Optimal. Leaf size=204 \[ -\frac {2 d e \sqrt {d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^2 (2 m+5) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+4) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {\left (d^2-e^2 x^2\right )^{3/2} (g x)^{m+1}}{g (m+4)} \]
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Rubi [A] time = 0.22, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {852, 1809, 808, 365, 364} \[ -\frac {2 d e \sqrt {d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {d^2 (2 m+5) \sqrt {d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+4) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {\left (d^2-e^2 x^2\right )^{3/2} (g x)^{m+1}}{g (m+4)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rule 852
Rule 1809
Rubi steps
\begin {align*} \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{(d+e x)^2} \, dx &=\int (g x)^m (d-e x)^2 \sqrt {d^2-e^2 x^2} \, dx\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{3/2}}{g (4+m)}-\frac {\int (g x)^m \left (-d^2 e^2 (5+2 m)+2 d e^3 (4+m) x\right ) \sqrt {d^2-e^2 x^2} \, dx}{e^2 (4+m)}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{3/2}}{g (4+m)}-\frac {(2 d e) \int (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, dx}{g}+\frac {\left (d^2 (5+2 m)\right ) \int (g x)^m \sqrt {d^2-e^2 x^2} \, dx}{4+m}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{3/2}}{g (4+m)}-\frac {\left (2 d e \sqrt {d^2-e^2 x^2}\right ) \int (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, dx}{g \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {\left (d^2 (5+2 m) \sqrt {d^2-e^2 x^2}\right ) \int (g x)^m \sqrt {1-\frac {e^2 x^2}{d^2}} \, dx}{(4+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}\\ &=-\frac {(g x)^{1+m} \left (d^2-e^2 x^2\right )^{3/2}}{g (4+m)}+\frac {d^2 (5+2 m) (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (4+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {2 d e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {1}{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 = 0.11, size = 173, normalized size = 0.85 \[ \frac {x \sqrt {d^2-e^2 x^2} (g x)^m \left (d^2 \left (m^2+5 m+6\right ) \, _2F_1\left (-\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )-e (m+1) x \left (2 d (m+3) \, _2F_1\left (-\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )-e (m+2) x \, _2F_1\left (-\frac {1}{2},\frac {m+3}{2};\frac {m+5}{2};\frac {e^2 x^2}{d^2}\right )\right )\right )}{(m+1) (m+2) (m+3) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} \sqrt {-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}, x\right ) \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}} \left (g x \right )^{m}}{\left (e x +d \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {5}{2}} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m}{{\left (d+e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 91.47, size = 185, normalized size = 0.91 \[ \frac {d^{3} g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} - \frac {d^{2} e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{\Gamma \left (\frac {m}{2} + 2\right )} + \frac {d e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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