Optimal. Leaf size=68 \[ -\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 308, 211}
\begin {gather*} \frac {b^{7/2} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}}-\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 269
Rule 308
Rubi steps
\begin {align*} \int \frac {x^6}{a+\frac {b}{x^2}} \, dx &=\int \frac {x^8}{b+a x^2} \, dx\\ &=\int \left (-\frac {b^3}{a^4}+\frac {b^2 x^2}{a^3}-\frac {b x^4}{a^2}+\frac {x^6}{a}+\frac {b^4}{a^4 \left (b+a x^2\right )}\right ) \, dx\\ &=-\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}+\frac {b^4 \int \frac {1}{b+a x^2} \, dx}{a^4}\\ &=-\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 68, normalized size = 1.00 \begin {gather*} -\frac {b^3 x}{a^4}+\frac {b^2 x^3}{3 a^3}-\frac {b x^5}{5 a^2}+\frac {x^7}{7 a}+\frac {b^{7/2} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b}}\right )}{a^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 60, normalized size = 0.88
method | result | size |
default | \(\frac {\frac {1}{7} a^{3} x^{7}-\frac {1}{5} a^{2} b \,x^{5}+\frac {1}{3} a \,b^{2} x^{3}-b^{3} x}{a^{4}}+\frac {b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{a^{4} \sqrt {a b}}\) | \(60\) |
risch | \(\frac {x^{7}}{7 a}-\frac {b \,x^{5}}{5 a^{2}}+\frac {b^{2} x^{3}}{3 a^{3}}-\frac {x \,b^{3}}{a^{4}}+\frac {\sqrt {-a b}\, b^{3} \ln \left (-\sqrt {-a b}\, x +b \right )}{2 a^{5}}-\frac {\sqrt {-a b}\, b^{3} \ln \left (\sqrt {-a b}\, x +b \right )}{2 a^{5}}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 60, normalized size = 0.88 \begin {gather*} \frac {b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {15 \, a^{3} x^{7} - 21 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} - 105 \, b^{3} x}{105 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 148, normalized size = 2.18 \begin {gather*} \left [\frac {30 \, a^{3} x^{7} - 42 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt {-\frac {b}{a}} \log \left (\frac {a x^{2} + 2 \, a x \sqrt {-\frac {b}{a}} - b}{a x^{2} + b}\right ) - 210 \, b^{3} x}{210 \, a^{4}}, \frac {15 \, a^{3} x^{7} - 21 \, a^{2} b x^{5} + 35 \, a b^{2} x^{3} + 105 \, b^{3} \sqrt {\frac {b}{a}} \arctan \left (\frac {a x \sqrt {\frac {b}{a}}}{b}\right ) - 105 \, b^{3} x}{105 \, a^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 107, normalized size = 1.57 \begin {gather*} - \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (- \frac {a^{4} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac {\sqrt {- \frac {b^{7}}{a^{9}}} \log {\left (\frac {a^{4} \sqrt {- \frac {b^{7}}{a^{9}}}}{b^{3}} + x \right )}}{2} + \frac {x^{7}}{7 a} - \frac {b x^{5}}{5 a^{2}} + \frac {b^{2} x^{3}}{3 a^{3}} - \frac {b^{3} x}{a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.36, size = 65, normalized size = 0.96 \begin {gather*} \frac {b^{4} \arctan \left (\frac {a x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{4}} + \frac {15 \, a^{6} x^{7} - 21 \, a^{5} b x^{5} + 35 \, a^{4} b^{2} x^{3} - 105 \, a^{3} b^{3} x}{105 \, a^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 54, normalized size = 0.79 \begin {gather*} \frac {x^7}{7\,a}-\frac {b\,x^5}{5\,a^2}-\frac {b^3\,x}{a^4}+\frac {b^{7/2}\,\mathrm {atan}\left (\frac {\sqrt {a}\,x}{\sqrt {b}}\right )}{a^{9/2}}+\frac {b^2\,x^3}{3\,a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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