Optimal. Leaf size=32 \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4 \]
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Rubi [A] time = 0.0340129, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 54, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019, Rules used = {1586} \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4 \]
Antiderivative was successfully verified.
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Rule 1586
Rubi steps
\begin{align*} \int \frac{a^2 c+a^2 d x+2 a b c x^2+2 a b d x^3+b^2 c x^4+b^2 d x^5}{a+b x^2} \, dx &=\int \left (a c+a d x+b c x^2+b d x^3\right ) \, dx\\ &=a c x+\frac{1}{2} a d x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4\\ \end{align*}
Mathematica [A] time = 0.0018921, size = 32, normalized size = 1. \[ a c x+\frac{1}{2} a d x^2+\frac{1}{3} b c x^3+\frac{1}{4} b d x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 27, normalized size = 0.8 \begin{align*} acx+{\frac{ad{x}^{2}}{2}}+{\frac{bc{x}^{3}}{3}}+{\frac{bd{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02178, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{4} \, b d x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{2} \, a d x^{2} + a c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.25276, size = 66, normalized size = 2.06 \begin{align*} \frac{1}{4} \, b d x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{2} \, a d x^{2} + a c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.080349, size = 29, normalized size = 0.91 \begin{align*} a c x + \frac{a d x^{2}}{2} + \frac{b c x^{3}}{3} + \frac{b d x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13044, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{4} \, b d x^{4} + \frac{1}{3} \, b c x^{3} + \frac{1}{2} \, a d x^{2} + a c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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