Optimal. Leaf size=120 \[ -\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]
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Rubi [A] time = 0.0391954, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {271, 264} \[ -\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n} \]
Antiderivative was successfully verified.
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Rule 271
Rule 264
Rubi steps
\begin{align*} \int \frac{x^{-1-\frac{7 n}{2}}}{\sqrt{a+b x^n}} \, dx &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}-\frac{(6 b) \int \frac{x^{-1-\frac{5 n}{2}}}{\sqrt{a+b x^n}} \, dx}{7 a}\\ &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}+\frac{\left (24 b^2\right ) \int \frac{x^{-1-\frac{3 n}{2}}}{\sqrt{a+b x^n}} \, dx}{35 a^2}\\ &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}-\frac{\left (16 b^3\right ) \int \frac{x^{-1-\frac{n}{2}}}{\sqrt{a+b x^n}} \, dx}{35 a^3}\\ &=-\frac{2 x^{-7 n/2} \sqrt{a+b x^n}}{7 a n}+\frac{12 b x^{-5 n/2} \sqrt{a+b x^n}}{35 a^2 n}-\frac{16 b^2 x^{-3 n/2} \sqrt{a+b x^n}}{35 a^3 n}+\frac{32 b^3 x^{-n/2} \sqrt{a+b x^n}}{35 a^4 n}\\ \end{align*}
Mathematica [A] time = 0.0217774, size = 64, normalized size = 0.53 \[ -\frac{2 x^{-7 n/2} \sqrt{a+b x^n} \left (-6 a^2 b x^n+5 a^3+8 a b^2 x^{2 n}-16 b^3 x^{3 n}\right )}{35 a^4 n} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.078, size = 0, normalized size = 0. \begin{align*} \int{{x}^{-1-{\frac{7\,n}{2}}}{\frac{1}{\sqrt{a+b{x}^{n}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 17.4394, size = 605, normalized size = 5.04 \begin{align*} - \frac{10 a^{6} b^{\frac{19}{2}} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac{18 a^{5} b^{\frac{21}{2}} x^{n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} - \frac{10 a^{4} b^{\frac{23}{2}} x^{2 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{10 a^{3} b^{\frac{25}{2}} x^{3 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{60 a^{2} b^{\frac{27}{2}} x^{4 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{80 a b^{\frac{29}{2}} x^{5 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} + \frac{32 b^{\frac{31}{2}} x^{6 n} \sqrt{\frac{a x^{- n}}{b} + 1}}{35 a^{7} b^{9} n x^{3 n} + 105 a^{6} b^{10} n x^{4 n} + 105 a^{5} b^{11} n x^{5 n} + 35 a^{4} b^{12} n x^{6 n}} \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 \frac{x^{-\frac{7}{2} \, n - 1}}{\sqrt{b x^{n} + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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