Optimal. Leaf size=112 \[ \frac{\left (a+b x^n\right )^7 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left (a b^2+b^3 x^n\right )}-\frac{a \left (a+b x^n\right )^6 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left (a b^2+b^3 x^n\right )} \]
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Rubi [A] time = 0.0416955, 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} \[ \frac{\left (a+b x^n\right )^7 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left (a b^2+b^3 x^n\right )}-\frac{a \left (a+b x^n\right )^6 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left (a b^2+b^3 x^n\right )} \]
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 )^{5/2} \, 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 )^5 \, dx}{b^4 \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 )^5 \, dx,x,x^n\right )}{b^4 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 )^5}{b}+\frac{\left (a b+b^2 x\right )^6}{b^2}\right ) \, dx,x,x^n\right )}{b^4 n \left (a b+b^2 x^n\right )}\\ &=-\frac{a \left (a+b x^n\right )^6 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{6 n \left (a b^2+b^3 x^n\right )}+\frac{\left (a+b x^n\right )^7 \sqrt{a^2+2 a b x^n+b^2 x^{2 n}}}{7 n \left (a b^2+b^3 x^n\right )}\\ \end{align*}
Mathematica [A] time = 0.0573805, size = 40, normalized size = 0.36 \[ -\frac{\left (a-6 b x^n\right ) \left (a+b x^n\right )^5 \sqrt{\left (a+b x^n\right )^2}}{42 b^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 208, normalized size = 1.9 \begin{align*}{\frac{{b}^{5} \left ({x}^{n} \right ) ^{7}}{ \left ( 7\,a+7\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{5\,a{b}^{4} \left ({x}^{n} \right ) ^{6}}{ \left ( 6\,a+6\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+2\,{\frac{\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}{a}^{2}{b}^{3} \left ({x}^{n} \right ) ^{5}}{ \left ( a+b{x}^{n} \right ) n}}+{\frac{5\,{a}^{3}{b}^{2} \left ({x}^{n} \right ) ^{4}}{ \left ( 2\,a+2\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{5\,{a}^{4}b \left ({x}^{n} \right ) ^{3}}{ \left ( 3\,a+3\,b{x}^{n} \right ) n}\sqrt{ \left ( a+b{x}^{n} \right ) ^{2}}}+{\frac{{a}^{5} \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 = 1.00644, size = 100, normalized size = 0.89 \begin{align*} \frac{6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59756, size = 165, normalized size = 1.47 \begin{align*} \frac{6 \, b^{5} x^{7 \, n} + 35 \, a b^{4} x^{6 \, n} + 84 \, a^{2} b^{3} x^{5 \, n} + 105 \, a^{3} b^{2} x^{4 \, n} + 70 \, a^{4} b x^{3 \, n} + 21 \, a^{5} x^{2 \, n}}{42 \, 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{5}{2}} x^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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