^{6}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a^{2} d +2 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a b c -3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a b d \,x^{6}+2 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) b^{2} c \,x^{6}}{12 b^{3} \left (b \,x^{6}+a \right )} \] Input:
int(x^8*(d*x^6+c)/(b*x^6+a)^(3/2),x)
Output:
(6*sqrt(a + b*x**6)*a*b*d*x**3 - 4*sqrt(a + b*x**6)*b**2*c*x**3 + 2*sqrt(a + b*x**6)*b**2*d*x**9 + 3*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x**3)*a* *2*d - 2*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x**3)*a*b*c + 3*sqrt(b)*lo g(sqrt(a + b*x**6) - sqrt(b)*x**3)*a*b*d*x**6 - 2*sqrt(b)*log(sqrt(a + b*x **6) - sqrt(b)*x**3)*b**2*c*x**6 - 3*sqrt(b)*log(sqrt(a + b*x**6) + sqrt(b )*x**3)*a**2*d + 2*sqrt(b)*log(sqrt(a + b*x**6) + sqrt(b)*x**3)*a*b*c - 3* sqrt(b)*log(sqrt(a + b*x**6) + sqrt(b)*x**3)*a*b*d*x**6 + 2*sqrt(b)*log(sq rt(a + b*x**6) + sqrt(b)*x**3)*b**2*c*x**6)/(12*b**3*(a + b*x**6))