Optimal. Leaf size=84 \[ -\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}} \]
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Rubi [A]
time = 0.03, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 47, 37}
\begin {gather*} \frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {\left (a+b x^3\right )^9}{36 a x^{36}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^8}{x^{37}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^8}{x^{13}} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^8}{x^{12}} \, dx,x,x^3\right )}{12 a}\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{11}} \, dx,x,x^3\right )}{66 a^2}\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}-\frac {b^3 \text {Subst}\left (\int \frac {(a+b x)^8}{x^{10}} \, dx,x,x^3\right )}{660 a^3}\\ &=-\frac {\left (a+b x^3\right )^9}{36 a x^{36}}+\frac {b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac {b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac {b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 108, normalized size = 1.29 \begin {gather*} -\frac {a^8}{36 x^{36}}-\frac {8 a^7 b}{33 x^{33}}-\frac {14 a^6 b^2}{15 x^{30}}-\frac {56 a^5 b^3}{27 x^{27}}-\frac {35 a^4 b^4}{12 x^{24}}-\frac {8 a^3 b^5}{3 x^{21}}-\frac {14 a^2 b^6}{9 x^{18}}-\frac {8 a b^7}{15 x^{15}}-\frac {b^8}{12 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 91, normalized size = 1.08
method | result | size |
default | \(-\frac {8 a \,b^{7}}{15 x^{15}}-\frac {b^{8}}{12 x^{12}}-\frac {8 a^{3} b^{5}}{3 x^{21}}-\frac {a^{8}}{36 x^{36}}-\frac {8 b \,a^{7}}{33 x^{33}}-\frac {35 a^{4} b^{4}}{12 x^{24}}-\frac {14 a^{2} b^{6}}{9 x^{18}}-\frac {56 a^{5} b^{3}}{27 x^{27}}-\frac {14 a^{6} b^{2}}{15 x^{30}}\) | \(91\) |
norman | \(\frac {-\frac {8}{3} a^{3} b^{5} x^{15}-\frac {14}{9} a^{2} b^{6} x^{18}-\frac {8}{15} a \,b^{7} x^{21}-\frac {1}{12} b^{8} x^{24}-\frac {35}{12} a^{4} b^{4} x^{12}-\frac {8}{33} a^{7} b \,x^{3}-\frac {14}{15} a^{6} b^{2} x^{6}-\frac {56}{27} a^{5} b^{3} x^{9}-\frac {1}{36} a^{8}}{x^{36}}\) | \(92\) |
risch | \(\frac {-\frac {8}{3} a^{3} b^{5} x^{15}-\frac {14}{9} a^{2} b^{6} x^{18}-\frac {8}{15} a \,b^{7} x^{21}-\frac {1}{12} b^{8} x^{24}-\frac {35}{12} a^{4} b^{4} x^{12}-\frac {8}{33} a^{7} b \,x^{3}-\frac {14}{15} a^{6} b^{2} x^{6}-\frac {56}{27} a^{5} b^{3} x^{9}-\frac {1}{36} a^{8}}{x^{36}}\) | \(92\) |
gosper | \(-\frac {495 b^{8} x^{24}+3168 a \,b^{7} x^{21}+9240 a^{2} b^{6} x^{18}+15840 a^{3} b^{5} x^{15}+17325 a^{4} b^{4} x^{12}+12320 a^{5} b^{3} x^{9}+5544 a^{6} b^{2} x^{6}+1440 a^{7} b \,x^{3}+165 a^{8}}{5940 x^{36}}\) | \(93\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 92, normalized size = 1.10 \begin {gather*} -\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 92, normalized size = 1.10 \begin {gather*} -\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.58, size = 99, normalized size = 1.18 \begin {gather*} \frac {- 165 a^{8} - 1440 a^{7} b x^{3} - 5544 a^{6} b^{2} x^{6} - 12320 a^{5} b^{3} x^{9} - 17325 a^{4} b^{4} x^{12} - 15840 a^{3} b^{5} x^{15} - 9240 a^{2} b^{6} x^{18} - 3168 a b^{7} x^{21} - 495 b^{8} x^{24}}{5940 x^{36}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.49, size = 92, normalized size = 1.10 \begin {gather*} -\frac {495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 92, normalized size = 1.10 \begin {gather*} -\frac {\frac {a^8}{36}+\frac {8\,a^7\,b\,x^3}{33}+\frac {14\,a^6\,b^2\,x^6}{15}+\frac {56\,a^5\,b^3\,x^9}{27}+\frac {35\,a^4\,b^4\,x^{12}}{12}+\frac {8\,a^3\,b^5\,x^{15}}{3}+\frac {14\,a^2\,b^6\,x^{18}}{9}+\frac {8\,a\,b^7\,x^{21}}{15}+\frac {b^8\,x^{24}}{12}}{x^{36}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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