Optimal. Leaf size=250 \[ -\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}-\frac {3 d \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)}+\frac {d^7 (4 m+11) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^6 e (4 m+29) \sqrt {d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+9) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
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Rubi [A] time = 0.39, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1809, 808, 365, 364} \[ \frac {d^6 e (4 m+29) \sqrt {d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+9) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}+\frac {d^7 (4 m+11) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {3 d \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rule 1809
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}-\frac {\int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \left (-d^3 e^2 (9+m)-d^2 e^3 (29+4 m) x-3 d e^4 (9+m) x^2\right ) \, dx}{e^2 (9+m)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac {\int (g x)^m \left (d^3 e^4 (9+m) (11+4 m)+d^2 e^5 (8+m) (29+4 m) x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^4 (8+m) (9+m)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac {\left (d^3 (11+4 m)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx}{8+m}+\frac {\left (d^2 e (29+4 m)\right ) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{5/2} \, dx}{g (9+m)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac {\left (d^7 (11+4 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 {\left (d^6 e (29+4 m) \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 (9+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac {d^7 (11+4 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 {d^6 e (29+4 m) (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) (9+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 199, normalized size = 0.80 \[ \frac {d^4 x \sqrt {d^2-e^2 x^2} (g x)^m \left (e x \left (\frac {3 d^2 \, _2F_1\left (-\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{m+2}+e x \left (\frac {3 d \, _2F_1\left (-\frac {5}{2},\frac {m+3}{2};\frac {m+5}{2};\frac {e^2 x^2}{d^2}\right )}{m+3}+\frac {e x \, _2F_1\left (-\frac {5}{2},\frac {m+4}{2};\frac {m+6}{2};\frac {e^2 x^2}{d^2}\right )}{m+4}\right )\right )+\frac {d^3 \, _2F_1\left (-\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{m+1}\right )}{\sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{7} x^{7} + 3 \, d e^{6} x^{6} + d^{2} e^{5} x^{5} - 5 \, d^{3} e^{4} x^{4} - 5 \, d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} + 3 \, d^{6} e x + d^{7}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}, x\right ) \]
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 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} \left (g x \right )^{m}\, dx \]
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 \[ \int {\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{3} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 45.91, size = 513, normalized size = 2.05 \[ \frac {d^{8} 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 {3 d^{7} 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 )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {d^{6} 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 {5 d^{5} 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 )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} - \frac {5 d^{4} 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^{3} 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 )}}{2 \Gamma \left (\frac {m}{2} + 4\right )} + \frac {3 d^{2} 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 )} + \frac {d e^{7} g^{m} x^{8} x^{m} \Gamma \left (\frac {m}{2} + 4\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 4 \\ \frac {m}{2} + 5 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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