Optimal. Leaf size=97 \[ \frac {2 b x^{1+n} \sqrt {a x^j+b x^n} \, _2F_1\left (-\frac {3}{2},\frac {1+\frac {3 n}{2}}{j-n};\frac {2+2 j+n}{2 (j-n)};-\frac {a x^{j-n}}{b}\right )}{(2+3 n) \sqrt {1+\frac {a x^{j-n}}{b}}} \]
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Rubi [A]
time = 0.04, 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 = {2036, 372, 371}
\begin {gather*} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 2036
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 = 0.16, size = 177, normalized size = 1.82 \begin {gather*} \frac {2 x \left ((2+4 j-n) \left (a x^j+b x^n\right ) \left (a (2-j+4 n) x^j+b (2+2 j+n) x^n\right )+3 a^2 (j-n)^2 x^{2 j} \sqrt {1+\frac {a x^{j-n}}{b}} \, _2F_1\left (\frac {1}{2},\frac {2+4 j-n}{2 j-2 n};\frac {2+6 j-3 n}{2 j-2 n};-\frac {a x^{j-n}}{b}\right )\right )}{(2+4 j-n) (2+2 j+n) (2+3 n) \sqrt {a x^j+b x^n}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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