Optimal. Leaf size=84 \[ -\frac {3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac {3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac {b^3 x \left (b^2-3 a c\right )^3}{c^3}+\frac {(b+c x)^{10}}{10 c^4} \]
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Rubi [A] time = 0.12, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2060, 194} \[ -\frac {3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac {3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac {b^3 x \left (b^2-3 a c\right )^3}{c^3}+\frac {(b+c x)^{10}}{10 c^4} \]
Antiderivative was successfully verified.
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Rule 194
Rule 2060
Rubi steps
\begin {align*} \int \left (3 a b+3 b^2 x+3 b c x^2+c^2 x^3\right )^3 \, dx &=\frac {1}{27} \operatorname {Subst}\left (\int \left (3 b \left (3 a-\frac {b^2}{c}\right )+3 c^2 x^3\right )^3 \, dx,x,\frac {b}{c}+x\right )\\ &=\frac {1}{27} \operatorname {Subst}\left (\int \left (\frac {27 \left (-b^3+3 a b c\right )^3}{c^3}+81 \left (b^3-3 a b c\right )^2 x^3-81 b c^3 \left (b^2-3 a c\right ) x^6+27 c^6 x^9\right ) \, dx,x,\frac {b}{c}+x\right )\\ &=-\frac {b^3 \left (b^2-3 a c\right )^3 x}{c^3}+\frac {3 b^2 \left (b^2-3 a c\right )^2 (b+c x)^4}{4 c^4}-\frac {3 b \left (b^2-3 a c\right ) (b+c x)^7}{7 c^4}+\frac {(b+c x)^{10}}{10 c^4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 159, normalized size = 1.89 \[ 27 a^3 b^3 x+\frac {81}{2} a^2 b^4 x^2+\frac {27}{4} b^2 x^4 \left (a^2 c^2+6 a b^2 c+b^4\right )+\frac {9}{7} b c^3 x^7 \left (a c+9 b^2\right )+9 b^2 c^2 x^6 \left (a c+2 b^2\right )+\frac {27}{5} b^3 c x^5 \left (5 a c+3 b^2\right )+27 a b^3 x^3 \left (a c+b^2\right )+\frac {9}{2} b^2 c^4 x^8+b c^5 x^9+\frac {c^6 x^{10}}{10} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 166, normalized size = 1.98 \[ \frac {1}{10} x^{10} c^{6} + x^{9} c^{5} b + \frac {9}{2} x^{8} c^{4} b^{2} + \frac {81}{7} x^{7} c^{3} b^{3} + \frac {9}{7} x^{7} c^{4} b a + 18 x^{6} c^{2} b^{4} + 9 x^{6} c^{3} b^{2} a + \frac {81}{5} x^{5} c b^{5} + 27 x^{5} c^{2} b^{3} a + \frac {27}{4} x^{4} b^{6} + \frac {81}{2} x^{4} c b^{4} a + \frac {27}{4} x^{4} c^{2} b^{2} a^{2} + 27 x^{3} b^{5} a + 27 x^{3} c b^{3} a^{2} + \frac {81}{2} x^{2} b^{4} a^{2} + 27 x b^{3} a^{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.27, size = 166, normalized size = 1.98 \[ \frac {1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac {9}{2} \, b^{2} c^{4} x^{8} + \frac {81}{7} \, b^{3} c^{3} x^{7} + \frac {9}{7} \, a b c^{4} x^{7} + 18 \, b^{4} c^{2} x^{6} + 9 \, a b^{2} c^{3} x^{6} + \frac {81}{5} \, b^{5} c x^{5} + 27 \, a b^{3} c^{2} x^{5} + \frac {27}{4} \, b^{6} x^{4} + \frac {81}{2} \, a b^{4} c x^{4} + \frac {27}{4} \, a^{2} b^{2} c^{2} x^{4} + 27 \, a b^{5} x^{3} + 27 \, a^{2} b^{3} c x^{3} + \frac {81}{2} \, a^{2} b^{4} x^{2} + 27 \, a^{3} b^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 295, normalized size = 3.51 \[ \frac {c^{6} x^{10}}{10}+b \,c^{5} x^{9}+\frac {9 b^{2} c^{4} x^{8}}{2}+\frac {81 a^{2} b^{4} x^{2}}{2}+27 a^{3} b^{3} x +\frac {\left (3 a b \,c^{4}+63 b^{3} c^{3}+\left (6 a b \,c^{2}+18 b^{3} c \right ) c^{2}\right ) x^{7}}{7}+\frac {\left (18 a \,b^{2} c^{3}+45 b^{4} c^{2}+3 \left (6 a b \,c^{2}+18 b^{3} c \right ) b c +\left (18 a \,b^{2} c +9 b^{4}\right ) c^{2}\right ) x^{6}}{6}+\frac {\left (63 a \,b^{3} c^{2}+3 \left (6 a b \,c^{2}+18 b^{3} c \right ) b^{2}+3 \left (18 a \,b^{2} c +9 b^{4}\right ) b c \right ) x^{5}}{5}+\frac {\left (9 a^{2} b^{2} c^{2}+54 a \,b^{4} c +3 \left (6 a b \,c^{2}+18 b^{3} c \right ) a b +3 \left (18 a \,b^{2} c +9 b^{4}\right ) b^{2}\right ) x^{4}}{4}+\frac {\left (27 a^{2} b^{3} c +54 a \,b^{5}+3 \left (18 a \,b^{2} c +9 b^{4}\right ) a b \right ) x^{3}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 204, normalized size = 2.43 \[ \frac {1}{10} \, c^{6} x^{10} + b c^{5} x^{9} + \frac {27}{8} \, b^{2} c^{4} x^{8} + \frac {27}{7} \, b^{3} c^{3} x^{7} + \frac {27}{4} \, b^{6} x^{4} + 27 \, a^{3} b^{3} x + \frac {27}{4} \, {\left (c^{2} x^{4} + 4 \, b c x^{3} + 6 \, b^{2} x^{2}\right )} a^{2} b^{2} + \frac {9}{10} \, {\left (5 \, c^{2} x^{6} + 18 \, b c x^{5}\right )} b^{4} + \frac {9}{70} \, {\left (10 \, c^{4} x^{7} + 70 \, b c^{3} x^{6} + 126 \, b^{2} c^{2} x^{5} + 210 \, b^{4} x^{3} + 21 \, {\left (4 \, c^{2} x^{5} + 15 \, b c x^{4}\right )} b^{2}\right )} a b + \frac {9}{56} \, {\left (7 \, c^{4} x^{8} + 48 \, b c^{3} x^{7} + 84 \, b^{2} c^{2} x^{6}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.08, size = 149, normalized size = 1.77 \[ x^4\,\left (\frac {27\,a^2\,b^2\,c^2}{4}+\frac {81\,a\,b^4\,c}{2}+\frac {27\,b^6}{4}\right )+\frac {c^6\,x^{10}}{10}+27\,a^3\,b^3\,x+b\,c^5\,x^9+\frac {81\,a^2\,b^4\,x^2}{2}+\frac {9\,b^2\,c^4\,x^8}{2}+9\,b^2\,c^2\,x^6\,\left (2\,b^2+a\,c\right )+27\,a\,b^3\,x^3\,\left (b^2+a\,c\right )+\frac {27\,b^3\,c\,x^5\,\left (3\,b^2+5\,a\,c\right )}{5}+\frac {9\,b\,c^3\,x^7\,\left (9\,b^2+a\,c\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 175, normalized size = 2.08 \[ 27 a^{3} b^{3} x + \frac {81 a^{2} b^{4} x^{2}}{2} + \frac {9 b^{2} c^{4} x^{8}}{2} + b c^{5} x^{9} + \frac {c^{6} x^{10}}{10} + x^{7} \left (\frac {9 a b c^{4}}{7} + \frac {81 b^{3} c^{3}}{7}\right ) + x^{6} \left (9 a b^{2} c^{3} + 18 b^{4} c^{2}\right ) + x^{5} \left (27 a b^{3} c^{2} + \frac {81 b^{5} c}{5}\right ) + x^{4} \left (\frac {27 a^{2} b^{2} c^{2}}{4} + \frac {81 a b^{4} c}{2} + \frac {27 b^{6}}{4}\right ) + x^{3} \left (27 a^{2} b^{3} c + 27 a b^{5}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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