Optimal. Leaf size=112 \[ \frac{\left (a+b x^n\right )^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{5 n \left (a b^2+b^3 x^n\right )}-\frac{a \left (a+b x^n\right )^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{4 n \left (a b^2+b^3 x^n\right )} \]
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Rubi [A] time = 0.0396355, antiderivative size = 112, 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} \[ \frac{\left (a+b x^n\right )^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{5 n \left (a b^2+b^3 x^n\right )}-\frac{a \left (a+b x^n\right )^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{4 n \left (a b^2+b^3 x^n\right )} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^{-1+2 n} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2} \, dx &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \int x^{-1+2 n} \left (a b+b^2 x^n\right )^3 \, dx}{b^2 \left (a b+b^2 x^n\right )}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \operatorname{Subst}\left (\int x \left (a b+b^2 x\right )^3 \, dx,x,x^n\right )}{b^2 n \left (a b+b^2 x^n\right )}\\ &=\frac{\sqrt{a^2+2 a b x^n+b^2 x^{2 n}} \operatorname{Subst}\left (\int \left (-\frac{a \left (a b+b^2 x\right )^3}{b}+\frac{\left (a b+b^2 x\right )^4}{b^2}\right ) \, dx,x,x^n\right )}{b^2 n \left (a b+b^2 x^n\right )}\\ &=-\frac{a \left (a+b x^n\right )^4 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{4 n \left (a b^2+b^3 x^n\right )}+\frac{\left (a+b x^n\right )^5 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{5 n \left (a b^2+b^3 x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0379362, size = 40, normalized size = 0.36 \[ -\frac{\left (a-4 b x^n\right ) \left (a+b x^n\right )^3 \sqrt{\left (a+b x^n\right )^2}}{20 b^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 135, normalized size = 1.2 \begin{align*}{\frac{{b}^{3} \left ({x}^{n} \right ) ^{5}}{ \left ( 5\,a+5\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{3\,a{b}^{2} \left ({x}^{n} \right ) ^{4}}{ \left ( 4\,a+4\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{a}^{2}b \left ({x}^{n} \right ) ^{3}}{ \left ( a+b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{a}^{3} \left ({x}^{n} \right ) ^{2}}{ \left ( 2\,a+2\,b{x}^{n} \right ) 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 = 0.986123, size = 65, normalized size = 0.58 \begin{align*} \frac{4 \, b^{3} x^{5 \, n} + 15 \, a b^{2} x^{4 \, n} + 20 \, a^{2} b x^{3 \, n} + 10 \, a^{3} x^{2 \, n}}{20 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59965, size = 107, normalized size = 0.96 \begin{align*} \frac{4 \, b^{3} x^{5 \, n} + 15 \, a b^{2} x^{4 \, n} + 20 \, a^{2} b x^{3 \, n} + 10 \, a^{3} x^{2 \, n}}{20 \, n} \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{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}} x^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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