Optimal. Leaf size=117 \[ -\frac {(d x)^{-2 n (1+p)} \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{a d n (1+2 p)}+\frac {(d x)^{-2 n (1+p)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{1+p}}{2 a^2 d n (1+p) (1+2 p)} \]
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Rubi [A]
time = 0.04, antiderivative size = 124, normalized size of antiderivative = 1.06, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1370, 279, 270}
\begin {gather*} \frac {\left (\frac {b x^n}{a}+1\right )^2 (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 d n \left (2 p^2+3 p+1\right )}-\frac {\left (\frac {b x^n}{a}+1\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (2 p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 279
Rule 1370
Rubi steps
\begin {align*} \int (d x)^{-1-2 n (1+p)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p \, dx &=\left (\left (1+\frac {b x^n}{a}\right )^{-2 p} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p\right ) \int (d x)^{-1-2 n (1+p)} \left (1+\frac {b x^n}{a}\right )^{2 p} \, dx\\ &=-\frac {(d x)^{-2 n (1+p)} \left (1+\frac {b x^n}{a}\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (1+2 p)}+\frac {\left ((-2 n (1+p)+n (1+2 p)) \left (1+\frac {b x^n}{a}\right )^{-2 p} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p\right ) \int (d x)^{-1-2 n (1+p)} \left (1+\frac {b x^n}{a}\right )^{1+2 p} \, dx}{n (1+2 p)}\\ &=-\frac {(d x)^{-2 n (1+p)} \left (1+\frac {b x^n}{a}\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (1+2 p)}+\frac {(d x)^{-2 n (1+p)} \left (1+\frac {b x^n}{a}\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 d n \left (1+3 p+2 p^2\right )}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.07, size = 75, normalized size = 0.64 \begin {gather*} -\frac {x (d x)^{-1-2 n (1+p)} \left (\left (a+b x^n\right )^2\right )^p \left (1+\frac {b x^n}{a}\right )^{-2 p} \, _2F_1\left (-2 p,-2 (1+p);1-2 (1+p);-\frac {b x^n}{a}\right )}{2 n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{-1-2 n \left (1+p \right )} \left (a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 165, normalized size = 1.41 \begin {gather*} -\frac {{\left (2 \, a b p x x^{n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) - {\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )} - b^{2} x x^{2 \, n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) - {\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )} + {\left (2 \, a^{2} p + a^{2}\right )} x e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \left (d\right ) - {\left (2 \, n p + 2 \, n + 1\right )} \log \left (x\right )\right )}\right )} {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p}}{2 \, {\left (2 \, a^{2} n p^{2} + 3 \, a^{2} n p + a^{2} n\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^p}{{\left (d\,x\right )}^{2\,n\,\left (p+1\right )+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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