Optimal. Leaf size=214 \[ -\frac{e (25-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^{10} g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^9 g (m+1) \sqrt{d^2-e^2 x^2}}+\frac{4 (d-e x) (g x)^{m+1}}{11 g \left (d^2-e^2 x^2\right )^{11/2}} \]
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Rubi [A] time = 0.229159, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {852, 1806, 808, 365, 364} \[ -\frac{e (25-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+2} \, _2F_1\left (\frac{11}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^{10} g^2 (m+2) \sqrt{d^2-e^2 x^2}}+\frac{(7-4 m) \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^{m+1} \, _2F_1\left (\frac{11}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^9 g (m+1) \sqrt{d^2-e^2 x^2}}+\frac{4 (d-e x) (g x)^{m+1}}{11 g \left (d^2-e^2 x^2\right )^{11/2}} \]
Antiderivative was successfully verified.
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Rule 852
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 &=\int \frac{(g x)^m (d-e x)^3}{\left (d^2-e^2 x^2\right )^{13/2}} \, dx\\ &=\frac{4 (g x)^{1+m} (d-e x)}{11 g \left (d^2-e^2 x^2\right )^{11/2}}-\frac{\int \frac{(g x)^m \left (-d^3 (7-4 m)+d^2 e (25-4 m) x\right )}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx}{11 d^2}\\ &=\frac{4 (g x)^{1+m} (d-e x)}{11 g \left (d^2-e^2 x^2\right )^{11/2}}+\frac{1}{11} (d (7-4 m)) \int \frac{(g x)^m}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx-\frac{(e (25-4 m)) \int \frac{(g x)^{1+m}}{\left (d^2-e^2 x^2\right )^{11/2}} \, dx}{11 g}\\ &=\frac{4 (g x)^{1+m} (d-e x)}{11 g \left (d^2-e^2 x^2\right )^{11/2}}+\frac{\left ((7-4 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 )^{11/2}} \, dx}{11 d^9 \sqrt{d^2-e^2 x^2}}-\frac{\left (e (25-4 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 )^{11/2}} \, dx}{11 d^{10} g \sqrt{d^2-e^2 x^2}}\\ &=\frac{4 (g x)^{1+m} (d-e x)}{11 g \left (d^2-e^2 x^2\right )^{11/2}}+\frac{(7-4 m) (g x)^{1+m} \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{11}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^9 g (1+m) \sqrt{d^2-e^2 x^2}}-\frac{e (25-4 m) (g x)^{2+m} \sqrt{1-\frac{e^2 x^2}{d^2}} \, _2F_1\left (\frac{11}{2},\frac{2+m}{2};\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{11 d^{10} g^2 (2+m) \sqrt{d^2-e^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.224017, size = 200, normalized size = 0.93 \[ \frac{x \sqrt{1-\frac{e^2 x^2}{d^2}} (g x)^m \left (\frac{d^3 \, _2F_1\left (\frac{13}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}+e x \left (e x \left (\frac{3 d \, _2F_1\left (\frac{13}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}-\frac{e x \, _2F_1\left (\frac{13}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right )}{m+4}\right )-\frac{3 d^2 \, _2F_1\left (\frac{13}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{m+2}\right )\right )}{d^{12} \sqrt{d^2-e^2 x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.645, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \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 (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{3}}\,{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^{11} x^{11} + 3 \, d e^{10} x^{10} - d^{2} e^{9} x^{9} - 11 \, d^{3} e^{8} x^{8} - 6 \, d^{4} e^{7} x^{7} + 14 \, d^{5} e^{6} x^{6} + 14 \, d^{6} e^{5} x^{5} - 6 \, d^{7} e^{4} x^{4} - 11 \, d^{8} e^{3} x^{3} - d^{9} e^{2} x^{2} + 3 \, d^{10} e x + d^{11}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (g x\right )^{m}}{{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{7}{2}}{\left (e x + d\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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