Optimal. Leaf size=250 \[ \frac{d^6 e (4 m+29) \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+9) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}+\frac{d^7 (4 m+11) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 d \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)} \]
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Rubi [A] time = 0.385916, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1809, 808, 365, 364} \[ \frac{d^6 e (4 m+29) \sqrt{d^2-e^2 x^2} (g x)^{m+2} \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+9) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{e \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+2}}{g^2 (m+9)}+\frac{d^7 (4 m+11) \sqrt{d^2-e^2 x^2} (g x)^{m+1} \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{g (m+1) (m+8) \sqrt{1-\frac{e^2 x^2}{d^2}}}-\frac{3 d \left (d^2-e^2 x^2\right )^{7/2} (g x)^{m+1}}{g (m+8)} \]
Antiderivative was successfully verified.
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Rule 1809
Rule 808
Rule 365
Rule 364
Rubi steps
\begin{align*} \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^{5/2} \, dx &=-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}-\frac{\int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \left (-d^3 e^2 (9+m)-d^2 e^3 (29+4 m) x-3 d e^4 (9+m) x^2\right ) \, dx}{e^2 (9+m)}\\ &=-\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac{\int (g x)^m \left (d^3 e^4 (9+m) (11+4 m)+d^2 e^5 (8+m) (29+4 m) x\right ) \left (d^2-e^2 x^2\right )^{5/2} \, dx}{e^4 (8+m) (9+m)}\\ &=-\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac{\left (d^3 (11+4 m)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^{5/2} \, dx}{8+m}+\frac{\left (d^2 e (29+4 m)\right ) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^{5/2} \, dx}{g (9+m)}\\ &=-\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac{\left (d^7 (11+4 m) \sqrt{d^2-e^2 x^2}\right ) \int (g x)^m \left (1-\frac{e^2 x^2}{d^2}\right )^{5/2} \, dx}{(8+m) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{\left (d^6 e (29+4 m) \sqrt{d^2-e^2 x^2}\right ) \int (g x)^{1+m} \left (1-\frac{e^2 x^2}{d^2}\right )^{5/2} \, dx}{g (9+m) \sqrt{1-\frac{e^2 x^2}{d^2}}}\\ &=-\frac{3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{7/2}}{g (8+m)}-\frac{e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{7/2}}{g^2 (9+m)}+\frac{d^7 (11+4 m) (g x)^{1+m} \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},\frac{1+m}{2};\frac{3+m}{2};\frac{e^2 x^2}{d^2}\right )}{g (1+m) (8+m) \sqrt{1-\frac{e^2 x^2}{d^2}}}+\frac{d^6 e (29+4 m) (g x)^{2+m} \sqrt{d^2-e^2 x^2} \, _2F_1\left (-\frac{5}{2},\frac{2+m}{2};\frac{4+m}{2};\frac{e^2 x^2}{d^2}\right )}{g^2 (2+m) (9+m) \sqrt{1-\frac{e^2 x^2}{d^2}}}\\ \end{align*}
Mathematica [A] time = 0.214215, size = 199, normalized size = 0.8 \[ \frac{d^4 x \sqrt{d^2-e^2 x^2} (g x)^m \left (\frac{d^3 \, _2F_1\left (-\frac{5}{2},\frac{m+1}{2};\frac{m+3}{2};\frac{e^2 x^2}{d^2}\right )}{m+1}+e x \left (\frac{3 d^2 \, _2F_1\left (-\frac{5}{2},\frac{m+2}{2};\frac{m+4}{2};\frac{e^2 x^2}{d^2}\right )}{m+2}+e x \left (\frac{3 d \, _2F_1\left (-\frac{5}{2},\frac{m+3}{2};\frac{m+5}{2};\frac{e^2 x^2}{d^2}\right )}{m+3}+\frac{e x \, _2F_1\left (-\frac{5}{2},\frac{m+4}{2};\frac{m+6}{2};\frac{e^2 x^2}{d^2}\right )}{m+4}\right )\right )\right )}{\sqrt{1-\frac{e^2 x^2}{d^2}}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.572, 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{5}{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{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{3} \left (g x\right )^{m}\,{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 ({\left (e^{7} x^{7} + 3 \, d e^{6} x^{6} + d^{2} e^{5} x^{5} - 5 \, d^{3} e^{4} x^{4} - 5 \, d^{4} e^{3} x^{3} + d^{5} e^{2} x^{2} + 3 \, d^{6} e x + d^{7}\right )} \sqrt{-e^{2} x^{2} + d^{2}} \left (g x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 134.441, size = 513, normalized size = 2.05 \begin{align*} \text{result too large to display} \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{\left (-e^{2} x^{2} + d^{2}\right )}^{\frac{5}{2}}{\left (e x + d\right )}^{3} \left (g x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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