Optimal. Leaf size=101 \[ \frac{2 x^{1-n} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{1-\frac{3 n}{2}}{j-n};\frac{1-\frac{3 n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt{a x^j+b x^n}} \]
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Rubi [A] time = 0.0568401, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2011, 365, 364} \[ \frac{2 x^{1-n} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{3}{2},\frac{1-\frac{3 n}{2}}{j-n};\frac{1-\frac{3 n}{2}}{j-n}+1;-\frac{a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Rule 2011
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\left (a x^j+b x^n\right )^{3/2}} \, dx &=\frac{\left (x^{n/2} \sqrt{b+a x^{j-n}}\right ) \int \frac{x^{-3 n/2}}{\left (b+a x^{j-n}\right )^{3/2}} \, dx}{\sqrt{a x^j+b x^n}}\\ &=\frac{\left (x^{n/2} \sqrt{1+\frac{a x^{j-n}}{b}}\right ) \int \frac{x^{-3 n/2}}{\left (1+\frac{a x^{j-n}}{b}\right )^{3/2}} \, dx}{b \sqrt{a x^j+b x^n}}\\ &=\frac{2 x^{1-n} \sqrt{1+\frac{a x^{j-n}}{b}} \, _2F_1\left (\frac{3}{2},\frac{1-\frac{3 n}{2}}{j-n};1+\frac{1-\frac{3 n}{2}}{j-n};-\frac{a x^{j-n}}{b}\right )}{b (2-3 n) \sqrt{a x^j+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0827265, size = 104, normalized size = 1.03 \[ \frac{2 x^{1-j} \left (\sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},-\frac{2 j+n-2}{2 (j-n)};\frac{2-3 n}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )-1\right )}{a (j-n) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.36, size = 0, normalized size = 0. \begin{align*} \int \left ( a{x}^{j}+b{x}^{n} \right ) ^{-{\frac{3}{2}}}\, 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 \frac{1}{{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}}}\,{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 \frac{1}{\left (a x^{j} + b x^{n}\right )^{\frac{3}{2}}}\, 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 \frac{1}{{\left (a x^{j} + b x^{n}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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