Optimal. Leaf size=17 \[ \frac {c^2 (a+b x)^8}{8 b} \]
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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^2 (a+b x)^8}{8 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)^2 \, dx &=c^2 \int (a+b x)^7 \, dx\\ &=\frac {c^2 (a+b x)^8}{8 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {c^2 (a+b x)^8}{8 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs.
\(2(15)=30\).
time = 0.14, size = 100, normalized size = 5.88
method | result | size |
gosper | \(\frac {x \left (b^{7} x^{7}+8 a \,b^{6} x^{6}+28 a^{2} b^{5} x^{5}+56 a^{3} b^{4} x^{4}+70 a^{4} b^{3} x^{3}+56 a^{5} b^{2} x^{2}+28 a^{6} b x +8 a^{7}\right ) c^{2}}{8}\) | \(80\) |
default | \(\frac {1}{8} b^{7} c^{2} x^{8}+a \,b^{6} c^{2} x^{7}+\frac {7}{2} a^{2} b^{5} c^{2} x^{6}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35}{4} a^{4} b^{3} c^{2} x^{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7}{2} a^{6} b \,c^{2} x^{2}+a^{7} c^{2} x\) | \(100\) |
norman | \(\frac {1}{8} b^{7} c^{2} x^{8}+a \,b^{6} c^{2} x^{7}+\frac {7}{2} a^{2} b^{5} c^{2} x^{6}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35}{4} a^{4} b^{3} c^{2} x^{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7}{2} a^{6} b \,c^{2} x^{2}+a^{7} c^{2} x\) | \(100\) |
risch | \(\frac {b^{7} c^{2} x^{8}}{8}+a \,b^{6} c^{2} x^{7}+\frac {7 a^{2} b^{5} c^{2} x^{6}}{2}+7 a^{3} b^{4} c^{2} x^{5}+\frac {35 a^{4} b^{3} c^{2} x^{4}}{4}+7 a^{5} b^{2} c^{2} x^{3}+\frac {7 a^{6} b \,c^{2} x^{2}}{2}+a^{7} c^{2} x +\frac {c^{2} a^{8}}{8 b}\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (15) = 30\).
time = 0.29, size = 99, normalized size = 5.82 \begin {gather*} \frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (15) = 30\).
time = 0.61, size = 99, normalized size = 5.82 \begin {gather*} \frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 110 vs.
\(2 (12) = 24\).
time = 0.02, size = 110, normalized size = 6.47 \begin {gather*} a^{7} c^{2} x + \frac {7 a^{6} b c^{2} x^{2}}{2} + 7 a^{5} b^{2} c^{2} x^{3} + \frac {35 a^{4} b^{3} c^{2} x^{4}}{4} + 7 a^{3} b^{4} c^{2} x^{5} + \frac {7 a^{2} b^{5} c^{2} x^{6}}{2} + a b^{6} c^{2} x^{7} + \frac {b^{7} c^{2} x^{8}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (15) = 30\).
time = 0.89, size = 99, normalized size = 5.82 \begin {gather*} \frac {1}{8} \, b^{7} c^{2} x^{8} + a b^{6} c^{2} x^{7} + \frac {7}{2} \, a^{2} b^{5} c^{2} x^{6} + 7 \, a^{3} b^{4} c^{2} x^{5} + \frac {35}{4} \, a^{4} b^{3} c^{2} x^{4} + 7 \, a^{5} b^{2} c^{2} x^{3} + \frac {7}{2} \, a^{6} b c^{2} x^{2} + a^{7} c^{2} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 99, normalized size = 5.82 \begin {gather*} a^7\,c^2\,x+\frac {7\,a^6\,b\,c^2\,x^2}{2}+7\,a^5\,b^2\,c^2\,x^3+\frac {35\,a^4\,b^3\,c^2\,x^4}{4}+7\,a^3\,b^4\,c^2\,x^5+\frac {7\,a^2\,b^5\,c^2\,x^6}{2}+a\,b^6\,c^2\,x^7+\frac {b^7\,c^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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