Optimal. Leaf size=89 \[ -\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}+\frac {8 b x^{-3 n/2} \sqrt {a+b x^n}}{15 a^2 n}-\frac {16 b^2 x^{-n/2} \sqrt {a+b x^n}}{15 a^3 n} \]
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Rubi [A]
time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {277, 270}
\begin {gather*} -\frac {16 b^2 x^{-n/2} \sqrt {a+b x^n}}{15 a^3 n}+\frac {8 b x^{-3 n/2} \sqrt {a+b x^n}}{15 a^2 n}-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 277
Rubi steps
\begin {align*} \int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}-\frac {(4 b) \int \frac {x^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx}{5 a}\\ &=-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}+\frac {8 b x^{-3 n/2} \sqrt {a+b x^n}}{15 a^2 n}+\frac {\left (8 b^2\right ) \int \frac {x^{-1-\frac {n}{2}}}{\sqrt {a+b x^n}} \, dx}{15 a^2}\\ &=-\frac {2 x^{-5 n/2} \sqrt {a+b x^n}}{5 a n}+\frac {8 b x^{-3 n/2} \sqrt {a+b x^n}}{15 a^2 n}-\frac {16 b^2 x^{-n/2} \sqrt {a+b x^n}}{15 a^3 n}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 51, normalized size = 0.57 \begin {gather*} -\frac {2 x^{-5 n/2} \sqrt {a+b x^n} \left (3 a^2-4 a b x^n+8 b^2 x^{2 n}\right )}{15 a^3 n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {x^{-1-\frac {5 n}{2}}}{\sqrt {a +b \,x^{n}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 354 vs.
\(2 (76) = 152\).
time = 4.12, size = 354, normalized size = 3.98 \begin {gather*} - \frac {6 a^{4} b^{\frac {9}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {4 a^{3} b^{\frac {11}{2}} x^{n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {6 a^{2} b^{\frac {13}{2}} x^{2 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {24 a b^{\frac {15}{2}} x^{3 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} - \frac {16 b^{\frac {17}{2}} x^{4 n} \sqrt {\frac {a x^{- n}}{b} + 1}}{15 a^{5} b^{4} n x^{2 n} + 30 a^{4} b^{5} n x^{3 n} + 15 a^{3} b^{6} n x^{4 n}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {1}{x^{\frac {5\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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