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 = 0.05, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 12
Rule 261
Rule 1815
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*}
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Mathematica [C] time = 0.11, size = 119, normalized size = 2.90 \[ \frac {f \left (d+f x^2\right )^p \left (\frac {f x^2}{d}+1\right )^{-p} \left ((2 p+3) \left (3 b \left (d+f x^2\right ) \left (\frac {f x^2}{d}+1\right )^p+2 c f (p+1) x^3 \, _2F_1\left (\frac {3}{2},-p;\frac {5}{2};-\frac {f x^2}{d}\right )\right )+6 c d (p+1) x \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {f x^2}{d}\right )\right )}{3 (p+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 75, normalized size = 1.83 \[ \frac {{\left (2 \, b d f p + 2 \, {\left (c f^{2} p + c f^{2}\right )} x^{3} + 3 \, b d f + {\left (2 \, b f^{2} p + 3 \, b f^{2}\right )} x^{2} + 2 \, {\left (c d f p + c d f\right )} x\right )} {\left (f x^{2} + d\right )}^{p}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 141, normalized size = 3.44 \[ \frac {2 \, {\left (f x^{2} + d\right )}^{p} c f^{2} p x^{3} + 2 \, {\left (f x^{2} + d\right )}^{p} b f^{2} p x^{2} + 2 \, {\left (f x^{2} + d\right )}^{p} c f^{2} x^{3} + 2 \, {\left (f x^{2} + d\right )}^{p} c d f p x + 3 \, {\left (f x^{2} + d\right )}^{p} b f^{2} x^{2} + 2 \, {\left (f x^{2} + d\right )}^{p} b d f p + 2 \, {\left (f x^{2} + d\right )}^{p} c d f x + 3 \, {\left (f x^{2} + d\right )}^{p} b d f}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 36, normalized size = 0.88 \[ \frac {\left (2 p c x +2 p b +2 c x +3 b \right ) f \left (f \,x^{2}+d \right )^{p +1}}{p +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 59, normalized size = 1.44 \[ \frac {{\left (2 \, c f^{2} {\left (p + 1\right )} x^{3} + b f^{2} {\left (2 \, p + 3\right )} x^{2} + 2 \, c d f {\left (p + 1\right )} x + b d f {\left (2 \, p + 3\right )}\right )} {\left (f x^{2} + d\right )}^{p}}{p + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.25, size = 58, normalized size = 1.41 \[ {\left (f\,x^2+d\right )}^p\,\left (2\,c\,f^2\,x^3+2\,c\,d\,f\,x+\frac {b\,f^2\,x^2\,\left (2\,p+3\right )}{p+1}+\frac {b\,d\,f\,\left (2\,p+3\right )}{p+1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.25, size = 221, normalized size = 5.39 \[ \begin {cases} \frac {2 b d f p \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {3 b d f \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {2 b f^{2} p x^{2} \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {3 b f^{2} x^{2} \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {2 c d f p x \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {2 c d f x \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {2 c f^{2} p x^{3} \left (d + f x^{2}\right )^{p}}{p + 1} + \frac {2 c f^{2} x^{3} \left (d + f x^{2}\right )^{p}}{p + 1} & \text {for}\: p \neq -1 \\b f \log {\left (- i \sqrt {d} \sqrt {\frac {1}{f}} + x \right )} + b f \log {\left (i \sqrt {d} \sqrt {\frac {1}{f}} + x \right )} + 2 c f x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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