Optimal. Leaf size=264 \[ -\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {2 d^3 (3+2 m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (3+m+2 p)}+\frac {2 d^2 e (7+2 m+3 p) (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (4+m+2 p)} \]
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Rubi [A]
time = 0.24, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {1823, 822, 372,
371} \begin {gather*} \frac {2 d^2 e (2 m+3 p+7) (g x)^{m+2} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (m+2) (m+2 p+4)}-\frac {e (g x)^{m+2} \left (d^2-e^2 x^2\right )^{p+1}}{g^2 (m+2 p+4)}-\frac {3 d (g x)^{m+1} \left (d^2-e^2 x^2\right )^{p+1}}{g (m+2 p+3)}+\frac {2 d^3 (2 m+p+3) (g x)^{m+1} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};\frac {e^2 x^2}{d^2}\right )}{g (m+1) (m+2 p+3)} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 372
Rule 822
Rule 1823
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^3 \left (d^2-e^2 x^2\right )^p \, dx &=-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}-\frac {\int (g x)^m \left (d^2-e^2 x^2\right )^p \left (-d^3 e^2 (4+m+2 p)-2 d^2 e^3 (7+2 m+3 p) x-3 d e^4 (4+m+2 p) x^2\right ) \, dx}{e^2 (4+m+2 p)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {\int (g x)^m \left (2 d^3 e^4 (3+2 m+p) (4+m+2 p)+2 d^2 e^5 (3+m+2 p) (7+2 m+3 p) x\right ) \left (d^2-e^2 x^2\right )^p \, dx}{e^4 (3+m+2 p) (4+m+2 p)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {\left (2 d^3 (3+2 m+p)\right ) \int (g x)^m \left (d^2-e^2 x^2\right )^p \, dx}{3+m+2 p}+\frac {\left (2 d^2 e (7+2 m+3 p)\right ) \int (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \, dx}{g (4+m+2 p)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {\left (2 d^3 (3+2 m+p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^m \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{3+m+2 p}+\frac {\left (2 d^2 e (7+2 m+3 p) \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p}\right ) \int (g x)^{1+m} \left (1-\frac {e^2 x^2}{d^2}\right )^p \, dx}{g (4+m+2 p)}\\ &=-\frac {3 d (g x)^{1+m} \left (d^2-e^2 x^2\right )^{1+p}}{g (3+m+2 p)}-\frac {e (g x)^{2+m} \left (d^2-e^2 x^2\right )^{1+p}}{g^2 (4+m+2 p)}+\frac {2 d^3 (3+2 m+p) (g x)^{1+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{g (1+m) (3+m+2 p)}+\frac {2 d^2 e (7+2 m+3 p) (g x)^{2+m} \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{g^2 (2+m) (4+m+2 p)}\\ \end {align*}
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Mathematica [A]
time = 0.21, size = 194, normalized size = 0.73 \begin {gather*} x (g x)^m \left (d^2-e^2 x^2\right )^p \left (1-\frac {e^2 x^2}{d^2}\right )^{-p} \left (\frac {d^3 \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};\frac {e^2 x^2}{d^2}\right )}{1+m}+e x \left (\frac {3 d^2 \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};\frac {e^2 x^2}{d^2}\right )}{2+m}+e x \left (\frac {3 d \, _2F_1\left (\frac {3+m}{2},-p;\frac {5+m}{2};\frac {e^2 x^2}{d^2}\right )}{3+m}+\frac {e x \, _2F_1\left (\frac {4+m}{2},-p;\frac {6+m}{2};\frac {e^2 x^2}{d^2}\right )}{4+m}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \left (g x \right )^{m} \left (e x +d \right )^{3} \left (-e^{2} x^{2}+d^{2}\right )^{p}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 11.27, size = 262, normalized size = 0.99 \begin {gather*} \frac {d^{3} d^{2 p} g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {3 d^{2} d^{2 p} e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} + \frac {3 d d^{2 p} e^{2} g^{m} x^{3} x^{m} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {d^{2 p} e^{3} g^{m} x^{4} x^{m} \Gamma \left (\frac {m}{2} + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 2 \\ \frac {m}{2} + 3 \end {matrix}\middle | {\frac {e^{2} x^{2} e^{2 i \pi }}{d^{2}}} \right )}}{2 \Gamma \left (\frac {m}{2} + 3\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (d^2-e^2\,x^2\right )}^p\,{\left (g\,x\right )}^m\,{\left (d+e\,x\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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