Optimal. Leaf size=105 \[ -\frac {a^8}{21 x^{21}}-\frac {4 a^7 b}{9 x^{18}}-\frac {28 a^6 b^2}{15 x^{15}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{3 x^6}-\frac {28 a^2 b^6}{3 x^3}+\frac {b^8 x^3}{3}+8 a b^7 \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} -\frac {a^8}{21 x^{21}}-\frac {4 a^7 b}{9 x^{18}}-\frac {28 a^6 b^2}{15 x^{15}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{3 x^6}-\frac {28 a^2 b^6}{3 x^3}+8 a b^7 \log (x)+\frac {b^8 x^3}{3} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{22}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^8} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (b^8+\frac {a^8}{x^8}+\frac {8 a^7 b}{x^7}+\frac {28 a^6 b^2}{x^6}+\frac {56 a^5 b^3}{x^5}+\frac {70 a^4 b^4}{x^4}+\frac {56 a^3 b^5}{x^3}+\frac {28 a^2 b^6}{x^2}+\frac {8 a b^7}{x}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{21 x^{21}}-\frac {4 a^7 b}{9 x^{18}}-\frac {28 a^6 b^2}{15 x^{15}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{3 x^6}-\frac {28 a^2 b^6}{3 x^3}+\frac {b^8 x^3}{3}+8 a b^7 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 105, normalized size = 1.00 \begin {gather*} -\frac {a^8}{21 x^{21}}-\frac {4 a^7 b}{9 x^{18}}-\frac {28 a^6 b^2}{15 x^{15}}-\frac {14 a^5 b^3}{3 x^{12}}-\frac {70 a^4 b^4}{9 x^9}-\frac {28 a^3 b^5}{3 x^6}-\frac {28 a^2 b^6}{3 x^3}+\frac {b^8 x^3}{3}+8 a b^7 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 90, normalized size = 0.86
method | result | size |
default | \(-\frac {a^{8}}{21 x^{21}}-\frac {4 a^{7} b}{9 x^{18}}-\frac {28 a^{6} b^{2}}{15 x^{15}}-\frac {14 a^{5} b^{3}}{3 x^{12}}-\frac {70 a^{4} b^{4}}{9 x^{9}}-\frac {28 a^{3} b^{5}}{3 x^{6}}-\frac {28 a^{2} b^{6}}{3 x^{3}}+\frac {b^{8} x^{3}}{3}+8 a \,b^{7} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{21} a^{8}+\frac {1}{3} b^{8} x^{24}-\frac {28}{3} a^{2} b^{6} x^{18}-\frac {28}{3} a^{3} b^{5} x^{15}-\frac {70}{9} a^{4} b^{4} x^{12}-\frac {14}{3} a^{5} b^{3} x^{9}-\frac {28}{15} a^{6} b^{2} x^{6}-\frac {4}{9} a^{7} b \,x^{3}}{x^{21}}+8 a \,b^{7} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{3}}{3}+\frac {-\frac {1}{21} a^{8}-\frac {4}{9} a^{7} b \,x^{3}-\frac {28}{15} a^{6} b^{2} x^{6}-\frac {14}{3} a^{5} b^{3} x^{9}-\frac {70}{9} a^{4} b^{4} x^{12}-\frac {28}{3} a^{3} b^{5} x^{15}-\frac {28}{3} a^{2} b^{6} x^{18}}{x^{21}}+8 a \,b^{7} \ln \left (x \right )\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 94, normalized size = 0.90 \begin {gather*} \frac {1}{3} \, b^{8} x^{3} + \frac {8}{3} \, a b^{7} \log \left (x^{3}\right ) - \frac {2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 94, normalized size = 0.90 \begin {gather*} \frac {105 \, b^{8} x^{24} + 2520 \, a b^{7} x^{21} \log \left (x\right ) - 2940 \, a^{2} b^{6} x^{18} - 2940 \, a^{3} b^{5} x^{15} - 2450 \, a^{4} b^{4} x^{12} - 1470 \, a^{5} b^{3} x^{9} - 588 \, a^{6} b^{2} x^{6} - 140 \, a^{7} b x^{3} - 15 \, a^{8}}{315 \, x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 99, normalized size = 0.94 \begin {gather*} 8 a b^{7} \log {\left (x \right )} + \frac {b^{8} x^{3}}{3} + \frac {- 15 a^{8} - 140 a^{7} b x^{3} - 588 a^{6} b^{2} x^{6} - 1470 a^{5} b^{3} x^{9} - 2450 a^{4} b^{4} x^{12} - 2940 a^{3} b^{5} x^{15} - 2940 a^{2} b^{6} x^{18}}{315 x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.43, size = 102, normalized size = 0.97 \begin {gather*} \frac {1}{3} \, b^{8} x^{3} + 8 \, a b^{7} \log \left ({\left | x \right |}\right ) - \frac {2178 \, a b^{7} x^{21} + 2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.99, size = 94, normalized size = 0.90 \begin {gather*} -\frac {15\,a^8-105\,b^8\,x^{24}+140\,a^7\,b\,x^3+588\,a^6\,b^2\,x^6+1470\,a^5\,b^3\,x^9+2450\,a^4\,b^4\,x^{12}+2940\,a^3\,b^5\,x^{15}+2940\,a^2\,b^6\,x^{18}-2520\,a\,b^7\,x^{21}\,\ln \left (x\right )}{315\,x^{21}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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