Optimal. Leaf size=55 \[ \frac {x^{m+1} \left (a+b x^n\right )^{5/2} \, _2F_1\left (1,\frac {m+1}{n}+\frac {5}{2};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{a (m+1)} \]
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Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {365, 364} \[ \frac {a x^{m+1} \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{n};\frac {m+n+1}{n};-\frac {b x^n}{a}\right )}{(m+1) \sqrt {\frac {b x^n}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rubi steps
\begin {align*} \int x^m \left (a+b x^n\right )^{3/2} \, dx &=\frac {\left (a \sqrt {a+b x^n}\right ) \int x^m \left (1+\frac {b x^n}{a}\right )^{3/2} \, dx}{\sqrt {1+\frac {b x^n}{a}}}\\ &=\frac {a x^{1+m} \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{n};\frac {1+m+n}{n};-\frac {b x^n}{a}\right )}{(1+m) \sqrt {1+\frac {b x^n}{a}}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 66, normalized size = 1.20 \[ \frac {a x^{m+1} \sqrt {a+b x^n} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{n};\frac {m+1}{n}+1;-\frac {b x^n}{a}\right )}{(m+1) \sqrt {\frac {b x^n}{a}+1}} \]
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 \[ \int {\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{\frac {3}{2}} x^{m}\, dx \]
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 \[ \int {\left (b x^{n} + a\right )}^{\frac {3}{2}} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,{\left (a+b\,x^n\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 15.81, size = 58, normalized size = 1.05 \[ \frac {a^{\frac {3}{2}} x x^{m} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{2}, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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