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 = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 1.02, size = 48, normalized size = 1.41 \[ \frac {{\left (c n + c\right )} x^{3} + {\left (c x^{3} + 3 \, a x\right )} {\left (\frac {1}{3} \, c x^{3} + a x\right )}^{n} + 3 \, {\left (a n + a\right )} x}{3 \, {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 30, normalized size = 0.88 \[ \frac {1}{3} \, c x^{3} + a x + \frac {{\left (\frac {1}{3} \, c x^{3} + a x\right )}^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 31, normalized size = 0.91 \[ \frac {c \,x^{3}}{3}+a x +\frac {\left (\frac {1}{3} c \,x^{3}+a x \right )^{n +1}}{n +1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 54, normalized size = 1.59 \[ \frac {1}{3} \, c x^{3} + a x + \frac {{\left (c x^{3} + 3 \, a x\right )} e^{\left (n \log \left (c x^{2} + 3 \, a\right ) + n \log \relax (x)\right )}}{3^{n + 1} n + 3^{n + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.11, size = 33, normalized size = 0.97 \[ \frac {x\,\left (c\,x^2+3\,a\right )\,\left (n+{\left (\frac {c\,x^3}{3}+a\,x\right )}^n+1\right )}{3\,\left (n+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 112.00, size = 201, normalized size = 5.91 \[ \begin {cases} \frac {3 \cdot 3^{n} a n x}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac {3 \cdot 3^{n} a x}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac {3^{n} c n x^{3}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac {3^{n} c x^{3}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac {3 a x \left (3 a x + c x^{3}\right )^{n}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} + \frac {c x^{3} \left (3 a x + c x^{3}\right )^{n}}{3 \cdot 3^{n} n + 3 \cdot 3^{n}} & \text {for}\: n \neq -1 \\a x + \frac {c x^{3}}{3} + \log {\relax (x )} + \log {\left (- \sqrt {3} i \sqrt {a} \sqrt {\frac {1}{c}} + x \right )} + \log {\left (\sqrt {3} i \sqrt {a} \sqrt {\frac {1}{c}} + x \right )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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