Optimal. Leaf size=85 \[ \frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{9/2}}-\frac {35 b x}{8 a^4}+\frac {35 x^3}{24 a^3}-\frac {7 x^5}{8 a^2 \left (a x^2+b\right )}-\frac {x^7}{4 a \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.04, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {263, 288, 302, 205} \[ \frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{9/2}}-\frac {7 x^5}{8 a^2 \left (a x^2+b\right )}-\frac {35 b x}{8 a^4}+\frac {35 x^3}{24 a^3}-\frac {x^7}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 263
Rule 288
Rule 302
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+\frac {b}{x^2}\right )^3} \, dx &=\int \frac {x^8}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac {x^7}{4 a \left (b+a x^2\right )^2}+\frac {7 \int \frac {x^6}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac {x^7}{4 a \left (b+a x^2\right )^2}-\frac {7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac {35 \int \frac {x^4}{b+a x^2} \, dx}{8 a^2}\\ &=-\frac {x^7}{4 a \left (b+a x^2\right )^2}-\frac {7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac {35 \int \left (-\frac {b}{a^2}+\frac {x^2}{a}+\frac {b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac {35 b x}{8 a^4}+\frac {35 x^3}{24 a^3}-\frac {x^7}{4 a \left (b+a x^2\right )^2}-\frac {7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac {\left (35 b^2\right ) \int \frac {1}{b+a x^2} \, dx}{8 a^4}\\ &=-\frac {35 b x}{8 a^4}+\frac {35 x^3}{24 a^3}-\frac {x^7}{4 a \left (b+a x^2\right )^2}-\frac {7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 77, normalized size = 0.91 \[ \frac {35 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{8 a^{9/2}}-\frac {-8 a^3 x^7+56 a^2 b x^5+175 a b^2 x^3+105 b^3 x}{24 a^4 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 230, normalized size = 2.71 \[ \left [\frac {16 \, a^{3} x^{7} - 112 \, a^{2} b x^{5} - 350 \, a b^{2} x^{3} - 210 \, b^{3} x + 105 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right )}{48 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}, \frac {8 \, a^{3} x^{7} - 56 \, a^{2} b x^{5} - 175 \, a b^{2} x^{3} - 105 \, b^{3} x + 105 \, {\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right )}{24 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 73, normalized size = 0.86 \[ \frac {35 \, b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} - \frac {13 \, a b^{2} x^{3} + 11 \, b^{3} x}{8 \, {\left (a x^{2} + b\right )}^{2} a^{4}} + \frac {a^{6} x^{3} - 9 \, a^{5} b x}{3 \, a^{9}} \]
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 \[ -\frac {13 b^{2} x^{3}}{8 \left (a \,x^{2}+b \right )^{2} a^{3}}+\frac {x^{3}}{3 a^{3}}-\frac {11 b^{3} x}{8 \left (a \,x^{2}+b \right )^{2} a^{4}}+\frac {35 b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \sqrt {a b}\, a^{4}}-\frac {3 b x}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.76, size = 82, normalized size = 0.96 \[ -\frac {13 \, a b^{2} x^{3} + 11 \, b^{3} x}{8 \, {\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}} + \frac {35 \, b^{2} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{8 \, \sqrt {a b} a^{4}} + \frac {a x^{3} - 9 \, b x}{3 \, a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 77, normalized size = 0.91 \[ \frac {x^3}{3\,a^3}-\frac {\frac {11\,b^3\,x}{8}+\frac {13\,a\,b^2\,x^3}{8}}{a^6\,x^4+2\,a^5\,b\,x^2+a^4\,b^2}+\frac {35\,b^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{8\,a^{9/2}}-\frac {3\,b\,x}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 133, normalized size = 1.56 \[ - \frac {35 \sqrt {- \frac {b^{3}}{a^{9}}} \log {\left (- \frac {a^{4} \sqrt {- \frac {b^{3}}{a^{9}}}}{b} + x \right )}}{16} + \frac {35 \sqrt {- \frac {b^{3}}{a^{9}}} \log {\left (\frac {a^{4} \sqrt {- \frac {b^{3}}{a^{9}}}}{b} + x \right )}}{16} + \frac {- 13 a b^{2} x^{3} - 11 b^{3} x}{8 a^{6} x^{4} + 16 a^{5} b x^{2} + 8 a^{4} b^{2}} + \frac {x^{3}}{3 a^{3}} - \frac {3 b x}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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