Optimal. Leaf size=70 \[ d x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+\frac{e \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.0197929, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {641, 246, 245} \[ d x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+\frac{e \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 641
Rule 246
Rule 245
Rubi steps
\begin{align*} \int (d+e x) \left (a+b x^2\right )^p \, dx &=\frac{e \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+d \int \left (a+b x^2\right )^p \, dx\\ &=\frac{e \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\left (d \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac{b x^2}{a}\right )^p \, dx\\ &=\frac{e \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+d x \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.0740842, size = 98, normalized size = 1.4 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (2 b d (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )+b e x^2 \left (\frac{b x^2}{a}+1\right )^p+a e \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )\right )}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.406, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) \left ( b{x}^{2}+a \right ) ^{p}\, 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 x + d\right )}{\left (b x^{2} + a\right )}^{p}\,{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 x + d\right )}{\left (b x^{2} + a\right )}^{p}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.82081, size = 61, normalized size = 0.87 \begin{align*} a^{p} d x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} + e \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 ) \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 x + d\right )}{\left (b x^{2} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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