Optimal. Leaf size=75 \[ -\frac {a^5}{6 x^6}-\frac {15 a^4 b}{17 x^{17/3}}-\frac {15 a^3 b^2}{8 x^{16/3}}-\frac {2 a^2 b^3}{x^5}-\frac {15 a b^4}{14 x^{14/3}}-\frac {3 b^5}{13 x^{13/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45}
\begin {gather*} -\frac {a^5}{6 x^6}-\frac {15 a^4 b}{17 x^{17/3}}-\frac {15 a^3 b^2}{8 x^{16/3}}-\frac {2 a^2 b^3}{x^5}-\frac {15 a b^4}{14 x^{14/3}}-\frac {3 b^5}{13 x^{13/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b \sqrt [3]{x}\right )^5}{x^7} \, dx &=3 \text {Subst}\left (\int \frac {(a+b x)^5}{x^{19}} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \text {Subst}\left (\int \left (\frac {a^5}{x^{19}}+\frac {5 a^4 b}{x^{18}}+\frac {10 a^3 b^2}{x^{17}}+\frac {10 a^2 b^3}{x^{16}}+\frac {5 a b^4}{x^{15}}+\frac {b^5}{x^{14}}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac {a^5}{6 x^6}-\frac {15 a^4 b}{17 x^{17/3}}-\frac {15 a^3 b^2}{8 x^{16/3}}-\frac {2 a^2 b^3}{x^5}-\frac {15 a b^4}{14 x^{14/3}}-\frac {3 b^5}{13 x^{13/3}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 0.89 \begin {gather*} \frac {-6188 a^5-32760 a^4 b \sqrt [3]{x}-69615 a^3 b^2 x^{2/3}-74256 a^2 b^3 x-39780 a b^4 x^{4/3}-8568 b^5 x^{5/3}}{37128 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 58, normalized size = 0.77
method | result | size |
derivativedivides | \(-\frac {a^{5}}{6 x^{6}}-\frac {15 a^{4} b}{17 x^{\frac {17}{3}}}-\frac {15 a^{3} b^{2}}{8 x^{\frac {16}{3}}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {15 a \,b^{4}}{14 x^{\frac {14}{3}}}-\frac {3 b^{5}}{13 x^{\frac {13}{3}}}\) | \(58\) |
default | \(-\frac {a^{5}}{6 x^{6}}-\frac {15 a^{4} b}{17 x^{\frac {17}{3}}}-\frac {15 a^{3} b^{2}}{8 x^{\frac {16}{3}}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {15 a \,b^{4}}{14 x^{\frac {14}{3}}}-\frac {3 b^{5}}{13 x^{\frac {13}{3}}}\) | \(58\) |
trager | \(\frac {\left (x -1\right ) \left (a^{3} x^{5}+12 b^{3} x^{5}+a^{3} x^{4}+12 b^{3} x^{4}+a^{3} x^{3}+12 b^{3} x^{3}+a^{3} x^{2}+12 b^{3} x^{2}+a^{3} x +12 b^{3} x +a^{3}\right ) a^{2}}{6 x^{6}}-\frac {15 \left (17 b^{3} x +14 a^{3}\right ) a b}{238 x^{\frac {17}{3}}}-\frac {3 \left (8 b^{3} x +65 a^{3}\right ) b^{2}}{104 x^{\frac {16}{3}}}\) | \(127\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 57, normalized size = 0.76 \begin {gather*} -\frac {8568 \, b^{5} x^{\frac {5}{3}} + 39780 \, a b^{4} x^{\frac {4}{3}} + 74256 \, a^{2} b^{3} x + 69615 \, a^{3} b^{2} x^{\frac {2}{3}} + 32760 \, a^{4} b x^{\frac {1}{3}} + 6188 \, a^{5}}{37128 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 59, normalized size = 0.79 \begin {gather*} -\frac {74256 \, a^{2} b^{3} x + 6188 \, a^{5} + 1071 \, {\left (8 \, b^{5} x + 65 \, a^{3} b^{2}\right )} x^{\frac {2}{3}} + 2340 \, {\left (17 \, a b^{4} x + 14 \, a^{4} b\right )} x^{\frac {1}{3}}}{37128 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.06, size = 75, normalized size = 1.00 \begin {gather*} - \frac {a^{5}}{6 x^{6}} - \frac {15 a^{4} b}{17 x^{\frac {17}{3}}} - \frac {15 a^{3} b^{2}}{8 x^{\frac {16}{3}}} - \frac {2 a^{2} b^{3}}{x^{5}} - \frac {15 a b^{4}}{14 x^{\frac {14}{3}}} - \frac {3 b^{5}}{13 x^{\frac {13}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.98, size = 57, normalized size = 0.76 \begin {gather*} -\frac {8568 \, b^{5} x^{\frac {5}{3}} + 39780 \, a b^{4} x^{\frac {4}{3}} + 74256 \, a^{2} b^{3} x + 69615 \, a^{3} b^{2} x^{\frac {2}{3}} + 32760 \, a^{4} b x^{\frac {1}{3}} + 6188 \, a^{5}}{37128 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 57, normalized size = 0.76 \begin {gather*} -\frac {\frac {a^5}{6}+\frac {3\,b^5\,x^{5/3}}{13}+2\,a^2\,b^3\,x+\frac {15\,a^4\,b\,x^{1/3}}{17}+\frac {15\,a\,b^4\,x^{4/3}}{14}+\frac {15\,a^3\,b^2\,x^{2/3}}{8}}{x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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