Optimal. Leaf size=29 \[ \frac {\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1468, 629} \[ \frac {\left (-a+b x^n+c x^{2 n}\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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Rule 629
Rule 1468
Rubi steps
\begin {align*} \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^p \, dx &=\frac {\operatorname {Subst}\left (\int (b+2 c x) \left (-a+b x+c x^2\right )^p \, dx,x,x^n\right )}{n}\\ &=\frac {\left (-a+b x^n+c x^{2 n}\right )^{1+p}}{n (1+p)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.97 \[ \frac {\left (x^n \left (b+c x^n\right )-a\right )^{p+1}}{n (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 42, normalized size = 1.45 \[ \frac {{\left (c x^{2 \, n} + b x^{n} - a\right )} {\left (c x^{2 \, n} + b x^{n} - a\right )}^{p}}{n p + n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.82, size = 29, normalized size = 1.00 \[ \frac {{\left (c x^{2 \, n} + b x^{n} - a\right )}^{p + 1}}{n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 45, normalized size = 1.55 \[ -\frac {\left (-b \,x^{n}-c \,x^{2 n}+a \right ) \left (b \,x^{n}+c \,x^{2 n}-a \right )^{p}}{\left (p +1\right ) n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.71, size = 43, normalized size = 1.48 \[ \frac {{\left (c x^{2 \, n} + b x^{n} - a\right )} {\left (c x^{2 \, n} + b x^{n} - a\right )}^{p}}{n {\left (p + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.54, size = 59, normalized size = 2.03 \[ \left (\frac {b\,x^n}{n\,\left (p+1\right )}-\frac {a}{n\,\left (p+1\right )}+\frac {c\,x^{2\,n}}{n\,\left (p+1\right )}\right )\,{\left (b\,x^n-a+c\,x^{2\,n}\right )}^p \]
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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