Optimal. Leaf size=71 \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{c^2 (2-n)}-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x} \]
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Rubi [A] time = 0.08, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {12, 2028, 2008, 206} \[ \frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{c^2 (2-n)}-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 2008
Rule 2028
Rubi steps
\begin {align*} \int \frac {\sqrt {a x^2+b x^n}}{c^2 x^2} \, dx &=\frac {\int \frac {\sqrt {a x^2+b x^n}}{x^2} \, dx}{c^2}\\ &=-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x}+\frac {a \int \frac {1}{\sqrt {a x^2+b x^n}} \, dx}{c^2}\\ &=-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x}+\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^n}}\right )}{c^2 (2-n)}\\ &=-\frac {2 \sqrt {a x^2+b x^n}}{c^2 (2-n) x}+\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^n}}\right )}{c^2 (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 99, normalized size = 1.39 \[ \frac {2 \left (-\sqrt {a} \sqrt {b} x^{\frac {n}{2}+1} \sqrt {\frac {a x^{2-n}}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {a} x^{1-\frac {n}{2}}}{\sqrt {b}}\right )+a x^2+b x^n\right )}{c^2 (n-2) x \sqrt {a x^2+b x^n}} \]
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 \[ \int \frac {\sqrt {a x^{2} + b x^{n}}}{c^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \,x^{2}+b \,x^{n}}}{c^{2} x^{2}}\, dx \]
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 \[ \frac {\int \frac {\sqrt {a x^{2} + b x^{n}}}{x^{2}}\,{d x}}{c^{2}} \]
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 \[ \int \frac {\sqrt {b\,x^n+a\,x^2}}{c^2\,x^2} \,d x \]
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 \[ \frac {\int \frac {\sqrt {a x^{2} + b x^{n}}}{x^{2}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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