Optimal. Leaf size=164 \[ \frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} \left (a e^2 (m+n+2) (m+n+3)+c d^2 (m+1) (m+2)\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {e x}{d}\right )}{e^2 g (m+1) (m+n+2) (m+n+3)}-\frac {c d (m+2) (g x)^{m+1} (d+e x)^{n+1}}{e^2 g (m+n+2) (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1}}{e g^2 (m+n+3)} \]
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Rubi [A] time = 0.13, antiderivative size = 150, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {952, 80, 66, 64} \[ \frac {(g x)^{m+1} (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} \left (\frac {a}{m+1}+\frac {c d^2 (m+2)}{e^2 (m+n+2) (m+n+3)}\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {e x}{d}\right )}{g}-\frac {c d (m+2) (g x)^{m+1} (d+e x)^{n+1}}{e^2 g (m+n+2) (m+n+3)}+\frac {c (g x)^{m+2} (d+e x)^{n+1}}{e g^2 (m+n+3)} \]
Antiderivative was successfully verified.
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Rule 64
Rule 66
Rule 80
Rule 952
Rubi steps
\begin {align*} \int (g x)^m (d+e x)^n \left (a+c x^2\right ) \, dx &=\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\frac {\int (g x)^m (d+e x)^n \left (a e g^2 (3+m+n)-c d g^2 (2+m) x\right ) \, dx}{e g^2 (3+m+n)}\\ &=-\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\left (a+\frac {c d^2 (1+m) (2+m)}{e^2 (2+m+n) (3+m+n)}\right ) \int (g x)^m (d+e x)^n \, dx\\ &=-\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\left (\left (a+\frac {c d^2 (1+m) (2+m)}{e^2 (2+m+n) (3+m+n)}\right ) (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n}\right ) \int (g x)^m \left (1+\frac {e x}{d}\right )^n \, dx\\ &=-\frac {c d (2+m) (g x)^{1+m} (d+e x)^{1+n}}{e^2 g (2+m+n) (3+m+n)}+\frac {c (g x)^{2+m} (d+e x)^{1+n}}{e g^2 (3+m+n)}+\frac {\left (a+\frac {c d^2 (1+m) (2+m)}{e^2 (2+m+n) (3+m+n)}\right ) (g x)^{1+m} (d+e x)^n \left (1+\frac {e x}{d}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {e x}{d}\right )}{g (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 113, normalized size = 0.69 \[ \frac {x (g x)^m (d+e x)^n \left (\frac {e x}{d}+1\right )^{-n} \left (\left (a e^2+c d^2\right ) \, _2F_1\left (m+1,-n;m+2;-\frac {e x}{d}\right )+c d^2 \, _2F_1\left (m+1,-n-2;m+2;-\frac {e x}{d}\right )-2 c d^2 \, _2F_1\left (m+1,-n-1;m+2;-\frac {e x}{d}\right )\right )}{e^2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c x^{2} + a\right )} {\left (e x + d\right )}^{n} \left (g x\right )^{m}, x\right ) \]
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 \[ \int {\left (c x^{2} + a\right )} {\left (e x + d\right )}^{n} \left (g x\right )^{m}\,{d x} \]
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 \[ \int \left (c \,x^{2}+a \right ) \left (g x \right )^{m} \left (e x +d \right )^{n}\, dx \]
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 \[ \int {\left (c x^{2} + a\right )} {\left (e x + d\right )}^{n} \left (g x\right )^{m}\,{d x} \]
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 \[ \int {\left (g\,x\right )}^m\,\left (c\,x^2+a\right )\,{\left (d+e\,x\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 15.75, size = 82, normalized size = 0.50 \[ \frac {a d^{n} g^{m} x x^{m} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 2\right )} + \frac {c d^{n} g^{m} x^{3} x^{m} \Gamma \left (m + 3\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 3 \\ m + 4 \end {matrix}\middle | {\frac {e x e^{i \pi }}{d}} \right )}}{\Gamma \left (m + 4\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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