Optimal. Leaf size=26 \[ \frac {2 x^3 \sqrt {a+b x^{-2+m}}}{b (4+m)} \]
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Rubi [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, antiderivative size = 160, normalized size of antiderivative = 6.15, number of steps
used = 5, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {372, 371}
\begin {gather*} \frac {x^{m+1} \sqrt {\frac {b x^{m-2}}{a}+1} \, _2F_1\left (\frac {1}{2},-\frac {m+1}{2-m};\frac {1-2 m}{2-m};-\frac {b x^{m-2}}{a}\right )}{(m+1) \sqrt {a+b x^{m-2}}}+\frac {2 a x^3 \sqrt {\frac {b x^{m-2}}{a}+1} \, _2F_1\left (\frac {1}{2},-\frac {3}{2-m};-\frac {m+1}{2-m};-\frac {b x^{m-2}}{a}\right )}{b (m+4) \sqrt {a+b x^{m-2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rubi steps
\begin {align*} \int \left (\frac {6 a x^2}{b (4+m) \sqrt {a+b x^{-2+m}}}+\frac {x^m}{\sqrt {a+b x^{-2+m}}}\right ) \, dx &=\frac {(6 a) \int \frac {x^2}{\sqrt {a+b x^{-2+m}}} \, dx}{b (4+m)}+\int \frac {x^m}{\sqrt {a+b x^{-2+m}}} \, dx\\ &=\frac {\sqrt {1+\frac {b x^{-2+m}}{a}} \int \frac {x^m}{\sqrt {1+\frac {b x^{-2+m}}{a}}} \, dx}{\sqrt {a+b x^{-2+m}}}+\frac {\left (6 a \sqrt {1+\frac {b x^{-2+m}}{a}}\right ) \int \frac {x^2}{\sqrt {1+\frac {b x^{-2+m}}{a}}} \, dx}{b (4+m) \sqrt {a+b x^{-2+m}}}\\ &=\frac {2 a x^3 \sqrt {1+\frac {b x^{-2+m}}{a}} \, _2F_1\left (\frac {1}{2},-\frac {3}{2-m};-\frac {1+m}{2-m};-\frac {b x^{-2+m}}{a}\right )}{b (4+m) \sqrt {a+b x^{-2+m}}}+\frac {x^{1+m} \sqrt {1+\frac {b x^{-2+m}}{a}} \, _2F_1\left (\frac {1}{2},-\frac {1+m}{2-m};\frac {1-2 m}{2-m};-\frac {b x^{-2+m}}{a}\right )}{(1+m) \sqrt {a+b x^{-2+m}}}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 26, normalized size = 1.00 \begin {gather*} \frac {2 x^3 \sqrt {a+b x^{-2+m}}}{b (4+m)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 40, normalized size = 1.54
method | result | size |
risch | \(\frac {2 x \left (a \,x^{2}+b \,x^{m}\right )}{b \left (4+m \right ) \sqrt {\frac {a \,x^{2}+b \,x^{m}}{x^{2}}}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 37, normalized size = 1.42 \begin {gather*} \frac {2 \, {\left (a x^{4} + b x^{2} x^{m}\right )}}{\sqrt {a x^{2} + b x^{m}} b {\left (m + 4\right )}} \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 [C] Result contains complex when optimal does not.
time = 17.00, size = 170, normalized size = 6.54 \begin {gather*} \frac {6 a x^{3} \Gamma \left (\frac {3}{m - 2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{m - 2} \\ 1 + \frac {3}{m - 2} \end {matrix}\middle | {\frac {b x^{m} e^{i \pi }}{a x^{2}}} \right )}}{b \left (m + 4\right ) \left (\sqrt {a} m \Gamma \left (1 + \frac {3}{m - 2}\right ) - 2 \sqrt {a} \Gamma \left (1 + \frac {3}{m - 2}\right )\right )} + \frac {x x^{m} \Gamma \left (\frac {m}{m - 2} + \frac {1}{m - 2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {m}{m - 2} + \frac {1}{m - 2} \\ \frac {m}{m - 2} + 1 + \frac {1}{m - 2} \end {matrix}\middle | {\frac {b x^{m} e^{i \pi }}{a x^{2}}} \right )}}{\sqrt {a} m \Gamma \left (\frac {m}{m - 2} + 1 + \frac {1}{m - 2}\right ) - 2 \sqrt {a} \Gamma \left (\frac {m}{m - 2} + 1 + \frac {1}{m - 2}\right )} \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 [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x^m}{\sqrt {a+b\,x^{m-2}}}+\frac {6\,a\,x^2}{b\,\sqrt {a+b\,x^{m-2}}\,\left (m+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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