Optimal. Leaf size=85 \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}-\frac {35 a x}{8 b^4}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}-\frac {x^7}{4 b \left (a+b x^2\right )^2}+\frac {35 x^3}{24 b^3} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {288, 302, 205} \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}-\frac {35 a x}{8 b^4}-\frac {x^7}{4 b \left (a+b x^2\right )^2}+\frac {35 x^3}{24 b^3} \]
Antiderivative was successfully verified.
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Rule 205
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^2\right )^3} \, dx &=-\frac {x^7}{4 b \left (a+b x^2\right )^2}+\frac {7 \int \frac {x^6}{\left (a+b x^2\right )^2} \, dx}{4 b}\\ &=-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {35 \int \frac {x^4}{a+b x^2} \, dx}{8 b^2}\\ &=-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {35 \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{8 b^2}\\ &=-\frac {35 a x}{8 b^4}+\frac {35 x^3}{24 b^3}-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {\left (35 a^2\right ) \int \frac {1}{a+b x^2} \, dx}{8 b^4}\\ &=-\frac {35 a x}{8 b^4}+\frac {35 x^3}{24 b^3}-\frac {x^7}{4 b \left (a+b x^2\right )^2}-\frac {7 x^5}{8 b^2 \left (a+b x^2\right )}+\frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.91 \[ \frac {35 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 b^{9/2}}-\frac {105 a^3 x+175 a^2 b x^3+56 a b^2 x^5-8 b^3 x^7}{24 b^4 \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 230, normalized size = 2.71 \[ \left [\frac {16 \, b^{3} x^{7} - 112 \, a b^{2} x^{5} - 350 \, a^{2} b x^{3} - 210 \, a^{3} x + 105 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right )}{48 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}, \frac {8 \, b^{3} x^{7} - 56 \, a b^{2} x^{5} - 175 \, a^{2} b x^{3} - 105 \, a^{3} x + 105 \, {\left (a b^{2} x^{4} + 2 \, a^{2} b x^{2} + a^{3}\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right )}{24 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.63, size = 73, normalized size = 0.86 \[ \frac {35 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} - \frac {13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \, {\left (b x^{2} + a\right )}^{2} b^{4}} + \frac {b^{6} x^{3} - 9 \, a b^{5} x}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 77, normalized size = 0.91 \[ -\frac {13 a^{2} x^{3}}{8 \left (b \,x^{2}+a \right )^{2} b^{3}}-\frac {11 a^{3} x}{8 \left (b \,x^{2}+a \right )^{2} b^{4}}+\frac {x^{3}}{3 b^{3}}+\frac {35 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, b^{4}}-\frac {3 a x}{b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.97, size = 82, normalized size = 0.96 \[ -\frac {13 \, a^{2} b x^{3} + 11 \, a^{3} x}{8 \, {\left (b^{6} x^{4} + 2 \, a b^{5} x^{2} + a^{2} b^{4}\right )}} + \frac {35 \, a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} b^{4}} + \frac {b x^{3} - 9 \, a x}{3 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 77, normalized size = 0.91 \[ \frac {x^3}{3\,b^3}-\frac {\frac {11\,a^3\,x}{8}+\frac {13\,b\,a^2\,x^3}{8}}{a^2\,b^4+2\,a\,b^5\,x^2+b^6\,x^4}+\frac {35\,a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{8\,b^{9/2}}-\frac {3\,a\,x}{b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.44, size = 133, normalized size = 1.56 \[ - \frac {3 a x}{b^{4}} - \frac {35 \sqrt {- \frac {a^{3}}{b^{9}}} \log {\left (x - \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac {35 \sqrt {- \frac {a^{3}}{b^{9}}} \log {\left (x + \frac {b^{4} \sqrt {- \frac {a^{3}}{b^{9}}}}{a} \right )}}{16} + \frac {- 11 a^{3} x - 13 a^{2} b x^{3}}{8 a^{2} b^{4} + 16 a b^{5} x^{2} + 8 b^{6} x^{4}} + \frac {x^{3}}{3 b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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