Optimal. Leaf size=213 \[ \frac {4 (d+e x) (g x)^{m+1}}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {e (7-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^4 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {(1-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^3 g (m+1) \sqrt {d^2-e^2 x^2}} \]
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Rubi [A] time = 0.21, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {1806, 808, 365, 364} \[ \frac {e (7-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac {5}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^4 g^2 (m+2) \sqrt {d^2-e^2 x^2}}+\frac {(1-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac {5}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^3 g (m+1) \sqrt {d^2-e^2 x^2}}+\frac {4 (d+e x) (g x)^{m+1}}{5 g \left (d^2-e^2 x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rule 1806
Rubi steps
\begin {align*} \int \frac {(g x)^m (d+e x)^3}{\left (d^2-e^2 x^2\right )^{7/2}} \, dx &=\frac {4 (g x)^{1+m} (d+e x)}{5 g \left (d^2-e^2 x^2\right )^{5/2}}-\frac {\int \frac {(g x)^m \left (-d^3 (1-4 m)-d^2 e (7-4 m) x\right )}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 d^2}\\ &=\frac {4 (g x)^{1+m} (d+e x)}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {1}{5} (d (1-4 m)) \int \frac {(g x)^m}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx+\frac {(e (7-4 m)) \int \frac {(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{5/2}} \, dx}{5 g}\\ &=\frac {4 (g x)^{1+m} (d+e x)}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {\left ((1-4 m) \sqrt {1-\frac {e^2 x^2}{d^2}}\right ) \int \frac {(g x)^m}{\left (1-\frac {e^2 x^2}{d^2}\right )^{5/2}} \, dx}{5 d^3 \sqrt {d^2-e^2 x^2}}+\frac {\left (e (7-4 m) \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 )^{5/2}} \, dx}{5 d^4 g \sqrt {d^2-e^2 x^2}}\\ &=\frac {4 (g x)^{1+m} (d+e x)}{5 g \left (d^2-e^2 x^2\right )^{5/2}}+\frac {(1-4 m) (g x)^{1+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {5}{2},\frac {1+m}{2};\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^3 g (1+m) \sqrt {d^2-e^2 x^2}}+\frac {e (7-4 m) (g x)^{2+m} \sqrt {1-\frac {e^2 x^2}{d^2}} \, _2F_1\left (\frac {5}{2},\frac {2+m}{2};\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{5 d^4 g^2 (2+m) \sqrt {d^2-e^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 199, normalized size = 0.93 \[ \frac {x \sqrt {1-\frac {e^2 x^2}{d^2}} (g x)^m \left (e x \left (\frac {3 d^2 \, _2F_1\left (\frac {7}{2},\frac {m+2}{2};\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{m+2}+e x \left (\frac {3 d \, _2F_1\left (\frac {7}{2},\frac {m+3}{2};\frac {m+5}{2};\frac {e^2 x^2}{d^2}\right )}{m+3}+\frac {e x \, _2F_1\left (\frac {7}{2},\frac {m+4}{2};\frac {m+6}{2};\frac {e^2 x^2}{d^2}\right )}{m+4}\right )\right )+\frac {d^3 \, _2F_1\left (\frac {7}{2},\frac {m+1}{2};\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{m+1}\right )}{d^6 \sqrt {d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}}{e^{5} x^{5} - 3 \, d e^{4} x^{4} + 2 \, d^{2} e^{3} x^{3} + 2 \, d^{3} e^{2} x^{2} - 3 \, d^{4} e x + d^{5}}, x\right ) \]
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 \[ \int \frac {{\left (e x + d\right )}^{3} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (e x +d \right )^{3} \left (g x \right )^{m}}{\left (-e^{2} x^{2}+d^{2}\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {{\left (e x + d\right )}^{3} \left (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^3}{{\left (d^2-e^2\,x^2\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x\right )^{m} \left (d + e x\right )^{3}}{\left (- \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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