Optimal. Leaf size=56 \[ -\frac {a^4}{14 x^{14}}-\frac {a^3 b}{3 x^{12}}-\frac {3 a^2 b^2}{5 x^{10}}-\frac {a b^3}{2 x^8}-\frac {b^4}{6 x^6} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {28, 272, 45}
\begin {gather*} -\frac {a^4}{14 x^{14}}-\frac {a^3 b}{3 x^{12}}-\frac {3 a^2 b^2}{5 x^{10}}-\frac {a b^3}{2 x^8}-\frac {b^4}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 28
Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{15}} \, dx &=\frac {\int \frac {\left (a b+b^2 x^2\right )^4}{x^{15}} \, dx}{b^4}\\ &=\frac {\text {Subst}\left (\int \frac {\left (a b+b^2 x\right )^4}{x^8} \, dx,x,x^2\right )}{2 b^4}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^4 b^4}{x^8}+\frac {4 a^3 b^5}{x^7}+\frac {6 a^2 b^6}{x^6}+\frac {4 a b^7}{x^5}+\frac {b^8}{x^4}\right ) \, dx,x,x^2\right )}{2 b^4}\\ &=-\frac {a^4}{14 x^{14}}-\frac {a^3 b}{3 x^{12}}-\frac {3 a^2 b^2}{5 x^{10}}-\frac {a b^3}{2 x^8}-\frac {b^4}{6 x^6}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 56, normalized size = 1.00 \begin {gather*} -\frac {a^4}{14 x^{14}}-\frac {a^3 b}{3 x^{12}}-\frac {3 a^2 b^2}{5 x^{10}}-\frac {a b^3}{2 x^8}-\frac {b^4}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 47, normalized size = 0.84
method | result | size |
default | \(-\frac {a^{4}}{14 x^{14}}-\frac {a^{3} b}{3 x^{12}}-\frac {3 a^{2} b^{2}}{5 x^{10}}-\frac {a \,b^{3}}{2 x^{8}}-\frac {b^{4}}{6 x^{6}}\) | \(47\) |
norman | \(\frac {-\frac {1}{14} a^{4}-\frac {1}{3} a^{3} b \,x^{2}-\frac {3}{5} a^{2} b^{2} x^{4}-\frac {1}{2} a \,b^{3} x^{6}-\frac {1}{6} b^{4} x^{8}}{x^{14}}\) | \(48\) |
risch | \(\frac {-\frac {1}{14} a^{4}-\frac {1}{3} a^{3} b \,x^{2}-\frac {3}{5} a^{2} b^{2} x^{4}-\frac {1}{2} a \,b^{3} x^{6}-\frac {1}{6} b^{4} x^{8}}{x^{14}}\) | \(48\) |
gosper | \(-\frac {35 b^{4} x^{8}+105 a \,b^{3} x^{6}+126 a^{2} b^{2} x^{4}+70 a^{3} b \,x^{2}+15 a^{4}}{210 x^{14}}\) | \(49\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 48, normalized size = 0.86 \begin {gather*} -\frac {35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 48, normalized size = 0.86 \begin {gather*} -\frac {35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 51, normalized size = 0.91 \begin {gather*} \frac {- 15 a^{4} - 70 a^{3} b x^{2} - 126 a^{2} b^{2} x^{4} - 105 a b^{3} x^{6} - 35 b^{4} x^{8}}{210 x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.18, size = 48, normalized size = 0.86 \begin {gather*} -\frac {35 \, b^{4} x^{8} + 105 \, a b^{3} x^{6} + 126 \, a^{2} b^{2} x^{4} + 70 \, a^{3} b x^{2} + 15 \, a^{4}}{210 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.33, size = 48, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^4}{14}+\frac {a^3\,b\,x^2}{3}+\frac {3\,a^2\,b^2\,x^4}{5}+\frac {a\,b^3\,x^6}{2}+\frac {b^4\,x^8}{6}}{x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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