Optimal. Leaf size=41 \[ \frac{b f (2 p+3) \left (d+f x^2\right )^{p+1}}{p+1}+2 c f x \left (d+f x^2\right )^{p+1} \]
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Rubi [A] time = 0.0515075, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1815, 12, 261} \[ \frac{b f (2 p+3) \left (d+f x^2\right )^{p+1}}{p+1}+2 c f x \left (d+f x^2\right )^{p+1} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 12
Rule 261
Rubi steps
\begin{align*} \int \left (d+f x^2\right )^p \left (2 c d f+2 b f^2 (3+2 p) x+2 c f^2 (3+2 p) x^2\right ) \, dx &=2 c f x \left (d+f x^2\right )^{1+p}+\frac{\int 2 b f^3 (3+2 p)^2 x \left (d+f x^2\right )^p \, dx}{f (3+2 p)}\\ &=2 c f x \left (d+f x^2\right )^{1+p}+\left (2 b f^2 (3+2 p)\right ) \int x \left (d+f x^2\right )^p \, dx\\ &=\frac{b f (3+2 p) \left (d+f x^2\right )^{1+p}}{1+p}+2 c f x \left (d+f x^2\right )^{1+p}\\ \end{align*}
Mathematica [C] time = 0.0946046, size = 119, normalized size = 2.9 \[ \frac{f \left (d+f x^2\right )^p \left (\frac{f x^2}{d}+1\right )^{-p} \left ((2 p+3) \left (3 b \left (d+f x^2\right ) \left (\frac{f x^2}{d}+1\right )^p+2 c f (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{f x^2}{d}\right )\right )+6 c d (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{f x^2}{d}\right )\right )}{3 (p+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 36, normalized size = 0.9 \begin{align*}{\frac{f \left ( f{x}^{2}+d \right ) ^{1+p} \left ( 2\,cxp+2\,bp+2\,cx+3\,b \right ) }{1+p}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11117, size = 80, normalized size = 1.95 \begin{align*} \frac{{\left (2 \, c f^{2}{\left (p + 1\right )} x^{3} + b f^{2}{\left (2 \, p + 3\right )} x^{2} + 2 \, c d f{\left (p + 1\right )} x + b d f{\left (2 \, p + 3\right )}\right )}{\left (f x^{2} + d\right )}^{p}}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.39543, size = 166, normalized size = 4.05 \begin{align*} \frac{{\left (2 \, b d f p + 2 \,{\left (c f^{2} p + c f^{2}\right )} x^{3} + 3 \, b d f +{\left (2 \, b f^{2} p + 3 \, b f^{2}\right )} x^{2} + 2 \,{\left (c d f p + c d f\right )} x\right )}{\left (f x^{2} + d\right )}^{p}}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.56548, size = 221, normalized size = 5.39 \begin{align*} \begin{cases} \frac{2 b d f p \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{3 b d f \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{2 b f^{2} p x^{2} \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{3 b f^{2} x^{2} \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{2 c d f p x \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{2 c d f x \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{2 c f^{2} p x^{3} \left (d + f x^{2}\right )^{p}}{p + 1} + \frac{2 c f^{2} x^{3} \left (d + f x^{2}\right )^{p}}{p + 1} & \text{for}\: p \neq -1 \\b f \log{\left (- i \sqrt{d} \sqrt{\frac{1}{f}} + x \right )} + b f \log{\left (i \sqrt{d} \sqrt{\frac{1}{f}} + x \right )} + 2 c f x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22614, size = 190, normalized size = 4.63 \begin{align*} \frac{2 \,{\left (f x^{2} + d\right )}^{p} c f^{2} p x^{3} + 2 \,{\left (f x^{2} + d\right )}^{p} b f^{2} p x^{2} + 2 \,{\left (f x^{2} + d\right )}^{p} c f^{2} x^{3} + 2 \,{\left (f x^{2} + d\right )}^{p} c d f p x + 3 \,{\left (f x^{2} + d\right )}^{p} b f^{2} x^{2} + 2 \,{\left (f x^{2} + d\right )}^{p} b d f p + 2 \,{\left (f x^{2} + d\right )}^{p} c d f x + 3 \,{\left (f x^{2} + d\right )}^{p} b d f}{p + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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