Optimal. Leaf size=99 \[ \frac {a x^{2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a+b x^n\right )}+\frac {b^2 x^{3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1355, 14} \[ \frac {a x^{2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a+b x^n\right )}+\frac {b^2 x^{3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )} \]
Antiderivative was successfully verified.
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Rule 14
Rule 1355
Rubi steps
\begin {align*} \int x^{-1+2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \, 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 ) \, dx}{a b+b^2 x^n}\\ &=\frac {\sqrt {a^2+2 a b x^n+b^2 x^{2 n}} \int \left (a b x^{-1+2 n}+b^2 x^{-1+3 n}\right ) \, dx}{a b+b^2 x^n}\\ &=\frac {a x^{2 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{2 n \left (a+b x^n\right )}+\frac {b^2 x^{3 n} \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}{3 n \left (a b+b^2 x^n\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.44 \[ \frac {x^{2 n} \sqrt {\left (a+b x^n\right )^2} \left (3 a+2 b x^n\right )}{6 n \left (a+b x^n\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 22, normalized size = 0.22 \[ \frac {2 \, b x^{3 \, n} + 3 \, a x^{2 \, n}}{6 \, n} \]
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 \[ \int \sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 64, normalized size = 0.65 \[ \frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, a \,x^{2 n}}{2 \left (b \,x^{n}+a \right ) n}+\frac {\sqrt {\left (b \,x^{n}+a \right )^{2}}\, b \,x^{3 n}}{3 \left (b \,x^{n}+a \right ) n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.89, size = 22, normalized size = 0.22 \[ \frac {2 \, b x^{3 \, n} + 3 \, a x^{2 \, n}}{6 \, n} \]
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 \[ \int x^{2\,n-1}\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2 n - 1} \sqrt {\left (a + b x^{n}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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