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

Optimal. Leaf size=29 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]

[Out]

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

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Rubi [A]  time = 0.0692443, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 13.2978, size = 20, normalized size = 0.69 \[ \frac{\left (- a + b x^{n} + c x^{2 n}\right )^{p + 1}}{n \left (p + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0875493, size = 28, normalized size = 0.97 \[ \frac{\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n p+n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.094, size = 45, normalized size = 1.6 \[ -{\frac{ \left ( -c \left ({x}^{n} \right ) ^{2}-b{x}^{n}+a \right ) \left ( -a+b{x}^{n}+c \left ({x}^{n} \right ) ^{2} \right ) ^{p}}{n \left ( 1+p \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.04987, size = 58, normalized size = 2. \[ \frac{{\left (c x^{2 \, n} + b x^{n} - a\right )}{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p}}{n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.307078, size = 57, normalized size = 1.97 \[ \frac{{\left (c x^{2 \, n} + b x^{n} - a\right )}{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p}}{n p + n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.286956, size = 39, normalized size = 1.34 \[ \frac{{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p + 1}}{n{\left (p + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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