Optimal. Leaf size=44 \[ \frac {2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}-\frac {2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {266, 43} \[ \frac {2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}-\frac {2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^{-1+2 n} \sqrt {a+b x^n} \, dx &=\frac {\operatorname {Subst}\left (\int x \sqrt {a+b x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+b x}}{b}+\frac {(a+b x)^{3/2}}{b}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac {2 a \left (a+b x^n\right )^{3/2}}{3 b^2 n}+\frac {2 \left (a+b x^n\right )^{5/2}}{5 b^2 n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 31, normalized size = 0.70 \[ \frac {2 \left (a+b x^n\right )^{3/2} \left (3 b x^n-2 a\right )}{15 b^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 39, normalized size = 0.89 \[ \frac {2 \, {\left (3 \, b^{2} x^{2 \, n} + a b x^{n} - 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{15 \, b^{2} n} \]
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 \[ \int \sqrt {b x^{n} + a} x^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 41, normalized size = 0.93 \[ -\frac {2 \left (-a b \,x^{n}-3 b^{2} x^{2 n}+2 a^{2}\right ) \sqrt {b \,x^{n}+a}}{15 b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 39, normalized size = 0.89 \[ \frac {2 \, {\left (3 \, b^{2} x^{2 \, n} + a b x^{n} - 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{15 \, b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^{2\,n-1}\,\sqrt {a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.99, size = 338, normalized size = 7.68 \[ - \frac {4 a^{\frac {11}{2}} b^{\frac {3}{2}} x^{\frac {3 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} n x^{n} + 15 a^{\frac {5}{2}} b^{4} n x^{2 n}} - \frac {2 a^{\frac {9}{2}} b^{\frac {5}{2}} x^{\frac {5 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} n x^{n} + 15 a^{\frac {5}{2}} b^{4} n x^{2 n}} + \frac {8 a^{\frac {7}{2}} b^{\frac {7}{2}} x^{\frac {7 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} n x^{n} + 15 a^{\frac {5}{2}} b^{4} n x^{2 n}} + \frac {6 a^{\frac {5}{2}} b^{\frac {9}{2}} x^{\frac {9 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{\frac {7}{2}} b^{3} n x^{n} + 15 a^{\frac {5}{2}} b^{4} n x^{2 n}} + \frac {4 a^{6} b x^{n}}{15 a^{\frac {7}{2}} b^{3} n x^{n} + 15 a^{\frac {5}{2}} b^{4} n x^{2 n}} + \frac {4 a^{5} b^{2} x^{2 n}}{15 a^{\frac {7}{2}} b^{3} n x^{n} + 15 a^{\frac {5}{2}} b^{4} n x^{2 n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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