Optimal. Leaf size=30 \[ \frac {1}{6} \left (a x+\frac {c x^3}{3}\right )^6+a x+\frac {c x^3}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 30, 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} \[ \frac {1}{6} \left (a x+\frac {c x^3}{3}\right )^6+a x+\frac {c x^3}{3} \]
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 )^5\right ) \, dx &=\operatorname {Subst}\left (\int \left (1+x^5\right ) \, dx,x,a x+\frac {c x^3}{3}\right )\\ &=a x+\frac {c x^3}{3}+\frac {1}{6} \left (a x+\frac {c x^3}{3}\right )^6\\ \end {align*}
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Mathematica [B] time = 0.01, size = 93, normalized size = 3.10 \[ \frac {a^6 x^6}{6}+\frac {1}{3} a^5 c x^8+\frac {5}{18} a^4 c^2 x^{10}+\frac {10}{81} a^3 c^3 x^{12}+\frac {5}{162} a^2 c^4 x^{14}+\frac {1}{243} a c^5 x^{16}+a x+\frac {c^6 x^{18}}{4374}+\frac {c x^3}{3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 77, normalized size = 2.57 \[ \frac {1}{4374} x^{18} c^{6} + \frac {1}{243} x^{16} c^{5} a + \frac {5}{162} x^{14} c^{4} a^{2} + \frac {10}{81} x^{12} c^{3} a^{3} + \frac {5}{18} x^{10} c^{2} a^{4} + \frac {1}{3} x^{8} c a^{5} + \frac {1}{6} x^{6} a^{6} + \frac {1}{3} x^{3} c + x a \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 24, normalized size = 0.80 \[ \frac {1}{4374} \, {\left (c x^{3} + 3 \, a x\right )}^{6} + \frac {1}{3} \, c x^{3} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 78, normalized size = 2.60 \[ \frac {1}{4374} c^{6} x^{18}+\frac {1}{243} a \,c^{5} x^{16}+\frac {5}{162} a^{2} c^{4} x^{14}+\frac {10}{81} a^{3} c^{3} x^{12}+\frac {5}{18} a^{4} c^{2} x^{10}+\frac {1}{3} a^{5} c \,x^{8}+\frac {1}{6} a^{6} x^{6}+\frac {1}{3} c \,x^{3}+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 77, normalized size = 2.57 \[ \frac {1}{4374} \, c^{6} x^{18} + \frac {1}{243} \, a c^{5} x^{16} + \frac {5}{162} \, a^{2} c^{4} x^{14} + \frac {10}{81} \, a^{3} c^{3} x^{12} + \frac {5}{18} \, a^{4} c^{2} x^{10} + \frac {1}{3} \, a^{5} c x^{8} + \frac {1}{6} \, a^{6} x^{6} + \frac {1}{3} \, c x^{3} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 77, normalized size = 2.57 \[ \frac {a^6\,x^6}{6}+\frac {a^5\,c\,x^8}{3}+\frac {5\,a^4\,c^2\,x^{10}}{18}+\frac {10\,a^3\,c^3\,x^{12}}{81}+\frac {5\,a^2\,c^4\,x^{14}}{162}+\frac {a\,c^5\,x^{16}}{243}+a\,x+\frac {c^6\,x^{18}}{4374}+\frac {c\,x^3}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.09, size = 87, normalized size = 2.90 \[ \frac {a^{6} x^{6}}{6} + \frac {a^{5} c x^{8}}{3} + \frac {5 a^{4} c^{2} x^{10}}{18} + \frac {10 a^{3} c^{3} x^{12}}{81} + \frac {5 a^{2} c^{4} x^{14}}{162} + \frac {a c^{5} x^{16}}{243} + a x + \frac {c^{6} x^{18}}{4374} + \frac {c x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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