Optimal. Leaf size=163 \[ \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {9}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{d^7 g (m+1) \sqrt {d^2-e^2 x^2}}-\frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {9}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (m+2) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.15, 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 = {892, 82, 126, 365, 364} \[ \frac {\sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {9}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{d^7 g (m+1) \sqrt {d^2-e^2 x^2}}-\frac {e \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {9}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (m+2) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 82
Rule 126
Rule 364
Rule 365
Rule 892
Rubi steps
\begin {align*} \int \frac {(g x)^m}{(d+e x) \left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {\left (\sqrt {d-e x} \sqrt {d+e x}\right ) \int \frac {(g x)^m}{(d-e x)^{7/2} (d+e x)^{9/2}} \, dx}{\sqrt {d^2-e^2 x^2}}\\ &=\frac {\left (d \sqrt {d-e x} \sqrt {d+e x}\right ) \int \frac {(g x)^m}{(d-e x)^{9/2} (d+e x)^{9/2}} \, dx}{\sqrt {d^2-e^2 x^2}}-\frac {\left (e \sqrt {d-e x} \sqrt {d+e x}\right ) \int \frac {(g x)^{1+m}}{(d-e x)^{9/2} (d+e x)^{9/2}} \, dx}{g \sqrt {d^2-e^2 x^2}}\\ &=d \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{9/2}} \, dx-\frac {e \int \frac {(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{9/2}} \, 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 )^{9/2}} \, dx}{d^7 \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 )^{9/2}} \, dx}{d^8 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 {9}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^7 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 {9}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{d^8 g^2 (2+m) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 122, normalized size = 0.75 \[ \frac {x \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^m \left (d (m+2) \, _2F_1\left (\frac {9}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )-e (m+1) x \, _2F_1\left (\frac {9}{2},\frac {m}{2}+1;\frac {m}{2}+2;\frac {e^2 x^2}{d^2}\right )\right )}{d^8 (m+1) (m+2) \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.02, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}}{e^{9} x^{9} + d e^{8} x^{8} - 4 \, d^{2} e^{7} x^{7} - 4 \, d^{3} e^{6} x^{6} + 6 \, d^{4} e^{5} x^{5} + 6 \, d^{5} e^{4} x^{4} - 4 \, d^{6} e^{3} x^{3} - 4 \, d^{7} e^{2} x^{2} + d^{8} e x + d^{9}}, x\right ) \]
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 \[ \int \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}}\,{d x} \]
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 \[ \int \frac {\left (g x \right )^{m}}{\left (e x +d \right ) \left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {\left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}} {\left (e x + d\right )}}\,{d x} \]
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 \[ \int \frac {{\left (g\,x\right )}^m}{{\left (d^2-e^2\,x^2\right )}^{7/2}\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x\right )^{m}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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