Optimal. Leaf size=75 \[ \frac{d \left (a+b x^2\right )^{p+1}}{2 b (p+1)}+\frac{1}{3} e x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right ) \]
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Rubi [A] time = 0.0326914, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {764, 261, 365, 364} \[ \frac{d \left (a+b x^2\right )^{p+1}}{2 b (p+1)}+\frac{1}{3} e x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 764
Rule 261
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x (d+e x) \left (a+b x^2\right )^p \, dx &=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*}
Mathematica [A] time = 0.0343881, size = 71, normalized size = 0.95 \[ \frac{1}{6} \left (a+b x^2\right )^p \left (\frac{3 d \left (a+b x^2\right )}{b (p+1)}+2 e x^3 \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.405, size = 0, normalized size = 0. \begin{align*} \int x \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{2} + d x\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 = 14.3415, size = 65, normalized size = 0.87 \begin{align*} \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 ) \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} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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