Optimal. Leaf size=213 \[ \frac{e (7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^4 g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(1-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{5 d^3 g (m+1) \sqrt{d^2-e^2 x^2}}+\frac{4 (d+e x) (g x)^{m+1}}{5 g \left (d^2-e^2 x^2\right )^{5/2}} \]
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Rubi [A] time = 0.21292, antiderivative size = 213, normalized size of antiderivative = 1., 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 1806
Rule 808
Rule 365
Rule 364
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*}
Mathematica [A] time = 0.203517, size = 199, normalized size = 0.93 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (\frac{d^3 \, _2F_1\left (\frac{7}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}+e x \left (\frac{3 d^2 \, _2F_1\left (\frac{7}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{m+2}+e x \left (\frac{3 d \, _2F_1\left (\frac{7}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}+\frac{e x \, _2F_1\left (\frac{7}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right )}{m+4}\right )\right )\right )}{d^6 \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.522, size = 0, normalized size = 0. \begin{align*} \int{ \left ( gx \right ) ^{m} \left ( ex+d \right ) ^{3} \left ( -{x}^{2}{e}^{2}+{d}^{2} \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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