Optimal. Leaf size=163 \[ \frac {d^3 (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {d^2 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {3}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) \sqrt {1-\frac {e^2 x^2}{d^2}}} \]
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Rubi [A]
time = 0.09, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {906, 83, 127,
372, 371} \begin {gather*} \frac {d^3 \sqrt {d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac {3}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {d^2 e \sqrt {d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac {3}{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 83
Rule 127
Rule 371
Rule 372
Rule 906
Rubi steps
\begin {align*} \int \frac {(g x)^m \left (d^2-e^2 x^2\right )^{5/2}}{d+e x} \, dx &=\frac {\sqrt {d^2-e^2 x^2} \int (g x)^m (d-e x)^{5/2} (d+e x)^{3/2} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}\\ &=\frac {\left (d \sqrt {d^2-e^2 x^2}\right ) \int (g x)^m (d-e x)^{3/2} (d+e x)^{3/2} \, dx}{\sqrt {d-e x} \sqrt {d+e x}}-\frac {\left (e \sqrt {d^2-e^2 x^2}\right ) \int (g x)^{1+m} (d-e x)^{3/2} (d+e x)^{3/2} \, dx}{g \sqrt {d-e x} \sqrt {d+e x}}\\ &=d \int (g x)^m \left (d^2-e^2 x^2\right )^{3/2} \, dx-\frac {e \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{3/2} \, dx}{g}\\ &=\frac {\left (d^3 \sqrt {d^2-e^2 x^2}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^{3/2} \, dx}{\sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {\left (d^2 e \sqrt {d^2-e^2 x^2}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^{3/2} \, dx}{g \sqrt {1-\frac {e^2 x^2}{d^2}}}\\ &=\frac {d^3 (g x)^{1+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) \sqrt {1-\frac {e^2 x^2}{d^2}}}-\frac {d^2 e (g x)^{2+m} \sqrt {d^2-e^2 x^2} \, _2F_1\left (-\frac {3}{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 = 0.63, size = 122, normalized size = 0.75 \begin {gather*} \frac {d^2 x (g x)^m \sqrt {d^2-e^2 x^2} \left (-e (1+m) x \, _2F_1\left (-\frac {3}{2},1+\frac {m}{2};2+\frac {m}{2};\frac {e^2 x^2}{d^2}\right )+d (2+m) \, _2F_1\left (-\frac {3}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )\right )}{(1+m) (2+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 \[\int \frac {\left (g x \right )^{m} \left (-e^{2} x^{2}+d^{2}\right )^{\frac {5}{2}}}{e x +d}\, dx\]
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 = 10.84, size = 248, normalized size = 1.52 \begin {gather*} \frac {d^{4} 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^{3} 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^{2} 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 {d 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 )} \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 = -0.01 \begin {gather*} \int \frac {{\left (d^2-e^2\,x^2\right )}^{5/2}\,{\left (g\,x\right )}^m}{d+e\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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