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 = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 364
Rule 365
Rule 2011
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*}
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Mathematica [A] time = 0.23, 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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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 \[ \int {\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.81, size = 0, normalized size = 0.00 \[ \int \left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.25, size = 82, normalized size = 0.85 \[ \frac {x\,{\left (a\,x^j+b\,x^n\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {\frac {3\,n}{2}+1}{j-n};\ \frac {\frac {3\,n}{2}+1}{j-n}+1;\ -\frac {a\,x^{j-n}}{b}\right )}{\left (\frac {3\,n}{2}+1\right )\,{\left (\frac {a\,x^{j-n}}{b}+1\right )}^{3/2}} \]
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 \[ \int \left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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