Optimal. Leaf size=75 \[ -\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n} \]
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Rubi [A]
time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2053, 2054,
212} \begin {gather*} \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2053
Rule 2054
Rubi steps
\begin {align*} \int x^{-1-\frac {j}{2}} \sqrt {a x^j+b x^n} \, dx &=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+a \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx\\ &=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}\\ &=-\frac {2 x^{-j/2} \sqrt {a x^j+b x^n}}{j-n}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{j-n}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 104, normalized size = 1.39 \begin {gather*} -\frac {2 x^{-j/2} \left (a x^j+b x^n-\sqrt {a} \sqrt {b} x^{\frac {j+n}{2}} \sqrt {1+\frac {a x^{j-n}}{b}} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{(j-n) \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int x^{-1-\frac {j}{2}} \sqrt {a \,x^{j}+b \,x^{n}}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{- \frac {j}{2} - 1} \sqrt {a x^{j} + b x^{n}}\, dx \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 {\sqrt {a\,x^j+b\,x^n}}{x^{\frac {j}{2}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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