Optimal. Leaf size=206 \[ -\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}+\frac {d^6 (9+2 m) (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (8+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {2 d^5 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
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Rubi [A]
time = 0.14, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1823, 822, 372,
371} \begin {gather*} -\frac {\left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}+\frac {d^6 (2 m+9) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {2 d^5 e \sqrt {d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 822
Rule 1823
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^2 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac {\int (g x)^m \left (-d^2 e^2 (9+2 m)-2 d e^3 (8+m) x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^2 (8+m)}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}+\frac {(2 d e) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{5/2} \, dx}{g}+\frac {\left (d^2 (9+2 m)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx}{8+m}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}+\frac {\left (2 d^5 e \sqrt {d^2-e^2 x^2}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2} \, dx}{g \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {\left (d^6 (9+2 m) \sqrt {d^2-e^2 x^2}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{5/2} \, dx}{(8+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}+\frac {d^6 (9+2 m) (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (8+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {2 d^5 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 174, normalized size = 0.84 \begin {gather*} \frac {d^4 x (g x)^m \sqrt {d^2-e^2 x^2} \left (d^2 \left (6+5 m+m^2\right ) \, _2F_1\left (-\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )+e (1+m) x \left (2 d (3+m) \, _2F_1\left (-\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )+e (2+m) x \, _2F_1\left (-\frac {5}{2},\frac {3+m}{2};\frac {5+m}{2};\frac {e^2 x^2}{d^2}\right )\right )\right )}{(1+m) (2+m) (3+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{m} \left (e x +d \right )^{2} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}\, dx\]
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]
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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Sympy [C] Result contains complex when optimal does not.
time = 15.52, size = 442, normalized size = 2.15 \begin {gather*} \frac {d^{7} g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {d^{6} e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} - \frac {d^{5} e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {2 d^{4} e^{3} g^{m} x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + 3\right )} - \frac {d^{3} e^{4} g^{m} x^{5} x^{m} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {5}{2} \\ \frac {m}{2} + \frac {7}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} + \frac {d^{2} e^{5} g^{m} x^{6} x^{m} \Gamma \left (\frac {m}{2} + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 3 \\ \frac {m}{2} + 4 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + 4\right )} + \frac {d e^{6} g^{m} x^{7} x^{m} \Gamma \left (\frac {m}{2} + \frac {7}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {7}{2} \\ \frac {m}{2} + \frac {9}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {9}{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 = -0.00 \begin {gather*} \int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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