Optimal. Leaf size=49 \[ \frac {\left (a+b x^n\right )^{p+2}}{b^2 n (p+2)}-\frac {a \left (a+b x^n\right )^{p+1}}{b^2 n (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {266, 43} \[ \frac {\left (a+b x^n\right )^{p+2}}{b^2 n (p+2)}-\frac {a \left (a+b x^n\right )^{p+1}}{b^2 n (p+1)} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+2 n} \left (a+b x^n\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int x (a+b x)^p \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^p}{b}+\frac {(a+b x)^{1+p}}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {a \left (a+b x^n\right )^{1+p}}{b^2 n (1+p)}+\frac {\left (a+b x^n\right )^{2+p}}{b^2 n (2+p)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 0.82 \[ \frac {\left (a+b x^n\right )^{p+1} \left (b (p+1) x^n-a\right )}{b^2 n (p+1) (p+2)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 62, normalized size = 1.27 \[ \frac {{\left (a b p x^{n} - a^{2} + {\left (b^{2} p + b^{2}\right )} x^{2 \, n}\right )} {\left (b x^{n} + a\right )}^{p}}{b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b x^{n} + a\right )}^{p} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 61, normalized size = 1.24 \[ -\frac {\left (-a b p \,x^{n}-b^{2} p \,x^{2 n}-b^{2} x^{2 n}+a^{2}\right ) \left (b \,x^{n}+a \right )^{p}}{\left (p +1\right ) \left (p +2\right ) b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 51, normalized size = 1.04 \[ \frac {{\left (b^{2} {\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )} {\left (b x^{n} + a\right )}^{p}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{2\,n-1}\,{\left (a+b\,x^n\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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