Optimal. Leaf size=60 \[ -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{x}+10 a^3 b^2 x+\frac {10}{3} a^2 b^3 x^3+a b^4 x^5+\frac {b^5 x^7}{7} \]
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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {276}
\begin {gather*} -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{x}+10 a^3 b^2 x+\frac {10}{3} a^2 b^3 x^3+a b^4 x^5+\frac {b^5 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 276
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x^4} \, dx &=\int \left (10 a^3 b^2+\frac {a^5}{x^4}+\frac {5 a^4 b}{x^2}+10 a^2 b^3 x^2+5 a b^4 x^4+b^5 x^6\right ) \, dx\\ &=-\frac {a^5}{3 x^3}-\frac {5 a^4 b}{x}+10 a^3 b^2 x+\frac {10}{3} a^2 b^3 x^3+a b^4 x^5+\frac {b^5 x^7}{7}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 60, normalized size = 1.00 \begin {gather*} -\frac {a^5}{3 x^3}-\frac {5 a^4 b}{x}+10 a^3 b^2 x+\frac {10}{3} a^2 b^3 x^3+a b^4 x^5+\frac {b^5 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 55, normalized size = 0.92
method | result | size |
default | \(-\frac {a^{5}}{3 x^{3}}-\frac {5 a^{4} b}{x}+10 a^{3} b^{2} x +\frac {10 a^{2} b^{3} x^{3}}{3}+a \,b^{4} x^{5}+\frac {b^{5} x^{7}}{7}\) | \(55\) |
risch | \(\frac {b^{5} x^{7}}{7}+a \,b^{4} x^{5}+\frac {10 a^{2} b^{3} x^{3}}{3}+10 a^{3} b^{2} x +\frac {-5 a^{4} b \,x^{2}-\frac {1}{3} a^{5}}{x^{3}}\) | \(57\) |
norman | \(\frac {\frac {1}{7} b^{5} x^{10}+a \,b^{4} x^{8}+\frac {10}{3} a^{2} b^{3} x^{6}+10 a^{3} b^{2} x^{4}-5 a^{4} b \,x^{2}-\frac {1}{3} a^{5}}{x^{3}}\) | \(58\) |
gosper | \(-\frac {-3 b^{5} x^{10}-21 a \,b^{4} x^{8}-70 a^{2} b^{3} x^{6}-210 a^{3} b^{2} x^{4}+105 a^{4} b \,x^{2}+7 a^{5}}{21 x^{3}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 55, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, b^{5} x^{7} + a b^{4} x^{5} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x - \frac {15 \, a^{4} b x^{2} + a^{5}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 59, normalized size = 0.98 \begin {gather*} \frac {3 \, b^{5} x^{10} + 21 \, a b^{4} x^{8} + 70 \, a^{2} b^{3} x^{6} + 210 \, a^{3} b^{2} x^{4} - 105 \, a^{4} b x^{2} - 7 \, a^{5}}{21 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 60, normalized size = 1.00 \begin {gather*} 10 a^{3} b^{2} x + \frac {10 a^{2} b^{3} x^{3}}{3} + a b^{4} x^{5} + \frac {b^{5} x^{7}}{7} + \frac {- a^{5} - 15 a^{4} b x^{2}}{3 x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.97, size = 55, normalized size = 0.92 \begin {gather*} \frac {1}{7} \, b^{5} x^{7} + a b^{4} x^{5} + \frac {10}{3} \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x - \frac {15 \, a^{4} b x^{2} + a^{5}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 57, normalized size = 0.95 \begin {gather*} \frac {b^5\,x^7}{7}-\frac {\frac {a^5}{3}+5\,b\,a^4\,x^2}{x^3}+10\,a^3\,b^2\,x+a\,b^4\,x^5+\frac {10\,a^2\,b^3\,x^3}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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