3.2713 \(\int x^{-1+n} \left (a+b x^n\right )^p \, dx\)

Optimal. Leaf size=23 \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]

[Out]

(a + b*x^n)^(1 + p)/(b*n*(1 + p))

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Rubi [A]  time = 0.0284346, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a+b x^n\right )^{p+1}}{b n (p+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)*(a + b*x^n)^p,x]

[Out]

(a + b*x^n)^(1 + p)/(b*n*(1 + p))

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Rubi in Sympy [A]  time = 3.00776, size = 15, normalized size = 0.65 \[ \frac{\left (a + b x^{n}\right )^{p + 1}}{b n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**n)**(p + 1)/(b*n*(p + 1))

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Mathematica [A]  time = 0.0335867, size = 22, normalized size = 0.96 \[ \frac{\left (a+b x^n\right )^{p+1}}{b n p+b n} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^n)^(1 + p)/(b*n + b*n*p)

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Maple [A]  time = 0.069, size = 29, normalized size = 1.3 \[{\frac{ \left ( a+b{x}^{n} \right ) \left ( a+b{x}^{n} \right ) ^{p}}{b \left ( 1+p \right ) n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(a+b*x^n)^p,x)

[Out]

(a+b*x^n)/b/(1+p)/n*(a+b*x^n)^p

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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)^p*x^(n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237404, size = 36, normalized size = 1.57 \[ \frac{{\left (b x^{n} + a\right )}{\left (b x^{n} + a\right )}^{p}}{b n p + b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^n + a)*(b*x^n + a)^p/(b*n*p + b*n)

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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+n)*(a+b*x**n)**p,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.215972, size = 31, normalized size = 1.35 \[ \frac{{\left (b x^{n} + a\right )}^{p + 1}}{b n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x^n + a)^(p + 1)/(b*n*(p + 1))