Integrand size = 16, antiderivative size = 75 \[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\frac {d \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {1}{3} e x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {778, 267, 372, 371} \[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\frac {d \left (a+b x^2\right )^{p+1}}{2 b (p+1)}+\frac {1}{3} e x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right ) \]
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Rule 267
Rule 371
Rule 372
Rule 778
Rubi steps \begin{align*} \text {integral}& = d \int x \left (a+b x^2\right )^p \, dx+e \int x^2 \left (a+b x^2\right )^p \, dx \\ & = \frac {d \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\left (e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int x^2 \left (1+\frac {b x^2}{a}\right )^p \, dx \\ & = \frac {d \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {1}{3} e x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {b x^2}{a}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95 \[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\frac {1}{6} \left (a+b x^2\right )^p \left (\frac {3 d \left (a+b x^2\right )}{b (1+p)}+2 e x^3 \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right )\right ) \]
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\[\int x \left (e x +d \right ) \left (b \,x^{2}+a \right )^{p}d x\]
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\[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p} x \,d x } \]
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Time = 3.69 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.87 \[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\frac {a^{p} e x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} + d \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) \]
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\[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p} x \,d x } \]
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\[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\int { {\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p} x \,d x } \]
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Timed out. \[ \int x (d+e x) \left (a+b x^2\right )^p \, dx=\int x\,{\left (b\,x^2+a\right )}^p\,\left (d+e\,x\right ) \,d x \]
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