Optimal. Leaf size=17 \[ \frac {c^3 (a+b x)^9}{9 b} \]
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Rubi [A] time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {21, 32} \begin {gather*} \frac {c^3 (a+b x)^9}{9 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 32
Rubi steps
\begin {align*} \int (a+b x)^5 (a c+b c x)^3 \, dx &=c^3 \int (a+b x)^8 \, dx\\ &=\frac {c^3 (a+b x)^9}{9 b}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^3 (a+b x)^9}{9 b} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b x)^5 (a c+b c x)^3 \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 1.25, size = 113, normalized size = 6.65 \begin {gather*} \frac {1}{9} x^{9} c^{3} b^{8} + x^{8} c^{3} b^{7} a + 4 x^{7} c^{3} b^{6} a^{2} + \frac {28}{3} x^{6} c^{3} b^{5} a^{3} + 14 x^{5} c^{3} b^{4} a^{4} + 14 x^{4} c^{3} b^{3} a^{5} + \frac {28}{3} x^{3} c^{3} b^{2} a^{6} + 4 x^{2} c^{3} b a^{7} + x c^{3} a^{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.96, size = 113, normalized size = 6.65 \begin {gather*} \frac {1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 114, normalized size = 6.71 \begin {gather*} \frac {1}{9} b^{8} c^{3} x^{9}+a \,b^{7} c^{3} x^{8}+4 a^{2} b^{6} c^{3} x^{7}+\frac {28}{3} a^{3} b^{5} c^{3} x^{6}+14 a^{4} b^{4} c^{3} x^{5}+14 a^{5} b^{3} c^{3} x^{4}+\frac {28}{3} a^{6} b^{2} c^{3} x^{3}+4 a^{7} b \,c^{3} x^{2}+a^{8} c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.31, size = 113, normalized size = 6.65 \begin {gather*} \frac {1}{9} \, b^{8} c^{3} x^{9} + a b^{7} c^{3} x^{8} + 4 \, a^{2} b^{6} c^{3} x^{7} + \frac {28}{3} \, a^{3} b^{5} c^{3} x^{6} + 14 \, a^{4} b^{4} c^{3} x^{5} + 14 \, a^{5} b^{3} c^{3} x^{4} + \frac {28}{3} \, a^{6} b^{2} c^{3} x^{3} + 4 \, a^{7} b c^{3} x^{2} + a^{8} c^{3} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 113, normalized size = 6.65 \begin {gather*} a^8\,c^3\,x+4\,a^7\,b\,c^3\,x^2+\frac {28\,a^6\,b^2\,c^3\,x^3}{3}+14\,a^5\,b^3\,c^3\,x^4+14\,a^4\,b^4\,c^3\,x^5+\frac {28\,a^3\,b^5\,c^3\,x^6}{3}+4\,a^2\,b^6\,c^3\,x^7+a\,b^7\,c^3\,x^8+\frac {b^8\,c^3\,x^9}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 124, normalized size = 7.29 \begin {gather*} a^{8} c^{3} x + 4 a^{7} b c^{3} x^{2} + \frac {28 a^{6} b^{2} c^{3} x^{3}}{3} + 14 a^{5} b^{3} c^{3} x^{4} + 14 a^{4} b^{4} c^{3} x^{5} + \frac {28 a^{3} b^{5} c^{3} x^{6}}{3} + 4 a^{2} b^{6} c^{3} x^{7} + a b^{7} c^{3} x^{8} + \frac {b^{8} c^{3} x^{9}}{9} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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