Optimal. Leaf size=109 \[ -\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{7/2} b^{3/2}}-\frac {5 x}{128 a^3 b \left (a-b x^2\right )}-\frac {5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac {x}{48 a b \left (a-b x^2\right )^3}+\frac {x}{8 b \left (a-b x^2\right )^4} \]
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Rubi [A] time = 0.04, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {288, 199, 208} \[ -\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{7/2} b^{3/2}}-\frac {5 x}{128 a^3 b \left (a-b x^2\right )}-\frac {5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac {x}{48 a b \left (a-b x^2\right )^3}+\frac {x}{8 b \left (a-b x^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 199
Rule 208
Rule 288
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a-b x^2\right )^5} \, dx &=\frac {x}{8 b \left (a-b x^2\right )^4}-\frac {\int \frac {1}{\left (a-b x^2\right )^4} \, dx}{8 b}\\ &=\frac {x}{8 b \left (a-b x^2\right )^4}-\frac {x}{48 a b \left (a-b x^2\right )^3}-\frac {5 \int \frac {1}{\left (a-b x^2\right )^3} \, dx}{48 a b}\\ &=\frac {x}{8 b \left (a-b x^2\right )^4}-\frac {x}{48 a b \left (a-b x^2\right )^3}-\frac {5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac {5 \int \frac {1}{\left (a-b x^2\right )^2} \, dx}{64 a^2 b}\\ &=\frac {x}{8 b \left (a-b x^2\right )^4}-\frac {x}{48 a b \left (a-b x^2\right )^3}-\frac {5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac {5 x}{128 a^3 b \left (a-b x^2\right )}-\frac {5 \int \frac {1}{a-b x^2} \, dx}{128 a^3 b}\\ &=\frac {x}{8 b \left (a-b x^2\right )^4}-\frac {x}{48 a b \left (a-b x^2\right )^3}-\frac {5 x}{192 a^2 b \left (a-b x^2\right )^2}-\frac {5 x}{128 a^3 b \left (a-b x^2\right )}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{7/2} b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 81, normalized size = 0.74 \[ \frac {15 a^3 x+73 a^2 b x^3-55 a b^2 x^5+15 b^3 x^7}{384 a^3 b \left (a-b x^2\right )^4}-\frac {5 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{7/2} b^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 324, normalized size = 2.97 \[ \left [\frac {30 \, a b^{4} x^{7} - 110 \, a^{2} b^{3} x^{5} + 146 \, a^{3} b^{2} x^{3} + 30 \, a^{4} b x + 15 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {a b} x + a}{b x^{2} - a}\right )}{768 \, {\left (a^{4} b^{6} x^{8} - 4 \, a^{5} b^{5} x^{6} + 6 \, a^{6} b^{4} x^{4} - 4 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}}, \frac {15 \, a b^{4} x^{7} - 55 \, a^{2} b^{3} x^{5} + 73 \, a^{3} b^{2} x^{3} + 15 \, a^{4} b x + 15 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b} x}{a}\right )}{384 \, {\left (a^{4} b^{6} x^{8} - 4 \, a^{5} b^{5} x^{6} + 6 \, a^{6} b^{4} x^{4} - 4 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 77, normalized size = 0.71 \[ \frac {5 \, \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{128 \, \sqrt {-a b} a^{3} b} + \frac {15 \, b^{3} x^{7} - 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} + 15 \, a^{3} x}{384 \, {\left (b x^{2} - a\right )}^{4} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 72, normalized size = 0.66 \[ -\frac {5 \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{128 \sqrt {a b}\, a^{3} b}-\frac {-\frac {5 b^{2} x^{7}}{128 a^{3}}+\frac {55 b \,x^{5}}{384 a^{2}}-\frac {73 x^{3}}{384 a}-\frac {5 x}{128 b}}{\left (b \,x^{2}-a \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.98, size = 124, normalized size = 1.14 \[ \frac {15 \, b^{3} x^{7} - 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} + 15 \, a^{3} x}{384 \, {\left (a^{3} b^{5} x^{8} - 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} - 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} + \frac {5 \, \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.76, size = 96, normalized size = 0.88 \[ \frac {\frac {5\,x}{128\,b}+\frac {73\,x^3}{384\,a}-\frac {55\,b\,x^5}{384\,a^2}+\frac {5\,b^2\,x^7}{128\,a^3}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8}-\frac {5\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{128\,a^{7/2}\,b^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.50, size = 160, normalized size = 1.47 \[ \frac {5 \sqrt {\frac {1}{a^{7} b^{3}}} \log {\left (- a^{4} b \sqrt {\frac {1}{a^{7} b^{3}}} + x \right )}}{256} - \frac {5 \sqrt {\frac {1}{a^{7} b^{3}}} \log {\left (a^{4} b \sqrt {\frac {1}{a^{7} b^{3}}} + x \right )}}{256} - \frac {- 15 a^{3} x - 73 a^{2} b x^{3} + 55 a b^{2} x^{5} - 15 b^{3} x^{7}}{384 a^{7} b - 1536 a^{6} b^{2} x^{2} + 2304 a^{5} b^{3} x^{4} - 1536 a^{4} b^{4} x^{6} + 384 a^{3} b^{5} x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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