Optimal. Leaf size=72 \[ \frac {a^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {d x^2}}-\frac {a x^2}{b^2 \sqrt {d x^2}}+\frac {x^4}{3 b \sqrt {d x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {15, 302, 205} \[ \frac {a^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {d x^2}}-\frac {a x^2}{b^2 \sqrt {d x^2}}+\frac {x^4}{3 b \sqrt {d x^2}} \]
Antiderivative was successfully verified.
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Rule 15
Rule 205
Rule 302
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {d x^2} \left (a+b x^2\right )} \, dx &=\frac {x \int \frac {x^4}{a+b x^2} \, dx}{\sqrt {d x^2}}\\ &=\frac {x \int \left (-\frac {a}{b^2}+\frac {x^2}{b}+\frac {a^2}{b^2 \left (a+b x^2\right )}\right ) \, dx}{\sqrt {d x^2}}\\ &=-\frac {a x^2}{b^2 \sqrt {d x^2}}+\frac {x^4}{3 b \sqrt {d x^2}}+\frac {\left (a^2 x\right ) \int \frac {1}{a+b x^2} \, dx}{b^2 \sqrt {d x^2}}\\ &=-\frac {a x^2}{b^2 \sqrt {d x^2}}+\frac {x^4}{3 b \sqrt {d x^2}}+\frac {a^{3/2} x \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 56, normalized size = 0.78 \[ \frac {x \left (3 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )+\sqrt {b} x \left (b x^2-3 a\right )\right )}{3 b^{5/2} \sqrt {d x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 147, normalized size = 2.04 \[ \left [\frac {3 \, a d \sqrt {-\frac {a}{b d}} \log \left (\frac {b x^{2} + 2 \, \sqrt {d x^{2}} b \sqrt {-\frac {a}{b d}} - a}{b x^{2} + a}\right ) + 2 \, {\left (b x^{2} - 3 \, a\right )} \sqrt {d x^{2}}}{6 \, b^{2} d}, \frac {3 \, a d \sqrt {\frac {a}{b d}} \arctan \left (\frac {\sqrt {d x^{2}} b \sqrt {\frac {a}{b d}}}{a}\right ) + {\left (b x^{2} - 3 \, a\right )} \sqrt {d x^{2}}}{3 \, b^{2} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 70, normalized size = 0.97 \[ \frac {a^{2} \arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d} b^{2}} + \frac {\sqrt {d x^{2}} b^{2} d^{5} x^{2} - 3 \, \sqrt {d x^{2}} a b d^{5}}{3 \, b^{3} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 53, normalized size = 0.74 \[ \frac {\left (\sqrt {a b}\, b \,x^{3}+3 a^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )-3 \sqrt {a b}\, a x \right ) x}{3 \sqrt {d \,x^{2}}\, \sqrt {a b}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.83, size = 67, normalized size = 0.93 \[ \frac {\frac {3 \, a^{2} d^{3} \arctan \left (\frac {\sqrt {d x^{2}} b}{\sqrt {a b d}}\right )}{\sqrt {a b d} b^{2}} + \frac {\left (d x^{2}\right )^{\frac {3}{2}} b d - 3 \, \sqrt {d x^{2}} a d^{2}}{b^{2}}}{3 \, d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.65, size = 51, normalized size = 0.71 \[ \frac {{\left (x^2\right )}^{3/2}}{3\,b\,\sqrt {d}}+\frac {a^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {x^2}}{\sqrt {a}}\right )}{b^{5/2}\,\sqrt {d}}-\frac {a\,\sqrt {x^2}}{b^2\,\sqrt {d}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{5}}{\sqrt {d x^{2}} \left (a + b x^{2}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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