3.2732 \(\int x^m \left (a+b x^{1+m}\right )^n \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]

[Out]

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

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Rubi [A]  time = 0.0287156, antiderivative size = 27, 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 \[ \frac{\left (a+b x^{m+1}\right )^{n+1}}{b (m+1) (n+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^(1 + m))^n,x]

[Out]

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

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Rubi in Sympy [A]  time = 3.3963, size = 19, normalized size = 0.7 \[ \frac{\left (a + b x^{m + 1}\right )^{n + 1}}{b \left (m + 1\right ) \left (n + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0451166, size = 28, normalized size = 1.04 \[ \frac{\left (a+b x^{m+1}\right )^{n+1}}{b m n+b m+b n+b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.044, size = 60, normalized size = 2.2 \[{\frac{a{{\rm e}^{n\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }}}{b \left ( nm+m+n+1 \right ) }}+{\frac{x{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( a+bx{{\rm e}^{m\ln \left ( x \right ) }} \right ) }}}{nm+m+n+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^(1+m))^n,x)

[Out]

a/b/(m*n+m+n+1)*exp(n*ln(a+b*x*exp(m*ln(x))))+1/(m*n+m+n+1)*x*exp(m*ln(x))*exp(n
*ln(a+b*x*exp(m*ln(x))))

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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^(m + 1) + a)^n*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236822, size = 47, normalized size = 1.74 \[ \frac{{\left (b x^{m + 1} + a\right )}{\left (b x^{m + 1} + a\right )}^{n}}{b m +{\left (b m + b\right )} n + b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.214024, size = 36, normalized size = 1.33 \[ \frac{{\left (b x^{m + 1} + a\right )}^{n + 1}}{b{\left (m + 1\right )}{\left (n + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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