3.912 \(\int x^{-1+n (1+p)} \left (b+c x^n\right )^p \left (b+2 c x^n\right ) \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0493801, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ \frac{x^{n (p+1)} \left (b+c x^n\right )^{p+1}}{n (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.2432, size = 20, normalized size = 0.74 \[ \frac{x^{n \left (p + 1\right )} \left (b + c x^{n}\right )^{p + 1}}{n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+n*(1+p))*(b+c*x**n)**p*(b+2*c*x**n),x)

[Out]

x**(n*(p + 1))*(b + c*x**n)**(p + 1)/(n*(p + 1))

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Mathematica [A]  time = 0.0935243, size = 26, normalized size = 0.96 \[ \frac{x^{n p+n} \left (b+c x^n\right )^{p+1}}{n p+n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.124, size = 0, normalized size = 0. \[ \int{x}^{-1+n \left ( 1+p \right ) } \left ( b+c{x}^{n} \right ) ^{p} \left ( b+2\,c{x}^{n} \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n*(1+p))*(b+c*x^n)^p*(b+2*c*x^n),x)

[Out]

int(x^(-1+n*(1+p))*(b+c*x^n)^p*(b+2*c*x^n),x)

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Maxima [A]  time = 1.87113, size = 53, normalized size = 1.96 \[ \frac{{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (n p \log \left (x\right ) + p \log \left (c x^{n} + b\right )\right )}}{n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*(c*x^n + b)^p*x^(n*(p + 1) - 1),x, algorithm="maxima")

[Out]

(c*x^(2*n) + b*x^n)*e^(n*p*log(x) + p*log(c*x^n + b))/(n*(p + 1))

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Fricas [A]  time = 0.231182, size = 47, normalized size = 1.74 \[ \frac{{\left (c x x^{n} + b x\right )}{\left (c x^{n} + b\right )}^{p} x^{n p + n - 1}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x*x^n + b*x)*(c*x^n + b)^p*x^(n*p + n - 1)/(n*p + 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*(1+p))*(b+c*x**n)**p*(b+2*c*x**n),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.218784, size = 95, normalized size = 3.52 \[ \frac{c x e^{\left (n p{\rm ln}\left (x\right ) + p{\rm ln}\left (c e^{\left (n{\rm ln}\left (x\right )\right )} + b\right ) + 2 \, n{\rm ln}\left (x\right ) -{\rm ln}\left (x\right )\right )} + b x e^{\left (n p{\rm ln}\left (x\right ) + p{\rm ln}\left (c e^{\left (n{\rm ln}\left (x\right )\right )} + b\right ) + n{\rm ln}\left (x\right ) -{\rm ln}\left (x\right )\right )}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(c*x*e^(n*p*ln(x) + p*ln(c*e^(n*ln(x)) + b) + 2*n*ln(x) - ln(x)) + b*x*e^(n*p*ln
(x) + p*ln(c*e^(n*ln(x)) + b) + n*ln(x) - ln(x)))/(n*p + n)