Optimal. Leaf size=47 \[ -\frac {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45}
\begin {gather*} -\frac {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rubi steps
\begin {align*} \int \frac {x^2}{(a+b x)^7} \, dx &=\int \left (\frac {a^2}{b^2 (a+b x)^7}-\frac {2 a}{b^2 (a+b x)^6}+\frac {1}{b^2 (a+b x)^5}\right ) \, dx\\ &=-\frac {a^2}{6 b^3 (a+b x)^6}+\frac {2 a}{5 b^3 (a+b x)^5}-\frac {1}{4 b^3 (a+b x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.66 \begin {gather*} -\frac {a^2+6 a b x+15 b^2 x^2}{60 b^3 (a+b x)^6} \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 2.37, size = 86, normalized size = 1.83 \begin {gather*} \frac {-a^2-6 a b x-15 b^2 x^2}{60 b^3 \left (a^6+6 a^5 b x+15 a^4 b^2 x^2+20 a^3 b^3 x^3+15 a^2 b^4 x^4+6 a b^5 x^5+b^6 x^6\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 42, normalized size = 0.89
method | result | size |
gosper | \(-\frac {15 x^{2} b^{2}+6 a b x +a^{2}}{60 \left (b x +a \right )^{6} b^{3}}\) | \(30\) |
norman | \(\frac {-\frac {x^{2}}{4 b}-\frac {a x}{10 b^{2}}-\frac {a^{2}}{60 b^{3}}}{\left (b x +a \right )^{6}}\) | \(33\) |
risch | \(\frac {-\frac {x^{2}}{4 b}-\frac {a x}{10 b^{2}}-\frac {a^{2}}{60 b^{3}}}{\left (b x +a \right )^{6}}\) | \(33\) |
default | \(-\frac {a^{2}}{6 b^{3} \left (b x +a \right )^{6}}+\frac {2 a}{5 b^{3} \left (b x +a \right )^{5}}-\frac {1}{4 b^{3} \left (b x +a \right )^{4}}\) | \(42\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (41) = 82\).
time = 0.25, size = 87, normalized size = 1.85 \begin {gather*} -\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 87 vs.
\(2 (41) = 82\).
time = 0.31, size = 87, normalized size = 1.85 \begin {gather*} -\frac {15 \, b^{2} x^{2} + 6 \, a b x + a^{2}}{60 \, {\left (b^{9} x^{6} + 6 \, a b^{8} x^{5} + 15 \, a^{2} b^{7} x^{4} + 20 \, a^{3} b^{6} x^{3} + 15 \, a^{4} b^{5} x^{2} + 6 \, a^{5} b^{4} x + a^{6} b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 92 vs.
\(2 (42) = 84\).
time = 0.22, size = 92, normalized size = 1.96 \begin {gather*} \frac {- a^{2} - 6 a b x - 15 b^{2} x^{2}}{60 a^{6} b^{3} + 360 a^{5} b^{4} x + 900 a^{4} b^{5} x^{2} + 1200 a^{3} b^{6} x^{3} + 900 a^{2} b^{7} x^{4} + 360 a b^{8} x^{5} + 60 b^{9} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 34, normalized size = 0.72 \begin {gather*} \frac {-15 x^{2} b^{2}-6 x b a-a^{2}}{60 b^{3} \left (x b+a\right )^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 31, normalized size = 0.66 \begin {gather*} -\frac {8\,a^2+48\,a\,b\,x+120\,b^2\,x^2}{480\,b^3\,{\left (a+b\,x\right )}^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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