Optimal. Leaf size=24 \[ \frac {(a c+b c x)^{n+6}}{b c^6 (n+6)} \]
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Rubi [A] time = 0.01, antiderivative size = 24, 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} \[ \frac {(a c+b c x)^{n+6}}{b c^6 (n+6)} \]
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)^n \, dx &=\frac {\int (a c+b c x)^{5+n} \, dx}{c^5}\\ &=\frac {(a c+b c x)^{6+n}}{b c^6 (6+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.04 \[ \frac {(a+b x)^6 (c (a+b x))^n}{b (n+6)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 80, normalized size = 3.33 \[ \frac {{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + 15 \, a^{2} b^{4} x^{4} + 20 \, a^{3} b^{3} x^{3} + 15 \, a^{4} b^{2} x^{2} + 6 \, a^{5} b x + a^{6}\right )} {\left (b c x + a c\right )}^{n}}{b n + 6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.13, size = 141, normalized size = 5.88 \[ \frac {{\left (b c x + a c\right )}^{n} b^{6} x^{6} + 6 \, {\left (b c x + a c\right )}^{n} a b^{5} x^{5} + 15 \, {\left (b c x + a c\right )}^{n} a^{2} b^{4} x^{4} + 20 \, {\left (b c x + a c\right )}^{n} a^{3} b^{3} x^{3} + 15 \, {\left (b c x + a c\right )}^{n} a^{4} b^{2} x^{2} + 6 \, {\left (b c x + a c\right )}^{n} a^{5} b x + {\left (b c x + a c\right )}^{n} a^{6}}{b n + 6 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 27, normalized size = 1.12 \[ \frac {\left (b x +a \right )^{6} \left (b c x +a c \right )^{n}}{\left (n +6\right ) b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.81, size = 649, normalized size = 27.04 \[ \frac {5 \, {\left (b^{2} c^{n} {\left (n + 1\right )} x^{2} + a b c^{n} n x - a^{2} c^{n}\right )} {\left (b x + a\right )}^{n} a^{4}}{{\left (n^{2} + 3 \, n + 2\right )} b} + \frac {10 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} b^{3} c^{n} x^{3} + {\left (n^{2} + n\right )} a b^{2} c^{n} x^{2} - 2 \, a^{2} b c^{n} n x + 2 \, a^{3} c^{n}\right )} {\left (b x + a\right )}^{n} a^{3}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b} + \frac {{\left (b c x + a c\right )}^{n + 1} a^{5}}{b c {\left (n + 1\right )}} + \frac {10 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} b^{4} c^{n} x^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a b^{3} c^{n} x^{3} - 3 \, {\left (n^{2} + n\right )} a^{2} b^{2} c^{n} x^{2} + 6 \, a^{3} b c^{n} n x - 6 \, a^{4} c^{n}\right )} {\left (b x + a\right )}^{n} a^{2}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b} + \frac {5 \, {\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} b^{5} c^{n} x^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a b^{4} c^{n} x^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{2} b^{3} c^{n} x^{3} + 12 \, {\left (n^{2} + n\right )} a^{3} b^{2} c^{n} x^{2} - 24 \, a^{4} b c^{n} n x + 24 \, a^{5} c^{n}\right )} {\left (b x + a\right )}^{n} a}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b} + \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} b^{6} c^{n} x^{6} + {\left (n^{5} + 10 \, n^{4} + 35 \, n^{3} + 50 \, n^{2} + 24 \, n\right )} a b^{5} c^{n} x^{5} - 5 \, {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} a^{2} b^{4} c^{n} x^{4} + 20 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} a^{3} b^{3} c^{n} x^{3} - 60 \, {\left (n^{2} + n\right )} a^{4} b^{2} c^{n} x^{2} + 120 \, a^{5} b c^{n} n x - 120 \, a^{6} c^{n}\right )} {\left (b x + a\right )}^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 107, normalized size = 4.46 \[ {\left (a\,c+b\,c\,x\right )}^n\,\left (\frac {a^6}{b\,\left (n+6\right )}+\frac {b^5\,x^6}{n+6}+\frac {6\,a^5\,x}{n+6}+\frac {15\,a^4\,b\,x^2}{n+6}+\frac {6\,a\,b^4\,x^5}{n+6}+\frac {20\,a^3\,b^2\,x^3}{n+6}+\frac {15\,a^2\,b^3\,x^4}{n+6}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.29, size = 212, normalized size = 8.83 \[ \begin {cases} \frac {x}{a c^{6}} & \text {for}\: b = 0 \wedge n = -6 \\a^{5} x \left (a c\right )^{n} & \text {for}\: b = 0 \\\frac {\log {\left (\frac {a}{b} + x \right )}}{b c^{6}} & \text {for}\: n = -6 \\\frac {a^{6} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {6 a^{5} b x \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {15 a^{4} b^{2} x^{2} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {20 a^{3} b^{3} x^{3} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {15 a^{2} b^{4} x^{4} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {6 a b^{5} x^{5} \left (a c + b c x\right )^{n}}{b n + 6 b} + \frac {b^{6} x^{6} \left (a c + b c x\right )^{n}}{b n + 6 b} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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