Optimal. Leaf size=23 \[ \frac {\left (a+b x+c x^2+d x^3\right )^{1+p}}{x^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 23, normalized size of antiderivative = 1.00, 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 = {1604}
\begin {gather*} \frac {\left (a+b x+c x^2+d x^3\right )^{p+1}}{x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1604
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*}
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Mathematica [A]
time = 0.79, size = 21, normalized size = 0.91 \begin {gather*} \frac {(a+x (b+x (c+d x)))^{1+p}}{x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 24, normalized size = 1.04
method | result | size |
gosper | \(\frac {\left (d \,x^{3}+c \,x^{2}+b x +a \right )^{1+p}}{x^{2}}\) | \(24\) |
risch | \(\frac {\left (d \,x^{3}+c \,x^{2}+b x +a \right ) \left (d \,x^{3}+c \,x^{2}+b x +a \right )^{p}}{x^{2}}\) | \(37\) |
norman | \(\frac {a \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+b x \,{\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+c \,x^{2} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}+d \,x^{3} {\mathrm e}^{p \ln \left (d \,x^{3}+c \,x^{2}+b x +a \right )}}{x^{2}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 36, normalized size = 1.57 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 36, normalized size = 1.57 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.32, size = 23, normalized size = 1.00 \begin {gather*} \frac {{\left (d\,x^3+c\,x^2+b\,x+a\right )}^{p+1}}{x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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