Optimal. Leaf size=133 \[ -\frac{x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (a e^2-c d^2 (2 p+3)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )}{c (2 p+3)}+\frac{e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \]
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Rubi [A] time = 0.0730234, antiderivative size = 125, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {743, 641, 246, 245} \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d^2-\frac{a e^2}{2 c p+3 c}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+\frac{e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \]
Antiderivative was successfully verified.
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Rule 743
Rule 641
Rule 246
Rule 245
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+c x^2\right )^p \, dx &=\frac{e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\frac{\int \left (-a e^2+c d^2 (3+2 p)+2 c d e (2+p) x\right ) \left (a+c x^2\right )^p \, dx}{c (3+2 p)}\\ &=\frac{d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac{e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\left (d^2-\frac{a e^2}{3 c+2 c p}\right ) \int \left (a+c x^2\right )^p \, dx\\ &=\frac{d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac{e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\left (\left (d^2-\frac{a e^2}{3 c+2 c p}\right ) \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p}\right ) \int \left (1+\frac{c x^2}{a}\right )^p \, dx\\ &=\frac{d e (2+p) \left (a+c x^2\right )^{1+p}}{c (1+p) (3+2 p)}+\frac{e (d+e x) \left (a+c x^2\right )^{1+p}}{c (3+2 p)}+\left (d^2-\frac{a e^2}{3 c+2 c p}\right ) x \left (a+c x^2\right )^p \left (1+\frac{c x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )\\ \end{align*}
Mathematica [A] time = 0.100026, size = 133, normalized size = 1. \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (3 c d^2 (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+e \left (3 d \left (c x^2 \left (\frac{c x^2}{a}+1\right )^p+a \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )\right )+c e (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right )\right )\right )}{3 c (p+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.45, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{2} \left ( c{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 )}^{2}{\left (c 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^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c 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 = 12.7396, size = 97, normalized size = 0.73 \begin{align*} a^{p} d^{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} + \frac{a^{p} e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{3} + 2 d e \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\begin{cases} \frac{\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + c x^{2} \right )} & \text{otherwise} \end{cases}}{2 c} & \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 (c x^{2} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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