3.2546 \(\int x^{-1+2 n} \left (a+b x^n\right )^5 \, dx\)

Optimal. Leaf size=40 \[ \frac{\left (a+b x^n\right )^7}{7 b^2 n}-\frac{a \left (a+b x^n\right )^6}{6 b^2 n} \]

[Out]

-(a*(a + b*x^n)^6)/(6*b^2*n) + (a + b*x^n)^7/(7*b^2*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0565317, antiderivative size = 40, 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 \[ \frac{\left (a+b x^n\right )^7}{7 b^2 n}-\frac{a \left (a+b x^n\right )^6}{6 b^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 2*n)*(a + b*x^n)^5,x]

[Out]

-(a*(a + b*x^n)^6)/(6*b^2*n) + (a + b*x^n)^7/(7*b^2*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.18729, size = 31, normalized size = 0.78 \[ - \frac{a \left (a + b x^{n}\right )^{6}}{6 b^{2} n} + \frac{\left (a + b x^{n}\right )^{7}}{7 b^{2} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+2*n)*(a+b*x**n)**5,x)

[Out]

-a*(a + b*x**n)**6/(6*b**2*n) + (a + b*x**n)**7/(7*b**2*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0314927, size = 74, normalized size = 1.85 \[ \frac{x^{2 n} \left (21 a^5+70 a^4 b x^n+105 a^3 b^2 x^{2 n}+84 a^2 b^3 x^{3 n}+35 a b^4 x^{4 n}+6 b^5 x^{5 n}\right )}{42 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 2*n)*(a + b*x^n)^5,x]

[Out]

(x^(2*n)*(21*a^5 + 70*a^4*b*x^n + 105*a^3*b^2*x^(2*n) + 84*a^2*b^3*x^(3*n) + 35*
a*b^4*x^(4*n) + 6*b^5*x^(5*n)))/(42*n)

_______________________________________________________________________________________

Maple [B]  time = 0.036, size = 88, normalized size = 2.2 \[{\frac{{b}^{5} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{6}}{6\,n}}+2\,{\frac{{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{5}}{n}}+{\frac{5\,{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{4}}{2\,n}}+{\frac{5\,{a}^{4}b \left ({x}^{n} \right ) ^{3}}{3\,n}}+{\frac{{a}^{5} \left ({x}^{n} \right ) ^{2}}{2\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+2*n)*(a+b*x^n)^5,x)

[Out]

1/7*b^5/n*(x^n)^7+5/6*a*b^4/n*(x^n)^6+2*a^2*b^3/n*(x^n)^5+5/2*a^3*b^2/n*(x^n)^4+
5/3*a^4*b/n*(x^n)^3+1/2*a^5/n*(x^n)^2

_______________________________________________________________________________________

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)^5*x^(2*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.226285, size = 100, normalized size = 2.5 \[ \frac{6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(2*n - 1),x, algorithm="fricas")

[Out]

1/42*(6*b^5*x^(7*n) + 35*a*b^4*x^(6*n) + 84*a^2*b^3*x^(5*n) + 105*a^3*b^2*x^(4*n
) + 70*a^4*b*x^(3*n) + 21*a^5*x^(2*n))/n

_______________________________________________________________________________________

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)**5,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{5} x^{2 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^5*x^(2*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^5*x^(2*n - 1), x)