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

Optimal. Leaf size=46 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+n+1}\right )^{p+1}}{b (p+1) (m p+n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0845235, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+n+1}\right )^{p+1}}{b (p+1) (m p+n+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^n*(a*x^m + b*x^(1 + m + n + m*p))^p,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.0193, size = 37, normalized size = 0.8 \[ \frac{x^{- m \left (p + 1\right )} \left (a x^{m} + b x^{m p + m + n + 1}\right )^{p + 1}}{b \left (p + 1\right ) \left (m p + n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**(-m*(p + 1))*(a*x**m + b*x**(m*p + m + n + 1))**(p + 1)/(b*(p + 1)*(m*p + n +
1))

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Mathematica [A]  time = 0.0366624, size = 45, normalized size = 0.98 \[ \frac{x^{-m (p+1)} \left (x^m \left (a+b x^{m p+n+1}\right )\right )^{p+1}}{b (p+1) (m p+n+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.345, size = 0, normalized size = 0. \[ \int{x}^{n} \left ( a{x}^{m}+b{x}^{mp+m+n+1} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^n*(a*x^m+b*x^(m*p+m+n+1))^p,x)

[Out]

int(x^n*(a*x^m+b*x^(m*p+m+n+1))^p,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{m p + m + n + 1} + a x^{m}\right )}^{p} x^{n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^(m*p + m + n + 1) + a*x^m)^p*x^n, x)

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Fricas [A]  time = 0.256261, size = 107, normalized size = 2.33 \[ \frac{{\left (b x x^{m p + m + n + 1} x^{n} + a x x^{m} x^{n}\right )}{\left (b x^{m p + m + n + 1} + a x^{m}\right )}^{p}}{{\left (b m p^{2} + b n +{\left (b m + b n + b\right )} p + b\right )} x^{m p + m + n + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*x*x^(m*p + m + n + 1)*x^n + a*x*x^m*x^n)*(b*x^(m*p + m + n + 1) + a*x^m)^p/((
b*m*p^2 + b*n + (b*m + b*n + b)*p + b)*x^(m*p + m + n + 1))

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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**n*(a*x**m+b*x**(m*p+m+n+1))**p,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^(m*p + m + n + 1) + a*x^m)^p*x^n, x)