Optimal. Leaf size=72 \[ -\frac {a^6}{5 x^5}-\frac {2 a^5 b}{x^3}-\frac {15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac {6}{5} a b^5 x^5+\frac {b^6 x^7}{7} \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {28, 270} \begin {gather*} 5 a^2 b^4 x^3+20 a^3 b^3 x-\frac {15 a^4 b^2}{x}-\frac {2 a^5 b}{x^3}-\frac {a^6}{5 x^5}+\frac {6}{5} a b^5 x^5+\frac {b^6 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 270
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^6} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^6} \, dx}{b^6}\\ &=\frac {\int \left (20 a^3 b^9+\frac {a^6 b^6}{x^6}+\frac {6 a^5 b^7}{x^4}+\frac {15 a^4 b^8}{x^2}+15 a^2 b^{10} x^2+6 a b^{11} x^4+b^{12} x^6\right ) \, dx}{b^6}\\ &=-\frac {a^6}{5 x^5}-\frac {2 a^5 b}{x^3}-\frac {15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac {6}{5} a b^5 x^5+\frac {b^6 x^7}{7}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 72, normalized size = 1.00 \begin {gather*} -\frac {a^6}{5 x^5}-\frac {2 a^5 b}{x^3}-\frac {15 a^4 b^2}{x}+20 a^3 b^3 x+5 a^2 b^4 x^3+\frac {6}{5} a b^5 x^5+\frac {b^6 x^7}{7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^3}{x^6} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.78, size = 70, normalized size = 0.97 \begin {gather*} \frac {5 \, b^{6} x^{12} + 42 \, a b^{5} x^{10} + 175 \, a^{2} b^{4} x^{8} + 700 \, a^{3} b^{3} x^{6} - 525 \, a^{4} b^{2} x^{4} - 70 \, a^{5} b x^{2} - 7 \, a^{6}}{35 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 67, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + \frac {6}{5} \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{3} + 20 \, a^{3} b^{3} x - \frac {75 \, a^{4} b^{2} x^{4} + 10 \, a^{5} b x^{2} + a^{6}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 67, normalized size = 0.93 \begin {gather*} \frac {b^{6} x^{7}}{7}+\frac {6 a \,b^{5} x^{5}}{5}+5 a^{2} b^{4} x^{3}+20 a^{3} b^{3} x -\frac {15 a^{4} b^{2}}{x}-\frac {2 a^{5} b}{x^{3}}-\frac {a^{6}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 67, normalized size = 0.93 \begin {gather*} \frac {1}{7} \, b^{6} x^{7} + \frac {6}{5} \, a b^{5} x^{5} + 5 \, a^{2} b^{4} x^{3} + 20 \, a^{3} b^{3} x - \frac {75 \, a^{4} b^{2} x^{4} + 10 \, a^{5} b x^{2} + a^{6}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 69, normalized size = 0.96 \begin {gather*} \frac {b^6\,x^7}{7}-\frac {\frac {a^6}{5}+2\,a^5\,b\,x^2+15\,a^4\,b^2\,x^4}{x^5}+20\,a^3\,b^3\,x+\frac {6\,a\,b^5\,x^5}{5}+5\,a^2\,b^4\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 73, normalized size = 1.01 \begin {gather*} 20 a^{3} b^{3} x + 5 a^{2} b^{4} x^{3} + \frac {6 a b^{5} x^{5}}{5} + \frac {b^{6} x^{7}}{7} + \frac {- a^{6} - 10 a^{5} b x^{2} - 75 a^{4} b^{2} x^{4}}{5 x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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