Optimal. Leaf size=124 \[ \frac {e (g x)^{m+2} \, _2F_1\left (1,\frac {m-3}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{d^2 g^2 (m+2) \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(g x)^{m+1} \, _2F_1\left (1,\frac {m-4}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{d g (m+1) \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.08, antiderivative size = 162, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {808, 365, 364} \[ \frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {7}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{d^6 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {7}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 g (m+1) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rubi steps
\begin {align*} \int \frac {(g x)^m (d+e x)}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=d \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx+\frac {e \int \frac {(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx}{g}\\ &=\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{7/2}} \, dx}{d^5 \sqrt {d^2-e^2 x^2}}+\frac {\left (e \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^{1+m}}{\left (1-\frac {e^2 x^2}{d^2}\right )^{7/2}} \, dx}{d^6 g \sqrt {d^2-e^2 x^2}}\\ &=\frac {(g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {7}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^5 g (1+m) \sqrt {d^2-e^2 x^2}}+\frac {e (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {7}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^6 g^2 (2+m) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 121, normalized size = 0.98 \[ \frac {x \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^m \left (d (m+2) \, _2F_1\left (\frac {7}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )+e (m+1) x \, _2F_1\left (\frac {7}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )\right )}{d^6 (m+1) (m+2) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}}{e^{7} x^{7} - d e^{6} x^{6} - 3 \, d^{2} e^{5} x^{5} + 3 \, d^{3} e^{4} x^{4} + 3 \, d^{4} e^{3} x^{3} - 3 \, d^{5} e^{2} x^{2} - d^{6} e x + d^{7}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right ) \left (g x \right )^{m}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (g\,x\right )}^m\,\left (d+e\,x\right )}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 63.25, size = 117, normalized size = 0.94 \[ \frac {g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{6} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{2}, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 d^{7} \Gamma \left (\frac {m}{2} + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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