Optimal. Leaf size=97 \[ \frac{2 b x^{n+1} \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{\frac{3 n}{2}+1}{j-n};\frac{2 j+n+2}{2 (j-n)};-\frac{a x^{j-n}}{b}\right )}{(3 n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]
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Rubi [A] time = 0.05521, antiderivative size = 97, 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} \[ \frac{2 b x^{n+1} \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{\frac{3 n}{2}+1}{j-n};\frac{2 j+n+2}{2 (j-n)};-\frac{a x^{j-n}}{b}\right )}{(3 n+2) \sqrt{\frac{a x^{j-n}}{b}+1}} \]
Antiderivative was successfully verified.
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Rule 2011
Rule 365
Rule 364
Rubi steps
\begin{align*} \int \left (a x^j+b x^n\right )^{3/2} \, dx &=\frac{\left (x^{-n/2} \sqrt{a x^j+b x^n}\right ) \int x^{3 n/2} \left (b+a x^{j-n}\right )^{3/2} \, dx}{\sqrt{b+a x^{j-n}}}\\ &=\frac{\left (b x^{-n/2} \sqrt{a x^j+b x^n}\right ) \int x^{3 n/2} \left (1+\frac{a x^{j-n}}{b}\right )^{3/2} \, dx}{\sqrt{1+\frac{a x^{j-n}}{b}}}\\ &=\frac{2 b x^{1+n} \sqrt{a x^j+b x^n} \, _2F_1\left (-\frac{3}{2},\frac{1+\frac{3 n}{2}}{j-n};\frac{2+2 j+n}{2 (j-n)};-\frac{a x^{j-n}}{b}\right )}{(2+3 n) \sqrt{1+\frac{a x^{j-n}}{b}}}\\ \end{align*}
Mathematica [A] time = 0.194312, size = 177, normalized size = 1.82 \[ \frac{2 x \left (3 a^2 (j-n)^2 x^{2 j} \sqrt{\frac{a x^{j-n}}{b}+1} \, _2F_1\left (\frac{1}{2},\frac{4 j-n+2}{2 j-2 n};\frac{6 j-3 n+2}{2 j-2 n};-\frac{a x^{j-n}}{b}\right )+(4 j-n+2) \left (a x^j+b x^n\right ) \left (a (-j+4 n+2) x^j+b (2 j+n+2) x^n\right )\right )}{(3 n+2) (4 j-n+2) (2 j+n+2) \sqrt{a x^j+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.398, 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{\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 \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{\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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