Optimal. Leaf size=60 \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]
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Rubi [A] time = 0.0380882, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036, Rules used = {1850} \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]
Antiderivative was successfully verified.
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Rule 1850
Rubi steps
\begin{align*} \int \left (a+b x^3\right ) \left (a c+a d x+b c x^3+b d x^4\right ) \, dx &=\int \left (a^2 c+a^2 d x+2 a b c x^3+2 a b d x^4+b^2 c x^6+b^2 d x^7\right ) \, dx\\ &=a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8\\ \end{align*}
Mathematica [A] time = 0.0021509, size = 60, normalized size = 1. \[ a^2 c x+\frac{1}{2} a^2 d x^2+\frac{1}{2} a b c x^4+\frac{2}{5} a b d x^5+\frac{1}{7} b^2 c x^7+\frac{1}{8} b^2 d x^8 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 51, normalized size = 0.9 \begin{align*}{a}^{2}cx+{\frac{{a}^{2}d{x}^{2}}{2}}+{\frac{abc{x}^{4}}{2}}+{\frac{2\,abd{x}^{5}}{5}}+{\frac{{b}^{2}c{x}^{7}}{7}}+{\frac{{b}^{2}d{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.990225, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.08037, size = 120, normalized size = 2. \begin{align*} \frac{1}{8} x^{8} d b^{2} + \frac{1}{7} x^{7} c b^{2} + \frac{2}{5} x^{5} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{2} x^{2} d a^{2} + x c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.076342, size = 58, normalized size = 0.97 \begin{align*} a^{2} c x + \frac{a^{2} d x^{2}}{2} + \frac{a b c x^{4}}{2} + \frac{2 a b d x^{5}}{5} + \frac{b^{2} c x^{7}}{7} + \frac{b^{2} d x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06269, size = 68, normalized size = 1.13 \begin{align*} \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{7} \, b^{2} c x^{7} + \frac{2}{5} \, a b d x^{5} + \frac{1}{2} \, a b c x^{4} + \frac{1}{2} \, a^{2} d x^{2} + a^{2} c x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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