Optimal. Leaf size=117 \[ \frac {(d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{p+1}}{2 a^2 d n (p+1) (2 p+1)}-\frac {\left (a+b x^n\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{a d n (2 p+1)} \]
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Rubi [A] time = 0.06, 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 = {1356, 273, 264} \[ \frac {\left (\frac {b x^n}{a}+1\right )^2 (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{2 d n \left (2 p^2+3 p+1\right )}-\frac {\left (\frac {b x^n}{a}+1\right ) (d x)^{-2 n (p+1)} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^p}{d n (2 p+1)} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rule 1356
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] time = 0.03, size = 75, normalized size = 0.64 \[ -\frac {x (d x)^{-2 n (p+1)-1} \left (\left (a+b x^n\right )^2\right )^p \left (\frac {b x^n}{a}+1\right )^{-2 p} \, _2F_1\left (-2 p,-2 (p+1);1-2 (p+1);-\frac {b x^n}{a}\right )}{2 n (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 165, normalized size = 1.41 \[ -\frac {{\left (2 \, a b p x x^{n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \relax (d) - {\left (2 \, n p + 2 \, n + 1\right )} \log \relax (x)\right )} - b^{2} x x^{2 \, n} e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \relax (d) - {\left (2 \, n p + 2 \, n + 1\right )} \log \relax (x)\right )} + {\left (2 \, a^{2} p + a^{2}\right )} x e^{\left (-{\left (2 \, n p + 2 \, n + 1\right )} \log \relax (d) - {\left (2 \, n p + 2 \, n + 1\right )} \log \relax (x)\right )}\right )} {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p}}{2 \, {\left (2 \, a^{2} n p^{2} + 3 \, a^{2} n p + a^{2} n\right )}} \]
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 \[ \int {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p} \left (d x\right )^{-2 \, n {\left (p + 1\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{-2 \left (p +1\right ) n -1} \left (2 a b \,x^{n}+b^{2} x^{2 n}+a^{2}\right )^{p}\, dx \]
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 \[ \int {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{p} \left (d x\right )^{-2 \, n {\left (p + 1\right )} - 1}\,{d x} \]
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 \[ \int \frac {{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^p}{{\left (d\,x\right )}^{2\,n\,\left (p+1\right )+1}} \,d x \]
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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