Optimal. Leaf size=113 \[ \frac{\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)}+\frac{e^2 \left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}+\frac{2}{3} d e x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right ) \]
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Rubi [A] time = 0.0970589, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {1652, 444, 43, 12, 365, 364} \[ \frac{\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{p+1}}{2 b^2 (p+1)}+\frac{e^2 \left (a+b x^2\right )^{p+2}}{2 b^2 (p+2)}+\frac{2}{3} d e x^3 \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 1652
Rule 444
Rule 43
Rule 12
Rule 365
Rule 364
Rubi steps
\begin{align*} \int x (d+e x)^2 \left (a+b x^2\right )^p \, dx &=\int 2 d e x^2 \left (a+b x^2\right )^p \, dx+\int x \left (a+b x^2\right )^p \left (d^2+e^2 x^2\right ) \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^p \left (d^2+e^2 x\right ) \, dx,x,x^2\right )+(2 d e) \int x^2 \left (a+b x^2\right )^p \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{\left (b d^2-a e^2\right ) (a+b x)^p}{b}+\frac{e^2 (a+b x)^{1+p}}{b}\right ) \, dx,x,x^2\right )+\left (2 d 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{\left (b d^2-a e^2\right ) \left (a+b x^2\right )^{1+p}}{2 b^2 (1+p)}+\frac{e^2 \left (a+b x^2\right )^{2+p}}{2 b^2 (2+p)}+\frac{2}{3} d 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 = 0.191503, size = 184, normalized size = 1.63 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (-3 a^2 e^2 \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )+3 b^2 x^2 \left (\frac{b x^2}{a}+1\right )^p \left (d^2 (p+2)+e^2 (p+1) x^2\right )+4 b^2 d e \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{b x^2}{a}\right )+3 a b \left (d^2 (p+2) \left (\left (\frac{b x^2}{a}+1\right )^p-1\right )+e^2 p x^2 \left (\frac{b x^2}{a}+1\right )^p\right )\right )}{6 b^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.498, size = 0, normalized size = 0. \begin{align*} \int x \left ( ex+d \right ) ^{2} \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^{2} x^{3} + 2 \, d e x^{2} + d^{2} 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 = 21.2082, size = 439, normalized size = 3.88 \begin{align*} \frac{2 a^{p} d 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^{2} \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 ) + e^{2} \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: b = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{a}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} + \frac{b x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 a b^{2} + 2 b^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{b}} + x \right )}}{2 b^{2}} + \frac{x^{2}}{2 b} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{a b p x^{2} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} p x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} + \frac{b^{2} x^{4} \left (a + b x^{2}\right )^{p}}{2 b^{2} p^{2} + 6 b^{2} p + 4 b^{2}} & \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 )}^{2}{\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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