Integrand size = 22, antiderivative size = 93 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=-\frac {(b c-a d) x^3}{9 b^2 \left (a+b x^6\right )^{3/2}}+\frac {(b c-4 a d) x^3}{9 a b^2 \sqrt {a+b x^6}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^3}{\sqrt {a+b x^6}}\right )}{3 b^{5/2}} \] Output:
-1/9*(-a*d+b*c)*x^3/b^2/(b*x^6+a)^(3/2)+1/9*(-4*a*d+b*c)*x^3/a/b^2/(b*x^6+ a)^(1/2)+1/3*d*arctanh(b^(1/2)*x^3/(b*x^6+a)^(1/2))/b^(5/2)
Time = 0.70 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-3 a^2 d x^3+b^2 c x^9-4 a b d x^9}{9 a b^2 \left (a+b x^6\right )^{3/2}}+\frac {d \log \left (\sqrt {b} x^3+\sqrt {a+b x^6}\right )}{3 b^{5/2}} \] Input:
Integrate[(x^8*(c + d*x^6))/(a + b*x^6)^(5/2),x]
Output:
(-3*a^2*d*x^3 + b^2*c*x^9 - 4*a*b*d*x^9)/(9*a*b^2*(a + b*x^6)^(3/2)) + (d* Log[Sqrt[b]*x^3 + Sqrt[a + b*x^6]])/(3*b^(5/2))
Time = 0.37 (sec) , antiderivative size = 88, 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 = {954, 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.
\(\displaystyle \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 954 |
\(\displaystyle \frac {d \int \frac {x^8}{\left (b x^6+a\right )^{3/2}}dx}{b}+\frac {x^9 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {d \int \frac {x^6}{\left (b x^6+a\right )^{3/2}}dx^3}{3 b}+\frac {x^9 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
\(\Big \downarrow \) 252 |
\(\displaystyle \frac {d \left (\frac {\int \frac {1}{\sqrt {b x^6+a}}dx^3}{b}-\frac {x^3}{b \sqrt {a+b x^6}}\right )}{3 b}+\frac {x^9 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {d \left (\frac {\int \frac {1}{1-b x^6}d\frac {x^3}{\sqrt {b x^6+a}}}{b}-\frac {x^3}{b \sqrt {a+b x^6}}\right )}{3 b}+\frac {x^9 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {d \left (\frac {\text {arctanh}\left (\frac {\sqrt {b} x^3}{\sqrt {a+b x^6}}\right )}{b^{3/2}}-\frac {x^3}{b \sqrt {a+b x^6}}\right )}{3 b}+\frac {x^9 (b c-a d)}{9 a b \left (a+b x^6\right )^{3/2}}\) |
Input:
Int[(x^8*(c + d*x^6))/(a + b*x^6)^(5/2),x]
Output:
((b*c - a*d)*x^9)/(9*a*b*(a + b*x^6)^(3/2)) + (d*(-(x^3/(b*Sqrt[a + b*x^6] )) + ArcTanh[(Sqrt[b]*x^3)/Sqrt[a + b*x^6]]/b^(3/2)))/(3*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[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b* e*(m + 1))), x] + Simp[d/b Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; Fre eQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 1, 0] && NeQ[m, -1]
Time = 4.75 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.88
method | result | size |
pseudoelliptic | \(\frac {d a \,\operatorname {arctanh}\left (\frac {\sqrt {b \,x^{6}+a}}{x^{3} \sqrt {b}}\right ) b^{2} \left (b \,x^{6}+a \right )^{\frac {3}{2}}+\frac {b^{\frac {5}{2}} \left (-4 a b d \,x^{6}+b^{2} c \,x^{6}-3 a^{2} d \right ) x^{3}}{3}}{3 \left (b \,x^{6}+a \right )^{\frac {3}{2}} b^{\frac {9}{2}} a}\) | \(82\) |
Input:
int(x^8*(d*x^6+c)/(b*x^6+a)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/3*(d*a*arctanh((b*x^6+a)^(1/2)/x^3/b^(1/2))*b^2*(b*x^6+a)^(3/2)+1/3*b^(5 /2)*(-4*a*b*d*x^6+b^2*c*x^6-3*a^2*d)*x^3)/(b*x^6+a)^(3/2)/b^(9/2)/a
Time = 0.09 (sec) , antiderivative size = 252, normalized size of antiderivative = 2.71 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\left [\frac {3 \, {\left (a b^{2} d x^{12} + 2 \, a^{2} b d x^{6} + a^{3} d\right )} \sqrt {b} \log \left (-2 \, b x^{6} - 2 \, \sqrt {b x^{6} + a} \sqrt {b} x^{3} - a\right ) + 2 \, {\left ({\left (b^{3} c - 4 \, a b^{2} d\right )} x^{9} - 3 \, a^{2} b d x^{3}\right )} \sqrt {b x^{6} + a}}{18 \, {\left (a b^{5} x^{12} + 2 \, a^{2} b^{4} x^{6} + a^{3} b^{3}\right )}}, -\frac {3 \, {\left (a b^{2} d x^{12} + 2 \, a^{2} b d x^{6} + a^{3} d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x^{3}}{\sqrt {b x^{6} + a}}\right ) - {\left ({\left (b^{3} c - 4 \, a b^{2} d\right )} x^{9} - 3 \, a^{2} b d x^{3}\right )} \sqrt {b x^{6} + a}}{9 \, {\left (a b^{5} x^{12} + 2 \, a^{2} b^{4} x^{6} + a^{3} b^{3}\right )}}\right ] \] Input:
integrate(x^8*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="fricas")
Output:
[1/18*(3*(a*b^2*d*x^12 + 2*a^2*b*d*x^6 + a^3*d)*sqrt(b)*log(-2*b*x^6 - 2*s qrt(b*x^6 + a)*sqrt(b)*x^3 - a) + 2*((b^3*c - 4*a*b^2*d)*x^9 - 3*a^2*b*d*x ^3)*sqrt(b*x^6 + a))/(a*b^5*x^12 + 2*a^2*b^4*x^6 + a^3*b^3), -1/9*(3*(a*b^ 2*d*x^12 + 2*a^2*b*d*x^6 + a^3*d)*sqrt(-b)*arctan(sqrt(-b)*x^3/sqrt(b*x^6 + a)) - ((b^3*c - 4*a*b^2*d)*x^9 - 3*a^2*b*d*x^3)*sqrt(b*x^6 + a))/(a*b^5* x^12 + 2*a^2*b^4*x^6 + a^3*b^3)]
Timed out. \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x**8*(d*x**6+c)/(b*x**6+a)**(5/2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {c x^{9}}{9 \, {\left (b x^{6} + a\right )}^{\frac {3}{2}} a} - \frac {1}{18} \, {\left (\frac {2 \, {\left (b + \frac {3 \, {\left (b x^{6} + a\right )}}{x^{6}}\right )} x^{9}}{{\left (b x^{6} + a\right )}^{\frac {3}{2}} b^{2}} + \frac {3 \, \log \left (-\frac {\sqrt {b} - \frac {\sqrt {b x^{6} + a}}{x^{3}}}{\sqrt {b} + \frac {\sqrt {b x^{6} + a}}{x^{3}}}\right )}{b^{\frac {5}{2}}}\right )} d \] Input:
integrate(x^8*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="maxima")
Output:
1/9*c*x^9/((b*x^6 + a)^(3/2)*a) - 1/18*(2*(b + 3*(b*x^6 + a)/x^6)*x^9/((b* x^6 + a)^(3/2)*b^2) + 3*log(-(sqrt(b) - sqrt(b*x^6 + a)/x^3)/(sqrt(b) + sq rt(b*x^6 + a)/x^3))/b^(5/2))*d
Exception generated. \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\text {Exception raised: NotImplementedError} \] Input:
integrate(x^8*(d*x^6+c)/(b*x^6+a)^(5/2),x, algorithm="giac")
Output:
Exception raised: NotImplementedError >> unable to parse Giac output: Recu rsive assumption sageVARa>=(-sageVARb/t_nostep^6) ignored2*(-(-9565938*sag eVARb^7*sageVARa^4*sageVARc+38263752*sageVARb^6*sageVARd*sageVARa^5)*1/172 186884/sageVARb^7/s
Timed out. \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\int \frac {x^8\,\left (d\,x^6+c\right )}{{\left (b\,x^6+a\right )}^{5/2}} \,d x \] Input:
int((x^8*(c + d*x^6))/(a + b*x^6)^(5/2),x)
Output:
int((x^8*(c + d*x^6))/(a + b*x^6)^(5/2), x)
Time = 0.31 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.63 \[ \int \frac {x^8 \left (c+d x^6\right )}{\left (a+b x^6\right )^{5/2}} \, dx=\frac {-6 \sqrt {b \,x^{6}+a}\, a^{2} b d \,x^{3}-8 \sqrt {b \,x^{6}+a}\, a \,b^{2} d \,x^{9}+2 \sqrt {b \,x^{6}+a}\, b^{3} c \,x^{9}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a^{3} d -6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a^{2} b d \,x^{6}-3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}-\sqrt {b}\, x^{3}\right ) a \,b^{2} d \,x^{12}+3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a^{3} d +6 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a^{2} b d \,x^{6}+3 \sqrt {b}\, \mathrm {log}\left (\sqrt {b \,x^{6}+a}+\sqrt {b}\, x^{3}\right ) a \,b^{2} d \,x^{12}}{18 a \,b^{3} \left (b^{2} x^{12}+2 a b \,x^{6}+a^{2}\right )} \] Input:
int(x^8*(d*x^6+c)/(b*x^6+a)^(5/2),x)
Output:
( - 6*sqrt(a + b*x**6)*a**2*b*d*x**3 - 8*sqrt(a + b*x**6)*a*b**2*d*x**9 + 2*sqrt(a + b*x**6)*b**3*c*x**9 - 3*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)* x**3)*a**3*d - 6*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x**3)*a**2*b*d*x** 6 - 3*sqrt(b)*log(sqrt(a + b*x**6) - sqrt(b)*x**3)*a*b**2*d*x**12 + 3*sqrt (b)*log(sqrt(a + b*x**6) + sqrt(b)*x**3)*a**3*d + 6*sqrt(b)*log(sqrt(a + b *x**6) + sqrt(b)*x**3)*a**2*b*d*x**6 + 3*sqrt(b)*log(sqrt(a + b*x**6) + sq rt(b)*x**3)*a*b**2*d*x**12)/(18*a*b**3*(a**2 + 2*a*b*x**6 + b**2*x**12))