Optimal. Leaf size=44 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \]
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Rubi [A] time = 0.0365398, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{x^{-m (p+1)} \left (a x^m+b x^{m p+m+1}\right )^{p+1}}{b (p+1) (m p+1)} \]
Antiderivative was successfully verified.
[In] Int[(x^m*(a + b*x^(1 + m*p)))^p,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (x^{m} \left (a + b x^{m p + 1}\right )\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**m*(a+b*x**(m*p+1)))**p,x)
[Out]
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Mathematica [A] time = 0.0277841, size = 43, normalized size = 0.98 \[ \frac{x^{-m (p+1)} \left (x^m \left (a+b x^{m p+1}\right )\right )^{p+1}}{b (p+1) (m p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(x^m*(a + b*x^(1 + m*p)))^p,x]
[Out]
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Maple [F] time = 0.104, size = 0, normalized size = 0. \[ \int \left ({x}^{m} \left ( a+b{x}^{mp+1} \right ) \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^m*(a+b*x^(m*p+1)))^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \left ({\left (b x^{m p + 1} + a\right )} x^{m}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x^(m*p + 1) + a)*x^m)^p,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254669, size = 82, normalized size = 1.86 \[ \frac{{\left (b x x^{m p + 1} + a x\right )}{\left (b x^{m p + 1} x^{m} + a x^{m}\right )}^{p}}{{\left (b m p^{2} +{\left (b m + b\right )} p + b\right )} x^{m p + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x^(m*p + 1) + a)*x^m)^p,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**m*(a+b*x**(m*p+1)))**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \left ({\left (b x^{m p + 1} + a\right )} x^{m}\right )^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x^(m*p + 1) + a)*x^m)^p,x, algorithm="giac")
[Out]