Optimal. Leaf size=65 \[ -\frac {2 (c x)^{-3 n/2} \sqrt {a+b x^n}}{a c n}+\frac {4 (c x)^{-3 n/2} \left (a+b x^n\right )^{3/2}}{3 a^2 c n} \]
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Rubi [A]
time = 0.01, 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 = {279, 270}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 270
Rule 279
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 = 0.01, size = 41, normalized size = 0.63 \begin {gather*} -\frac {2 (c x)^{-3 n/2} \left (a-2 b x^n\right ) \sqrt {a+b x^n}}{3 a^2 c n} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{-1-\frac {3 n}{2}}}{\sqrt {a +b \,x^{n}}}\, dx\]
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 [A]
time = 3.04, size = 68, normalized size = 1.05 \begin {gather*} - \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} \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 = -0.02 \begin {gather*} \int \frac {1}{{\left (c\,x\right )}^{\frac {3\,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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