Optimal. Leaf size=31 \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}+d\right )^6+a x+\frac{c x^3}{3} \]
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Rubi [A] time = 0.0384825, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1591} \[ \frac{1}{6} \left (a x+\frac{c x^3}{3}+d\right )^6+a x+\frac{c x^3}{3} \]
Antiderivative was successfully verified.
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Rule 1591
Rubi steps
\begin{align*} \int \left (a+c x^2\right ) \left (1+\left (d+a x+\frac{c x^3}{3}\right )^5\right ) \, dx &=\operatorname{Subst}\left (\int \left (1+x^5\right ) \, dx,x,d+a x+\frac{c x^3}{3}\right )\\ &=a x+\frac{c x^3}{3}+\frac{1}{6} \left (d+a x+\frac{c x^3}{3}\right )^6\\ \end{align*}
Mathematica [B] time = 0.0460005, size = 140, normalized size = 4.52 \[ \frac{x \left (3 a+c x^2\right ) \left (90 a^2 c^3 x^{11}+270 a^3 c^2 x^9+405 a^4 c x^7+243 a^5 x^5+15 a c^4 x^{13}+135 d^2 \left (3 a x+c x^3\right )^3+540 d^3 \left (3 a x+c x^3\right )^2+1215 d^4 \left (3 a x+c x^3\right )+18 d \left (3 a x+c x^3\right )^4+c^5 x^{15}+1458 d^5+1458\right )}{4374} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.002, size = 618, normalized size = 19.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06855, size = 378, normalized size = 12.19 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{1}{243} \, c^{5} d x^{15} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{5}{81} \, a c^{4} d x^{13} + \frac{10}{27} \, a^{2} c^{3} d x^{11} + \frac{5}{162} \,{\left (4 \, a^{3} c^{3} + c^{4} d^{2}\right )} x^{12} + \frac{5}{54} \,{\left (3 \, a^{4} c^{2} + 4 \, a c^{3} d^{2}\right )} x^{10} + \frac{10}{81} \,{\left (9 \, a^{3} c^{2} d + c^{3} d^{3}\right )} x^{9} + \frac{1}{3} \,{\left (a^{5} c + 5 \, a^{2} c^{2} d^{2}\right )} x^{8} + \frac{5}{2} \, a^{2} d^{4} x^{2} + \frac{5}{9} \,{\left (3 \, a^{4} c d + 2 \, a c^{2} d^{3}\right )} x^{7} + \frac{1}{18} \,{\left (3 \, a^{6} + 60 \, a^{3} c d^{2} + 5 \, c^{2} d^{4}\right )} x^{6} + \frac{1}{3} \,{\left (3 \, a^{5} d + 10 \, a^{2} c d^{3}\right )} x^{5} + \frac{5}{6} \,{\left (3 \, a^{4} d^{2} + 2 \, a c d^{4}\right )} x^{4} + \frac{1}{3} \,{\left (10 \, a^{3} d^{3} + c d^{5} + c\right )} x^{3} +{\left (a d^{5} + a\right )} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.16685, size = 710, normalized size = 22.9 \begin{align*} \frac{1}{4374} x^{18} c^{6} + \frac{1}{243} x^{16} c^{5} a + \frac{1}{243} x^{15} d c^{5} + \frac{5}{162} x^{14} c^{4} a^{2} + \frac{5}{81} x^{13} d c^{4} a + \frac{5}{162} x^{12} d^{2} c^{4} + \frac{10}{81} x^{12} c^{3} a^{3} + \frac{10}{27} x^{11} d c^{3} a^{2} + \frac{10}{27} x^{10} d^{2} c^{3} a + \frac{5}{18} x^{10} c^{2} a^{4} + \frac{10}{81} x^{9} d^{3} c^{3} + \frac{10}{9} x^{9} d c^{2} a^{3} + \frac{5}{3} x^{8} d^{2} c^{2} a^{2} + \frac{1}{3} x^{8} c a^{5} + \frac{10}{9} x^{7} d^{3} c^{2} a + \frac{5}{3} x^{7} d c a^{4} + \frac{5}{18} x^{6} d^{4} c^{2} + \frac{10}{3} x^{6} d^{2} c a^{3} + \frac{1}{6} x^{6} a^{6} + \frac{10}{3} x^{5} d^{3} c a^{2} + x^{5} d a^{5} + \frac{5}{3} x^{4} d^{4} c a + \frac{5}{2} x^{4} d^{2} a^{4} + \frac{1}{3} x^{3} d^{5} c + \frac{10}{3} x^{3} d^{3} a^{3} + \frac{5}{2} x^{2} d^{4} a^{2} + x d^{5} a + \frac{1}{3} x^{3} c + x a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.130175, size = 314, normalized size = 10.13 \begin{align*} \frac{5 a^{2} c^{4} x^{14}}{162} + \frac{10 a^{2} c^{3} d x^{11}}{27} + \frac{5 a^{2} d^{4} x^{2}}{2} + \frac{a c^{5} x^{16}}{243} + \frac{5 a c^{4} d x^{13}}{81} + \frac{c^{6} x^{18}}{4374} + \frac{c^{5} d x^{15}}{243} + x^{12} \left (\frac{10 a^{3} c^{3}}{81} + \frac{5 c^{4} d^{2}}{162}\right ) + x^{10} \left (\frac{5 a^{4} c^{2}}{18} + \frac{10 a c^{3} d^{2}}{27}\right ) + x^{9} \left (\frac{10 a^{3} c^{2} d}{9} + \frac{10 c^{3} d^{3}}{81}\right ) + x^{8} \left (\frac{a^{5} c}{3} + \frac{5 a^{2} c^{2} d^{2}}{3}\right ) + x^{7} \left (\frac{5 a^{4} c d}{3} + \frac{10 a c^{2} d^{3}}{9}\right ) + x^{6} \left (\frac{a^{6}}{6} + \frac{10 a^{3} c d^{2}}{3} + \frac{5 c^{2} d^{4}}{18}\right ) + x^{5} \left (a^{5} d + \frac{10 a^{2} c d^{3}}{3}\right ) + x^{4} \left (\frac{5 a^{4} d^{2}}{2} + \frac{5 a c d^{4}}{3}\right ) + x^{3} \left (\frac{10 a^{3} d^{3}}{3} + \frac{c d^{5}}{3} + \frac{c}{3}\right ) + x \left (a d^{5} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.23805, size = 393, normalized size = 12.68 \begin{align*} \frac{1}{4374} \, c^{6} x^{18} + \frac{1}{243} \, a c^{5} x^{16} + \frac{1}{243} \, c^{5} d x^{15} + \frac{5}{162} \, a^{2} c^{4} x^{14} + \frac{5}{81} \, a c^{4} d x^{13} + \frac{10}{81} \, a^{3} c^{3} x^{12} + \frac{5}{162} \, c^{4} d^{2} x^{12} + \frac{10}{27} \, a^{2} c^{3} d x^{11} + \frac{5}{18} \, a^{4} c^{2} x^{10} + \frac{10}{27} \, a c^{3} d^{2} x^{10} + \frac{10}{9} \, a^{3} c^{2} d x^{9} + \frac{10}{81} \, c^{3} d^{3} x^{9} + \frac{1}{3} \, a^{5} c x^{8} + \frac{5}{3} \, a^{2} c^{2} d^{2} x^{8} + \frac{5}{3} \, a^{4} c d x^{7} + \frac{10}{9} \, a c^{2} d^{3} x^{7} + \frac{1}{6} \, a^{6} x^{6} + \frac{10}{3} \, a^{3} c d^{2} x^{6} + \frac{5}{18} \, c^{2} d^{4} x^{6} + a^{5} d x^{5} + \frac{10}{3} \, a^{2} c d^{3} x^{5} + \frac{5}{2} \, a^{4} d^{2} x^{4} + \frac{5}{3} \, a c d^{4} x^{4} + \frac{10}{3} \, a^{3} d^{3} x^{3} + \frac{1}{3} \, c d^{5} x^{3} + \frac{5}{2} \, a^{2} d^{4} x^{2} + a d^{5} x + \frac{1}{3} \, c x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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