Optimal. Leaf size=99 \[ \frac{2 \sqrt{a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}-\frac{2 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)} \]
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Rubi [A] time = 0.157338, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2031, 2028, 2029, 206} \[ \frac{2 \sqrt{a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}-\frac{2 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)} \]
Antiderivative was successfully verified.
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Rule 2031
Rule 2028
Rule 2029
Rule 206
Rubi steps
\begin{align*} \int (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 x^{-1-\frac{j}{2}} \sqrt{a x^j+b x^n} \, dx}{c}\\ &=-\frac{2 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)}+\frac{\left (a 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{2 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)}+\frac{\left (2 a 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 (c x)^{-j/2} \sqrt{a x^j+b x^n}}{c (j-n)}+\frac{2 \sqrt{a} x^{j/2} (c x)^{-j/2} \tanh ^{-1}\left (\frac{\sqrt{a} x^{j/2}}{\sqrt{a x^j+b x^n}}\right )}{c (j-n)}\\ \end{align*}
Mathematica [A] time = 0.0644413, size = 109, normalized size = 1.1 \[ -\frac{2 (c x)^{-j/2} \left (-\sqrt{a} \sqrt{b} x^{\frac{j+n}{2}} \sqrt{\frac{a x^{j-n}}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{\frac{j-n}{2}}}{\sqrt{b}}\right )+a x^j+b x^n\right )}{c (j-n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.584, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{-1-{\frac{j}{2}}}\sqrt{a{x}^{j}+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*} \int \sqrt{a x^{j} + b x^{n}} \left (c x\right )^{-\frac{1}{2} \, j - 1}\,{d x} \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*} \int \left (c x\right )^{- \frac{j}{2} - 1} \sqrt{a x^{j} + b x^{n}}\, dx \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 \sqrt{a x^{j} + b x^{n}} \left (c x\right )^{-\frac{1}{2} \, j - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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