Optimal. Leaf size=34 \[ \frac{\left (a x+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{c x^3}{3} \]
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Rubi [A] time = 0.0090691, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {1591} \[ \frac{\left (a x+\frac{c x^3}{3}\right )^{n+1}}{n+1}+a x+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int \left (a+c x^2\right ) \left (1+\left (a x+\frac{c x^3}{3}\right )^n\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^n\right ) \, dx,x,a x+\frac{c x^3}{3}\right )\\ &=a x+\frac{c x^3}{3}+\frac{\left (a x+\frac{c x^3}{3}\right )^{1+n}}{1+n}\\ \end{align*}
Mathematica [A] time = 0.0616451, size = 36, normalized size = 1.06 \[ \frac{x \left (3 a+c x^2\right ) \left (\left (a x+\frac{c x^3}{3}\right )^n+n+1\right )}{3 (n+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 31, normalized size = 0.9 \begin{align*} ax+{\frac{c{x}^{3}}{3}}+{\frac{1}{1+n} \left ( ax+{\frac{c{x}^{3}}{3}} \right ) ^{1+n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65076, size = 73, normalized size = 2.15 \begin{align*} \frac{1}{3} \, c x^{3} + a x + \frac{{\left (c x^{3} + 3 \, a x\right )} e^{\left (n \log \left (c x^{2} + 3 \, a\right ) + n \log \left (x\right )\right )}}{3^{n + 1} n + 3^{n + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.35445, size = 112, normalized size = 3.29 \begin{align*} \frac{{\left (c n + c\right )} x^{3} +{\left (c x^{3} + 3 \, a x\right )}{\left (\frac{1}{3} \, c x^{3} + a x\right )}^{n} + 3 \,{\left (a n + a\right )} x}{3 \,{\left (n + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 175.955, size = 201, normalized size = 5.91 \begin{align*} \begin{cases} \frac{3 \cdot 3^{n} a n x}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3 \cdot 3^{n} a x}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3^{n} c n x^{3}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3^{n} c x^{3}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{3 a x \left (3 a x + c x^{3}\right )^{n}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac{c x^{3} \left (3 a x + c x^{3}\right )^{n}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} & \text{for}\: n \neq -1 \\a x + \frac{c x^{3}}{3} + \log{\left (x \right )} + \log{\left (- \sqrt{3} i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )} + \log{\left (\sqrt{3} i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17752, size = 41, normalized size = 1.21 \begin{align*} \frac{1}{3} \, c x^{3} + a x + \frac{{\left (\frac{1}{3} \, c x^{3} + a x\right )}^{n + 1}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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