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

Optimal. Leaf size=36 \[ \frac{x^{-(1-n) (p+1)} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.144703, antiderivative size = 36, 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 \[ \frac{x^{-(1-n) (p+1)} \left (b x+c x^{n+1}\right )^{p+1}}{n (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 12.8781, size = 26, normalized size = 0.72 \[ \frac{x^{\left (n - 1\right ) \left (p + 1\right )} \left (b x + c x^{n + 1}\right )^{p + 1}}{n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.10897, size = 31, normalized size = 0.86 \[ \frac{x^{(n-1) (p+1)} \left (x \left (b+c x^n\right )\right )^{p+1}}{n (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.113, size = 0, normalized size = 0. \[ \int{x}^{ \left ( -1+n \right ) \left ( 1+p \right ) } \left ( b+2\,c{x}^{n} \right ) \left ( bx+c{x}^{1+n} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.13246, size = 53, normalized size = 1.47 \[ \frac{{\left (c x^{2 \, n} + b x^{n}\right )} e^{\left (n p \log \left (x\right ) + p \log \left (c x^{n} + b\right )\right )}}{n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*(b*x + c*x^(n + 1))^p*x^((n - 1)*(p + 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 = 0.273908, size = 57, normalized size = 1.58 \[ \frac{{\left (b x + c x^{n + 1}\right )}{\left (b x + c x^{n + 1}\right )}^{p} x^{{\left (n - 1\right )} p + n - 1}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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