Optimal. Leaf size=62 \[ \frac {2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{\sqrt {a} c (j-n)} \]
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Rubi [A] time = 0.11, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2031, 2029, 206} \[ \frac {2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{\sqrt {a} c (j-n)} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2029
Rule 2031
Rubi steps
\begin {align*} \int \frac {(c x)^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx &=\frac {\left (x^{-j/2} (c x)^{j/2}\right ) \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx}{c}\\ &=\frac {\left (2 x^{-j/2} (c x)^{j/2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{c (j-n)}\\ &=\frac {2 x^{-j/2} (c x)^{j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{\sqrt {a} c (j-n)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 98, normalized size = 1.58 \[ \frac {2 \sqrt {b} (c x)^{j/2} x^{\frac {n-j}{2}} \sqrt {\frac {a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )}{\sqrt {a} c (j-n) \sqrt {a x^j+b x^n}} \]
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 \[ \int \frac {\left (c x\right )^{\frac {1}{2} \, j - 1}}{\sqrt {a x^{j} + b x^{n}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.79, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{\frac {j}{2}-1}}{\sqrt {a \,x^{j}+b \,x^{n}}}\, dx \]
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 \[ \int \frac {\left (c x\right )^{\frac {1}{2} \, j - 1}}{\sqrt {a x^{j} + b x^{n}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,x\right )}^{\frac {j}{2}-1}}{\sqrt {a\,x^j+b\,x^n}} \,d x \]
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 \[ \int \frac {\left (c x\right )^{\frac {j}{2} - 1}}{\sqrt {a x^{j} + b x^{n}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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