Optimal. Leaf size=100 \[ \frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac {35 x}{128 a^4 \left (a-b x^2\right )}+\frac {35 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b}} \]
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Rubi [A]
time = 0.02, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {205, 214}
\begin {gather*} \frac {35 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b}}+\frac {35 x}{128 a^4 \left (a-b x^2\right )}+\frac {35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {x}{8 a \left (a-b x^2\right )^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rubi steps
\begin {align*} \int \frac {1}{\left (a-b x^2\right )^5} \, dx &=\frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 \int \frac {1}{\left (a-b x^2\right )^4} \, dx}{8 a}\\ &=\frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {35 \int \frac {1}{\left (a-b x^2\right )^3} \, dx}{48 a^2}\\ &=\frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac {35 \int \frac {1}{\left (a-b x^2\right )^2} \, dx}{64 a^3}\\ &=\frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac {35 x}{128 a^4 \left (a-b x^2\right )}+\frac {35 \int \frac {1}{a-b x^2} \, dx}{128 a^4}\\ &=\frac {x}{8 a \left (a-b x^2\right )^4}+\frac {7 x}{48 a^2 \left (a-b x^2\right )^3}+\frac {35 x}{192 a^3 \left (a-b x^2\right )^2}+\frac {35 x}{128 a^4 \left (a-b x^2\right )}+\frac {35 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 79, normalized size = 0.79 \begin {gather*} \frac {\frac {\sqrt {a} x \left (279 a^3-511 a^2 b x^2+385 a b^2 x^4-105 b^3 x^6\right )}{\left (a-b x^2\right )^4}+\frac {105 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}{384 a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 103, normalized size = 1.03
method | result | size |
risch | \(\frac {-\frac {35 b^{3} x^{7}}{128 a^{4}}+\frac {385 b^{2} x^{5}}{384 a^{3}}-\frac {511 b \,x^{3}}{384 a^{2}}+\frac {93 x}{128 a}}{\left (-b \,x^{2}+a \right )^{4}}+\frac {35 \ln \left (b x +\sqrt {a b}\right )}{256 \sqrt {a b}\, a^{4}}-\frac {35 \ln \left (-b x +\sqrt {a b}\right )}{256 \sqrt {a b}\, a^{4}}\) | \(92\) |
default | \(\frac {x}{8 a \left (-b \,x^{2}+a \right )^{4}}+\frac {\frac {7 x}{48 a \left (-b \,x^{2}+a \right )^{3}}+\frac {7 \left (\frac {5 x}{24 a \left (-b \,x^{2}+a \right )^{2}}+\frac {5 \left (\frac {3 x}{8 a \left (-b \,x^{2}+a \right )}+\frac {3 \arctanh \left (\frac {b x}{\sqrt {a b}}\right )}{8 a \sqrt {a b}}\right )}{6 a}\right )}{8 a}}{a}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 117, normalized size = 1.17 \begin {gather*} -\frac {105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \, {\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac {35 \, \log \left (\frac {b x - \sqrt {a b}}{b x + \sqrt {a b}}\right )}{256 \, \sqrt {a b} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.08, size = 320, normalized size = 3.20 \begin {gather*} \left [-\frac {210 \, a b^{4} x^{7} - 770 \, a^{2} b^{3} x^{5} + 1022 \, a^{3} b^{2} x^{3} - 558 \, a^{4} b x - 105 \, {\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^{5} b^{5} x^{8} - 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} - 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}, -\frac {105 \, a b^{4} x^{7} - 385 \, a^{2} b^{3} x^{5} + 511 \, a^{3} b^{2} x^{3} - 279 \, a^{4} b x + 105 \, {\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^{5} b^{5} x^{8} - 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} - 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.23, size = 146, normalized size = 1.46 \begin {gather*} - \frac {35 \sqrt {\frac {1}{a^{9} b}} \log {\left (- a^{5} \sqrt {\frac {1}{a^{9} b}} + x \right )}}{256} + \frac {35 \sqrt {\frac {1}{a^{9} b}} \log {\left (a^{5} \sqrt {\frac {1}{a^{9} b}} + x \right )}}{256} - \frac {- 279 a^{3} x + 511 a^{2} b x^{3} - 385 a b^{2} x^{5} + 105 b^{3} x^{7}}{384 a^{8} - 1536 a^{7} b x^{2} + 2304 a^{6} b^{2} x^{4} - 1536 a^{5} b^{3} x^{6} + 384 a^{4} b^{4} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.56, size = 71, normalized size = 0.71 \begin {gather*} -\frac {35 \, \arctan \left (\frac {b x}{\sqrt {-a b}}\right )}{128 \, \sqrt {-a b} a^{4}} - \frac {105 \, b^{3} x^{7} - 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} - 279 \, a^{3} x}{384 \, {\left (b x^{2} - a\right )}^{4} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.79, size = 99, normalized size = 0.99 \begin {gather*} \frac {\frac {93\,x}{128\,a}-\frac {511\,b\,x^3}{384\,a^2}+\frac {385\,b^2\,x^5}{384\,a^3}-\frac {35\,b^3\,x^7}{128\,a^4}}{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 {35\,\mathrm {atanh}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{128\,a^{9/2}\,\sqrt {b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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