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 = 0.0755826, antiderivative size = 71, normalized size of antiderivative = 1., 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 2028
Rule 2008
Rule 206
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*}
Mathematica [A] time = 0.144159, 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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Maple [F] time = 0.341, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{c}^{2}{x}^{2}}\sqrt{a{x}^{2}+b{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{a x^{2} + b x^{n}}}{x^{2}}\,{d x}}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sqrt{a x^{2} + b x^{n}}}{x^{2}}\, dx}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{2} + b x^{n}}}{c^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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