Optimal. Leaf size=105 \[ -\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x) \]
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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}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+70 a^4 b^4 \log (x)+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12} \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^{13}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^5} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (56 a^3 b^5+\frac {a^8}{x^5}+\frac {8 a^7 b}{x^4}+\frac {28 a^6 b^2}{x^3}+\frac {56 a^5 b^3}{x^2}+\frac {70 a^4 b^4}{x}+28 a^2 b^6 x+8 a b^7 x^2+b^8 x^3\right ) \, dx,x,x^3\right )\\ &=-\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 105, normalized size = 1.00 \begin {gather*} -\frac {a^8}{12 x^{12}}-\frac {8 a^7 b}{9 x^9}-\frac {14 a^6 b^2}{3 x^6}-\frac {56 a^5 b^3}{3 x^3}+\frac {56}{3} a^3 b^5 x^3+\frac {14}{3} a^2 b^6 x^6+\frac {8}{9} a b^7 x^9+\frac {b^8 x^{12}}{12}+70 a^4 b^4 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 90, normalized size = 0.86
method | result | size |
default | \(-\frac {a^{8}}{12 x^{12}}-\frac {8 a^{7} b}{9 x^{9}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {56 a^{5} b^{3}}{3 x^{3}}+\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {14 a^{2} b^{6} x^{6}}{3}+\frac {8 a \,b^{7} x^{9}}{9}+\frac {b^{8} x^{12}}{12}+70 a^{4} b^{4} \ln \left (x \right )\) | \(90\) |
norman | \(\frac {-\frac {1}{12} a^{8}+\frac {1}{12} b^{8} x^{24}+\frac {8}{9} a \,b^{7} x^{21}+\frac {14}{3} a^{2} b^{6} x^{18}+\frac {56}{3} a^{3} b^{5} x^{15}-\frac {56}{3} a^{5} b^{3} x^{9}-\frac {14}{3} a^{6} b^{2} x^{6}-\frac {8}{9} a^{7} b \,x^{3}}{x^{12}}+70 a^{4} b^{4} \ln \left (x \right )\) | \(92\) |
risch | \(\frac {b^{8} x^{12}}{12}+\frac {8 a \,b^{7} x^{9}}{9}+\frac {14 a^{2} b^{6} x^{6}}{3}+\frac {56 a^{3} b^{5} x^{3}}{3}+\frac {-\frac {56}{3} a^{5} b^{3} x^{9}-\frac {14}{3} a^{6} b^{2} x^{6}-\frac {8}{9} a^{7} b \,x^{3}-\frac {1}{12} a^{8}}{x^{12}}+70 a^{4} b^{4} \ln \left (x \right )\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 94, normalized size = 0.90 \begin {gather*} \frac {1}{12} \, b^{8} x^{12} + \frac {8}{9} \, a b^{7} x^{9} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {56}{3} \, a^{3} b^{5} x^{3} + \frac {70}{3} \, a^{4} b^{4} \log \left (x^{3}\right ) - \frac {672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 94, normalized size = 0.90 \begin {gather*} \frac {3 \, b^{8} x^{24} + 32 \, a b^{7} x^{21} + 168 \, a^{2} b^{6} x^{18} + 672 \, a^{3} b^{5} x^{15} + 2520 \, a^{4} b^{4} x^{12} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{9} - 168 \, a^{6} b^{2} x^{6} - 32 \, a^{7} b x^{3} - 3 \, a^{8}}{36 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 104, normalized size = 0.99 \begin {gather*} 70 a^{4} b^{4} \log {\left (x \right )} + \frac {56 a^{3} b^{5} x^{3}}{3} + \frac {14 a^{2} b^{6} x^{6}}{3} + \frac {8 a b^{7} x^{9}}{9} + \frac {b^{8} x^{12}}{12} + \frac {- 3 a^{8} - 32 a^{7} b x^{3} - 168 a^{6} b^{2} x^{6} - 672 a^{5} b^{3} x^{9}}{36 x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.78, size = 104, normalized size = 0.99 \begin {gather*} \frac {1}{12} \, b^{8} x^{12} + \frac {8}{9} \, a b^{7} x^{9} + \frac {14}{3} \, a^{2} b^{6} x^{6} + \frac {56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4} \log \left ({\left | x \right |}\right ) - \frac {1750 \, a^{4} b^{4} x^{12} + 672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.96, size = 92, normalized size = 0.88 \begin {gather*} \frac {b^8\,x^{12}}{12}-\frac {\frac {a^8}{12}+\frac {8\,a^7\,b\,x^3}{9}+\frac {14\,a^6\,b^2\,x^6}{3}+\frac {56\,a^5\,b^3\,x^9}{3}}{x^{12}}+\frac {8\,a\,b^7\,x^9}{9}+\frac {56\,a^3\,b^5\,x^3}{3}+\frac {14\,a^2\,b^6\,x^6}{3}+70\,a^4\,b^4\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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