Optimal. Leaf size=89 \[ -\frac {c^6 (a-b x)^7}{10 x^{10}}-\frac {13 b c^6 (a-b x)^7}{90 a x^9}-\frac {13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac {13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7} \]
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Rubi [A]
time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {79, 47, 37}
\begin {gather*} -\frac {13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}-\frac {13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac {c^6 (a-b x)^7}{10 x^{10}}-\frac {13 b c^6 (a-b x)^7}{90 a x^9} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rubi steps
\begin {align*} \int \frac {(a+b x) (a c-b c x)^6}{x^{11}} \, dx &=-\frac {c^6 (a-b x)^7}{10 x^{10}}+\frac {1}{10} (13 b) \int \frac {(a c-b c x)^6}{x^{10}} \, dx\\ &=-\frac {c^6 (a-b x)^7}{10 x^{10}}-\frac {13 b c^6 (a-b x)^7}{90 a x^9}+\frac {\left (13 b^2\right ) \int \frac {(a c-b c x)^6}{x^9} \, dx}{45 a}\\ &=-\frac {c^6 (a-b x)^7}{10 x^{10}}-\frac {13 b c^6 (a-b x)^7}{90 a x^9}-\frac {13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}+\frac {\left (13 b^3\right ) \int \frac {(a c-b c x)^6}{x^8} \, dx}{360 a^2}\\ &=-\frac {c^6 (a-b x)^7}{10 x^{10}}-\frac {13 b c^6 (a-b x)^7}{90 a x^9}-\frac {13 b^2 c^6 (a-b x)^7}{360 a^2 x^8}-\frac {13 b^3 c^6 (a-b x)^7}{2520 a^3 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 119, normalized size = 1.34 \begin {gather*} -\frac {a^7 c^6}{10 x^{10}}+\frac {5 a^6 b c^6}{9 x^9}-\frac {9 a^5 b^2 c^6}{8 x^8}+\frac {5 a^4 b^3 c^6}{7 x^7}+\frac {5 a^3 b^4 c^6}{6 x^6}-\frac {9 a^2 b^5 c^6}{5 x^5}+\frac {5 a b^6 c^6}{4 x^4}-\frac {b^7 c^6}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 84, normalized size = 0.94
method | result | size |
gosper | \(-\frac {c^{6} \left (840 b^{7} x^{7}-3150 a \,b^{6} x^{6}+4536 a^{2} b^{5} x^{5}-2100 a^{3} b^{4} x^{4}-1800 a^{4} b^{3} x^{3}+2835 a^{5} b^{2} x^{2}-1400 a^{6} b x +252 a^{7}\right )}{2520 x^{10}}\) | \(83\) |
default | \(c^{6} \left (\frac {5 a^{6} b}{9 x^{9}}-\frac {9 a^{2} b^{5}}{5 x^{5}}+\frac {5 a \,b^{6}}{4 x^{4}}+\frac {5 a^{3} b^{4}}{6 x^{6}}-\frac {a^{7}}{10 x^{10}}-\frac {b^{7}}{3 x^{3}}-\frac {9 a^{5} b^{2}}{8 x^{8}}+\frac {5 a^{4} b^{3}}{7 x^{7}}\right )\) | \(84\) |
norman | \(\frac {-\frac {1}{10} a^{7} c^{6}-\frac {1}{3} b^{7} c^{6} x^{7}+\frac {5}{4} a \,b^{6} c^{6} x^{6}-\frac {9}{5} a^{2} b^{5} c^{6} x^{5}+\frac {5}{6} a^{3} b^{4} c^{6} x^{4}+\frac {5}{7} a^{4} b^{3} c^{6} x^{3}-\frac {9}{8} a^{5} b^{2} c^{6} x^{2}+\frac {5}{9} a^{6} b \,c^{6} x}{x^{10}}\) | \(103\) |
risch | \(\frac {-\frac {1}{10} a^{7} c^{6}-\frac {1}{3} b^{7} c^{6} x^{7}+\frac {5}{4} a \,b^{6} c^{6} x^{6}-\frac {9}{5} a^{2} b^{5} c^{6} x^{5}+\frac {5}{6} a^{3} b^{4} c^{6} x^{4}+\frac {5}{7} a^{4} b^{3} c^{6} x^{3}-\frac {9}{8} a^{5} b^{2} c^{6} x^{2}+\frac {5}{9} a^{6} b \,c^{6} x}{x^{10}}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 103, normalized size = 1.16 \begin {gather*} -\frac {840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.60, size = 103, normalized size = 1.16 \begin {gather*} -\frac {840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 110, normalized size = 1.24 \begin {gather*} \frac {- 252 a^{7} c^{6} + 1400 a^{6} b c^{6} x - 2835 a^{5} b^{2} c^{6} x^{2} + 1800 a^{4} b^{3} c^{6} x^{3} + 2100 a^{3} b^{4} c^{6} x^{4} - 4536 a^{2} b^{5} c^{6} x^{5} + 3150 a b^{6} c^{6} x^{6} - 840 b^{7} c^{6} x^{7}}{2520 x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.87, size = 103, normalized size = 1.16 \begin {gather*} -\frac {840 \, b^{7} c^{6} x^{7} - 3150 \, a b^{6} c^{6} x^{6} + 4536 \, a^{2} b^{5} c^{6} x^{5} - 2100 \, a^{3} b^{4} c^{6} x^{4} - 1800 \, a^{4} b^{3} c^{6} x^{3} + 2835 \, a^{5} b^{2} c^{6} x^{2} - 1400 \, a^{6} b c^{6} x + 252 \, a^{7} c^{6}}{2520 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 103, normalized size = 1.16 \begin {gather*} -\frac {\frac {a^7\,c^6}{10}-\frac {5\,a^6\,b\,c^6\,x}{9}+\frac {9\,a^5\,b^2\,c^6\,x^2}{8}-\frac {5\,a^4\,b^3\,c^6\,x^3}{7}-\frac {5\,a^3\,b^4\,c^6\,x^4}{6}+\frac {9\,a^2\,b^5\,c^6\,x^5}{5}-\frac {5\,a\,b^6\,c^6\,x^6}{4}+\frac {b^7\,c^6\,x^7}{3}}{x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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