Optimal. Leaf size=23 \[ \frac{\left (a+b x+c x^2+d x^3\right )^{p+1}}{x^2} \]
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Rubi [A] time = 0.0353046, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.021, Rules used = {1590} \[ \frac{\left (a+b x+c x^2+d x^3\right )^{p+1}}{x^2} \]
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin{align*} \int \frac{\left (a+b x+c x^2+d x^3\right )^p \left (-2 a+b (-1+p) x+2 c p x^2+d (1+3 p) x^3\right )}{x^3} \, dx &=\frac{\left (a+b x+c x^2+d x^3\right )^{1+p}}{x^2}\\ \end{align*}
Mathematica [A] time = 0.182003, size = 21, normalized size = 0.91 \[ \frac{(a+x (b+x (c+d x)))^{p+1}}{x^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 24, normalized size = 1. \begin{align*}{\frac{ \left ( d{x}^{3}+c{x}^{2}+bx+a \right ) ^{1+p}}{{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.31428, size = 49, normalized size = 2.13 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p}}{x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82445, size = 80, normalized size = 3.48 \begin{align*} \frac{{\left (d x^{3} + c x^{2} + b x + a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p}}{x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \frac{{\left (d{\left (3 \, p + 1\right )} x^{3} + 2 \, c p x^{2} + b{\left (p - 1\right )} x - 2 \, a\right )}{\left (d x^{3} + c x^{2} + b x + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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