Optimal. Leaf size=89 \[ -\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \]
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Rubi [A]
time = 0.02, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} -\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {(a+b x)^7}{x^8} \, dx &=\int \left (\frac {a^7}{x^8}+\frac {7 a^6 b}{x^7}+\frac {21 a^5 b^2}{x^6}+\frac {35 a^4 b^3}{x^5}+\frac {35 a^3 b^4}{x^4}+\frac {21 a^2 b^5}{x^3}+\frac {7 a b^6}{x^2}+\frac {b^7}{x}\right ) \, dx\\ &=-\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 89, normalized size = 1.00 \begin {gather*} -\frac {a^7}{7 x^7}-\frac {7 a^6 b}{6 x^6}-\frac {21 a^5 b^2}{5 x^5}-\frac {35 a^4 b^3}{4 x^4}-\frac {35 a^3 b^4}{3 x^3}-\frac {21 a^2 b^5}{2 x^2}-\frac {7 a b^6}{x}+b^7 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.38, size = 80, normalized size = 0.90 \begin {gather*} \frac {-\frac {a \left (60 a^6+490 a^5 b x+1764 a^4 b^2 x^2+3675 a^3 b^3 x^3+4900 a^2 b^4 x^4+4410 a b^5 x^5+2940 b^6 x^6\right )}{420}+b^7 x^7 \text {Log}\left [x\right ]}{x^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 78, normalized size = 0.88
method | result | size |
default | \(-\frac {a^{7}}{7 x^{7}}-\frac {7 a^{6} b}{6 x^{6}}-\frac {21 a^{5} b^{2}}{5 x^{5}}-\frac {35 a^{4} b^{3}}{4 x^{4}}-\frac {35 a^{3} b^{4}}{3 x^{3}}-\frac {21 a^{2} b^{5}}{2 x^{2}}-\frac {7 a \,b^{6}}{x}+b^{7} \ln \left (x \right )\) | \(78\) |
norman | \(\frac {-\frac {1}{7} a^{7}-7 a \,b^{6} x^{6}-\frac {21}{2} a^{2} b^{5} x^{5}-\frac {35}{3} a^{3} b^{4} x^{4}-\frac {35}{4} a^{4} b^{3} x^{3}-\frac {21}{5} a^{5} b^{2} x^{2}-\frac {7}{6} a^{6} b x}{x^{7}}+b^{7} \ln \left (x \right )\) | \(78\) |
risch | \(\frac {-\frac {1}{7} a^{7}-7 a \,b^{6} x^{6}-\frac {21}{2} a^{2} b^{5} x^{5}-\frac {35}{3} a^{3} b^{4} x^{4}-\frac {35}{4} a^{4} b^{3} x^{3}-\frac {21}{5} a^{5} b^{2} x^{2}-\frac {7}{6} a^{6} b x}{x^{7}}+b^{7} \ln \left (x \right )\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 78, normalized size = 0.88 \begin {gather*} b^{7} \log \left (x\right ) - \frac {2940 \, a b^{6} x^{6} + 4410 \, a^{2} b^{5} x^{5} + 4900 \, a^{3} b^{4} x^{4} + 3675 \, a^{4} b^{3} x^{3} + 1764 \, a^{5} b^{2} x^{2} + 490 \, a^{6} b x + 60 \, a^{7}}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.30, size = 81, normalized size = 0.91 \begin {gather*} \frac {420 \, b^{7} x^{7} \log \left (x\right ) - 2940 \, a b^{6} x^{6} - 4410 \, a^{2} b^{5} x^{5} - 4900 \, a^{3} b^{4} x^{4} - 3675 \, a^{4} b^{3} x^{3} - 1764 \, a^{5} b^{2} x^{2} - 490 \, a^{6} b x - 60 \, a^{7}}{420 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.26, size = 83, normalized size = 0.93 \begin {gather*} b^{7} \log {\left (x \right )} + \frac {- 60 a^{7} - 490 a^{6} b x - 1764 a^{5} b^{2} x^{2} - 3675 a^{4} b^{3} x^{3} - 4900 a^{3} b^{4} x^{4} - 4410 a^{2} b^{5} x^{5} - 2940 a b^{6} x^{6}}{420 x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 89, normalized size = 1.00 \begin {gather*} \frac {\frac {1}{420} \left (-2940 b^{6} a x^{6}-4410 b^{5} a^{2} x^{5}-4900 b^{4} a^{3} x^{4}-3675 b^{3} a^{4} x^{3}-1764 b^{2} a^{5} x^{2}-490 b a^{6} x-60 a^{7}\right )}{x^{7}}+b^{7} \ln \left |x\right | \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 78, normalized size = 0.88 \begin {gather*} b^7\,\ln \left (x\right )-\frac {\frac {a^7}{7}+\frac {7\,a^6\,b\,x}{6}+\frac {21\,a^5\,b^2\,x^2}{5}+\frac {35\,a^4\,b^3\,x^3}{4}+\frac {35\,a^3\,b^4\,x^4}{3}+\frac {21\,a^2\,b^5\,x^5}{2}+7\,a\,b^6\,x^6}{x^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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