Optimal. Leaf size=65 \[ \frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {273, 264} \[ \frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n} \]
Antiderivative was successfully verified.
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Rule 264
Rule 273
Rubi steps
\begin {align*} \int \frac {(c x)^{-1-\frac {3 n}{2}}}{\sqrt {a+b x^n}} \, dx &=-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n}-\frac {2 \int (c x)^{-1-\frac {3 n}{2}} \sqrt {a+b x^n} \, dx}{a}\\ &=-\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 41, normalized size = 0.63 \[ -\frac {2 (c x)^{-3 n/2} \left (a-2 b x^n\right ) \sqrt {a+b x^n}}{3 a^2 c n} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\left (c x\right )^{-\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{-\frac {3 n}{2}-1}}{\sqrt {b \,x^{n}+a}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{-\frac {3}{2} \, n - 1}}{\sqrt {b x^{n} + a}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,x\right )}^{\frac {3\,n}{2}+1}\,\sqrt {a+b\,x^n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.05, size = 68, normalized size = 1.05 \[ - \frac {2 \sqrt {b} c^{- \frac {3 n}{2}} x^{- n} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a c n} + \frac {4 b^{\frac {3}{2}} c^{- \frac {3 n}{2}} \sqrt {\frac {a x^{- n}}{b} + 1}}{3 a^{2} c n} \]
Verification of antiderivative is not currently implemented for this CAS.
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