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

Optimal. Leaf size=13 \[ x^m \left (a+b x^n\right )^p \]

[Out]

x^m*(a + b*x^n)^p

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Rubi [A]  time = 0.0573118, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032 \[ x^m \left (a+b x^n\right )^p \]

Antiderivative was successfully verified.

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

[Out]

x^m*(a + b*x^n)^p

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Rubi in Sympy [A]  time = 6.8081, size = 10, normalized size = 0.77 \[ x^{m} \left (a + b x^{n}\right )^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**m*(a + b*x**n)**p

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Mathematica [A]  time = 0.0774996, size = 13, normalized size = 1. \[ x^m \left (a+b x^n\right )^p \]

Antiderivative was successfully verified.

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

[Out]

x^m*(a + b*x^n)^p

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{x}^{-1+m} \left ( a+b{x}^{n} \right ) ^{-1+p} \left ( am+b \left ( np+m \right ){x}^{n} \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.82283, size = 22, normalized size = 1.69 \[ e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e^(p*log(b*x^n + a) + m*log(x))

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Fricas [A]  time = 0.232569, size = 43, normalized size = 3.31 \[{\left (b x x^{m - 1} x^{n} + a x x^{m - 1}\right )}{\left (b x^{n} + a\right )}^{p - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221463, size = 107, normalized size = 8.23 \[ b x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) -{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) -{\rm ln}\left (x\right )\right )} + a x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + m{\rm ln}\left (x\right ) -{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) -{\rm ln}\left (x\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x*e^(p*ln(b*e^(n*ln(x)) + a) + m*ln(x) + n*ln(x) - ln(b*e^(n*ln(x)) + a) - ln(
x)) + a*x*e^(p*ln(b*e^(n*ln(x)) + a) + m*ln(x) - ln(b*e^(n*ln(x)) + a) - ln(x))