Optimal. Leaf size=84 \[ -\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]
[Out]
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Rubi [A] time = 0.126183, antiderivative size = 84, 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 \[ -\frac{a^3 \left (a+b x^n\right )^9}{9 b^4 n}+\frac{3 a^2 \left (a+b x^n\right )^{10}}{10 b^4 n}+\frac{\left (a+b x^n\right )^{12}}{12 b^4 n}-\frac{3 a \left (a+b x^n\right )^{11}}{11 b^4 n} \]
Antiderivative was successfully verified.
[In] Int[x^(-1 + 4*n)*(a + b*x^n)^8,x]
[Out]
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Rubi in Sympy [A] time = 22.1104, size = 71, normalized size = 0.85 \[ - \frac{a^{3} \left (a + b x^{n}\right )^{9}}{9 b^{4} n} + \frac{3 a^{2} \left (a + b x^{n}\right )^{10}}{10 b^{4} n} - \frac{3 a \left (a + b x^{n}\right )^{11}}{11 b^{4} n} + \frac{\left (a + b x^{n}\right )^{12}}{12 b^{4} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(-1+4*n)*(a+b*x**n)**8,x)
[Out]
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Mathematica [A] time = 0.0462353, size = 113, normalized size = 1.35 \[ \frac{x^{4 n} \left (495 a^8+3168 a^7 b x^n+9240 a^6 b^2 x^{2 n}+15840 a^5 b^3 x^{3 n}+17325 a^4 b^4 x^{4 n}+12320 a^3 b^5 x^{5 n}+5544 a^2 b^6 x^{6 n}+1440 a b^7 x^{7 n}+165 b^8 x^{8 n}\right )}{1980 n} \]
Antiderivative was successfully verified.
[In] Integrate[x^(-1 + 4*n)*(a + b*x^n)^8,x]
[Out]
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Maple [A] time = 0.04, size = 136, normalized size = 1.6 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{12}}{12\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{14\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{8}}{4\,n}}+8\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{14\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{4}}{4\,n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(-1+4*n)*(a+b*x^n)^8,x)
[Out]
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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^(4*n - 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.22563, size = 153, normalized size = 1.82 \[ \frac{165 \, b^{8} x^{12 \, n} + 1440 \, a b^{7} x^{11 \, n} + 5544 \, a^{2} b^{6} x^{10 \, n} + 12320 \, a^{3} b^{5} x^{9 \, n} + 17325 \, a^{4} b^{4} x^{8 \, n} + 15840 \, a^{5} b^{3} x^{7 \, n} + 9240 \, a^{6} b^{2} x^{6 \, n} + 3168 \, a^{7} b x^{5 \, n} + 495 \, a^{8} x^{4 \, n}}{1980 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(4*n - 1),x, algorithm="fricas")
[Out]
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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+4*n)*(a+b*x**n)**8,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{4 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^n + a)^8*x^(4*n - 1),x, algorithm="giac")
[Out]