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