Optimal. Leaf size=40 \[ -\frac {\left (a+b x^2\right )^6}{14 a x^{14}}+\frac {b \left (a+b x^2\right )^6}{84 a^2 x^{12}} \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 47, 37}
\begin {gather*} \frac {b \left (a+b x^2\right )^6}{84 a^2 x^{12}}-\frac {\left (a+b x^2\right )^6}{14 a x^{14}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 272
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^5}{x^{15}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^5}{x^8} \, dx,x,x^2\right )\\ &=-\frac {\left (a+b x^2\right )^6}{14 a x^{14}}-\frac {b \text {Subst}\left (\int \frac {(a+b x)^5}{x^7} \, dx,x,x^2\right )}{14 a}\\ &=-\frac {\left (a+b x^2\right )^6}{14 a x^{14}}+\frac {b \left (a+b x^2\right )^6}{84 a^2 x^{12}}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 67, normalized size = 1.68 \begin {gather*} -\frac {a^5}{14 x^{14}}-\frac {5 a^4 b}{12 x^{12}}-\frac {a^3 b^2}{x^{10}}-\frac {5 a^2 b^3}{4 x^8}-\frac {5 a b^4}{6 x^6}-\frac {b^5}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 58, normalized size = 1.45
method | result | size |
default | \(-\frac {a^{3} b^{2}}{x^{10}}-\frac {b^{5}}{4 x^{4}}-\frac {5 a \,b^{4}}{6 x^{6}}-\frac {a^{5}}{14 x^{14}}-\frac {5 a^{2} b^{3}}{4 x^{8}}-\frac {5 a^{4} b}{12 x^{12}}\) | \(58\) |
norman | \(\frac {-\frac {1}{14} a^{5}-\frac {5}{12} a^{4} b \,x^{2}-a^{3} b^{2} x^{4}-\frac {5}{4} a^{2} b^{3} x^{6}-\frac {5}{6} a \,b^{4} x^{8}-\frac {1}{4} b^{5} x^{10}}{x^{14}}\) | \(59\) |
risch | \(\frac {-\frac {1}{14} a^{5}-\frac {5}{12} a^{4} b \,x^{2}-a^{3} b^{2} x^{4}-\frac {5}{4} a^{2} b^{3} x^{6}-\frac {5}{6} a \,b^{4} x^{8}-\frac {1}{4} b^{5} x^{10}}{x^{14}}\) | \(59\) |
gosper | \(-\frac {21 b^{5} x^{10}+70 a \,b^{4} x^{8}+105 a^{2} b^{3} x^{6}+84 a^{3} b^{2} x^{4}+35 a^{4} b \,x^{2}+6 a^{5}}{84 x^{14}}\) | \(60\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.46, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.21, size = 63, normalized size = 1.58 \begin {gather*} \frac {- 6 a^{5} - 35 a^{4} b x^{2} - 84 a^{3} b^{2} x^{4} - 105 a^{2} b^{3} x^{6} - 70 a b^{4} x^{8} - 21 b^{5} x^{10}}{84 x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 59, normalized size = 1.48 \begin {gather*} -\frac {21 \, b^{5} x^{10} + 70 \, a b^{4} x^{8} + 105 \, a^{2} b^{3} x^{6} + 84 \, a^{3} b^{2} x^{4} + 35 \, a^{4} b x^{2} + 6 \, a^{5}}{84 \, x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.74, size = 58, normalized size = 1.45 \begin {gather*} -\frac {\frac {a^5}{14}+\frac {5\,a^4\,b\,x^2}{12}+a^3\,b^2\,x^4+\frac {5\,a^2\,b^3\,x^6}{4}+\frac {5\,a\,b^4\,x^8}{6}+\frac {b^5\,x^{10}}{4}}{x^{14}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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