Optimal. Leaf size=135 \[ \frac {d (g x)^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {c x^2}{a}\right )}{g (m+1)}+\frac {e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};-\frac {c x^2}{a}\right )}{g^2 (m+2)} \]
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Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {808, 365, 364} \[ \frac {d (g x)^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {c x^2}{a}\right )}{g (m+1)}+\frac {e (g x)^{m+2} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};-\frac {c x^2}{a}\right )}{g^2 (m+2)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 365
Rule 808
Rubi steps
\begin {align*} \int (g x)^m (d+e x) \left (a+c x^2\right )^p \, dx &=d \int (g x)^m \left (a+c x^2\right )^p \, dx+\frac {e \int (g x)^{1+m} \left (a+c x^2\right )^p \, dx}{g}\\ &=\left (d \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^m \left (1+\frac {c x^2}{a}\right )^p \, dx+\frac {\left (e \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p}\right ) \int (g x)^{1+m} \left (1+\frac {c x^2}{a}\right )^p \, dx}{g}\\ &=\frac {d (g x)^{1+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};-\frac {c x^2}{a}\right )}{g (1+m)}+\frac {e (g x)^{2+m} \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \, _2F_1\left (\frac {2+m}{2},-p;\frac {4+m}{2};-\frac {c x^2}{a}\right )}{g^2 (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 106, normalized size = 0.79 \[ \frac {x (g x)^m \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \left (d (m+2) \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {c x^2}{a}\right )+e (m+1) x \, _2F_1\left (\frac {m+2}{2},-p;\frac {m+4}{2};-\frac {c x^2}{a}\right )\right )}{(m+1) (m+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (e x + d\right )} {\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}, x\right ) \]
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 \[ \int {\left (e x + d\right )} {\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (e x +d \right ) \left (g x \right )^{m} \left (c \,x^{2}+a \right )^{p}\, dx \]
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 \[ \int {\left (e x + d\right )} {\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^p\,\left (d+e\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 88.94, size = 109, normalized size = 0.81 \[ \frac {a^{p} d g^{m} x x^{m} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {a^{p} e g^{m} x^{2} x^{m} \Gamma \left (\frac {m}{2} + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + 1 \\ \frac {m}{2} + 2 \end {matrix}\middle | {\frac {c x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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