Optimal. Leaf size=135 \[ -\frac{a^8 x^{-2 n}}{2 n}-\frac{8 a^7 b x^{-n}}{n}+28 a^6 b^2 \log (x)+\frac{56 a^5 b^3 x^n}{n}+\frac{35 a^4 b^4 x^{2 n}}{n}+\frac{56 a^3 b^5 x^{3 n}}{3 n}+\frac{7 a^2 b^6 x^{4 n}}{n}+\frac{8 a b^7 x^{5 n}}{5 n}+\frac{b^8 x^{6 n}}{6 n} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.151508, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^8 x^{-2 n}}{2 n}-\frac{8 a^7 b x^{-n}}{n}+28 a^6 b^2 \log (x)+\frac{56 a^5 b^3 x^n}{n}+\frac{35 a^4 b^4 x^{2 n}}{n}+\frac{56 a^3 b^5 x^{3 n}}{3 n}+\frac{7 a^2 b^6 x^{4 n}}{n}+\frac{8 a b^7 x^{5 n}}{5 n}+\frac{b^8 x^{6 n}}{6 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 - 2*n)*(a + b*x^n)^8,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8} x^{- 2 n}}{2 n} - \frac{8 a^{7} b x^{- n}}{n} + \frac{28 a^{6} b^{2} \log{\left (x^{n} \right )}}{n} + \frac{56 a^{5} b^{3} x^{n}}{n} + \frac{70 a^{4} b^{4} \int ^{x^{n}} x\, dx}{n} + \frac{56 a^{3} b^{5} x^{3 n}}{3 n} + \frac{7 a^{2} b^{6} x^{4 n}}{n} + \frac{8 a b^{7} x^{5 n}}{5 n} + \frac{b^{8} x^{6 n}}{6 n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1-2*n)*(a+b*x**n)**8,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0519688, size = 116, normalized size = 0.86 \[ \frac{x^{-2 n} \left (-15 a^8-240 a^7 b x^n+840 a^6 b^2 n x^{2 n} \log (x)+1680 a^5 b^3 x^{3 n}+1050 a^4 b^4 x^{4 n}+560 a^3 b^5 x^{5 n}+210 a^2 b^6 x^{6 n}+48 a b^7 x^{7 n}+5 b^8 x^{8 n}\right )}{30 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 - 2*n)*(a + b*x^n)^8,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.045, size = 128, normalized size = 1. \[ 28\,{a}^{6}{b}^{2}\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{6}}{6\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{5}}{5\,n}}+7\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{3}}{3\,n}}+35\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{2}}{n}}+56\,{\frac{{a}^{5}{b}^{3}{x}^{n}}{n}}-8\,{\frac{b{a}^{7}}{n{x}^{n}}}-{\frac{{a}^{8}}{2\,n \left ({x}^{n} \right ) ^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1-2*n)*(a+b*x^n)^8,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(-2*n - 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.228129, size = 157, normalized size = 1.16 \[ \frac{840 \, a^{6} b^{2} n x^{2 \, n} \log \left (x\right ) + 5 \, b^{8} x^{8 \, n} + 48 \, a b^{7} x^{7 \, n} + 210 \, a^{2} b^{6} x^{6 \, n} + 560 \, a^{3} b^{5} x^{5 \, n} + 1050 \, a^{4} b^{4} x^{4 \, n} + 1680 \, a^{5} b^{3} x^{3 \, n} - 240 \, a^{7} b x^{n} - 15 \, a^{8}}{30 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(-2*n - 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**(-1-2*n)*(a+b*x**n)**8,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.233199, size = 167, normalized size = 1.24 \[ \frac{{\left (840 \, a^{6} b^{2} n e^{\left (2 \, n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 5 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 48 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 210 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 560 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 1050 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 1680 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} - 240 \, a^{7} b e^{\left (n{\rm ln}\left (x\right )\right )} - 15 \, a^{8}\right )} e^{\left (-2 \, n{\rm ln}\left (x\right )\right )}}{30 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(-2*n - 1),x, algorithm="giac")
[Out]