Optimal. Leaf size=88 \[ \frac{a}{4 b^2 n \left (a+b x^n\right )^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{3 b^2 n \left (a+b x^n\right )^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
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Rubi [A] time = 0.0515943, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {1355, 266, 43} \[ \frac{a}{4 b^2 n \left (a+b x^n\right )^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{3 b^2 n \left (a+b x^n\right )^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{-1+2 n}}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^n\right )\right ) \int \frac{x^{-1+2 n}}{\left (a b+b^2 x^n\right )^5} \, dx}{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^n\right )\right ) \operatorname{Subst}\left (\int \frac{x}{\left (a b+b^2 x\right )^5} \, dx,x,x^n\right )}{n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{\left (b^4 \left (a b+b^2 x^n\right )\right ) \operatorname{Subst}\left (\int \left (-\frac{a}{b^6 (a+b x)^5}+\frac{1}{b^6 (a+b x)^4}\right ) \, dx,x,x^n\right )}{n \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ &=\frac{a}{4 b^2 n \left (a+b x^n\right )^3 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}-\frac{1}{3 b^2 n \left (a+b x^n\right )^2 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}\\ \end{align*}
Mathematica [A] time = 0.0327514, size = 40, normalized size = 0.45 \[ -\frac{a+4 b x^n}{12 b^2 n \left (a+b x^n\right )^3 \sqrt{\left (a+b x^n\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 37, normalized size = 0.4 \begin{align*} -{\frac{4\,b{x}^{n}+a}{12\, \left ( a+b{x}^{n} \right ) ^{5}{b}^{2}n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02756, size = 93, normalized size = 1.06 \begin{align*} -\frac{4 \, b x^{n} + a}{12 \,{\left (b^{6} n x^{4 \, n} + 4 \, a b^{5} n x^{3 \, n} + 6 \, a^{2} b^{4} n x^{2 \, n} + 4 \, a^{3} b^{3} n x^{n} + a^{4} b^{2} n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56607, size = 147, normalized size = 1.67 \begin{align*} -\frac{4 \, b x^{n} + a}{12 \,{\left (b^{6} n x^{4 \, n} + 4 \, a b^{5} n x^{3 \, n} + 6 \, a^{2} b^{4} n x^{2 \, n} + 4 \, a^{3} b^{3} n x^{n} + a^{4} b^{2} n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2 \, n - 1}}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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