Optimal. Leaf size=72 \[ -\frac {a^6}{x}+6 a^5 b x+5 a^4 b^2 x^3+4 a^3 b^3 x^5+\frac {15}{7} a^2 b^4 x^7+\frac {2}{3} a b^5 x^9+\frac {b^6 x^{11}}{11} \]
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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*} \frac {15}{7} a^2 b^4 x^7+4 a^3 b^3 x^5+5 a^4 b^2 x^3+6 a^5 b x-\frac {a^6}{x}+\frac {2}{3} a b^5 x^9+\frac {b^6 x^{11}}{11} \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^2} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^6}{x^2} \, dx}{b^6}\\ &=\frac {\int \left (6 a^5 b^7+\frac {a^6 b^6}{x^2}+15 a^4 b^8 x^2+20 a^3 b^9 x^4+15 a^2 b^{10} x^6+6 a b^{11} x^8+b^{12} x^{10}\right ) \, dx}{b^6}\\ &=-\frac {a^6}{x}+6 a^5 b x+5 a^4 b^2 x^3+4 a^3 b^3 x^5+\frac {15}{7} a^2 b^4 x^7+\frac {2}{3} a b^5 x^9+\frac {b^6 x^{11}}{11}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 72, normalized size = 1.00 \begin {gather*} -\frac {a^6}{x}+6 a^5 b x+5 a^4 b^2 x^3+4 a^3 b^3 x^5+\frac {15}{7} a^2 b^4 x^7+\frac {2}{3} a b^5 x^9+\frac {b^6 x^{11}}{11} \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^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.91, size = 70, normalized size = 0.97 \begin {gather*} \frac {21 \, b^{6} x^{12} + 154 \, a b^{5} x^{10} + 495 \, a^{2} b^{4} x^{8} + 924 \, a^{3} b^{3} x^{6} + 1155 \, a^{4} b^{2} x^{4} + 1386 \, a^{5} b x^{2} - 231 \, a^{6}}{231 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 66, normalized size = 0.92 \begin {gather*} \frac {1}{11} \, b^{6} x^{11} + \frac {2}{3} \, a b^{5} x^{9} + \frac {15}{7} \, a^{2} b^{4} x^{7} + 4 \, a^{3} b^{3} x^{5} + 5 \, a^{4} b^{2} x^{3} + 6 \, a^{5} b x - \frac {a^{6}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 67, normalized size = 0.93 \begin {gather*} \frac {b^{6} x^{11}}{11}+\frac {2 a \,b^{5} x^{9}}{3}+\frac {15 a^{2} b^{4} x^{7}}{7}+4 a^{3} b^{3} x^{5}+5 a^{4} b^{2} x^{3}+6 a^{5} b x -\frac {a^{6}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.35, size = 66, normalized size = 0.92 \begin {gather*} \frac {1}{11} \, b^{6} x^{11} + \frac {2}{3} \, a b^{5} x^{9} + \frac {15}{7} \, a^{2} b^{4} x^{7} + 4 \, a^{3} b^{3} x^{5} + 5 \, a^{4} b^{2} x^{3} + 6 \, a^{5} b x - \frac {a^{6}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 66, normalized size = 0.92 \begin {gather*} \frac {b^6\,x^{11}}{11}-\frac {a^6}{x}+\frac {2\,a\,b^5\,x^9}{3}+5\,a^4\,b^2\,x^3+4\,a^3\,b^3\,x^5+\frac {15\,a^2\,b^4\,x^7}{7}+6\,a^5\,b\,x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 70, normalized size = 0.97 \begin {gather*} - \frac {a^{6}}{x} + 6 a^{5} b x + 5 a^{4} b^{2} x^{3} + 4 a^{3} b^{3} x^{5} + \frac {15 a^{2} b^{4} x^{7}}{7} + \frac {2 a b^{5} x^{9}}{3} + \frac {b^{6} x^{11}}{11} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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