Optimal. Leaf size=145 \[ \frac {x \left (a+b x^2\right ) (b d-a e)}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e x^3 \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {a} \left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.09, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1250, 459, 321, 205} \[ \frac {x \left (a+b x^2\right ) (b d-a e)}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {a} \left (a+b x^2\right ) (b d-a e) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e x^3 \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 321
Rule 459
Rule 1250
Rubi steps
\begin {align*} \int \frac {x^2 \left (d+e x^2\right )}{\sqrt {a^2+2 a b x^2+b^2 x^4}} \, dx &=\frac {\left (a b+b^2 x^2\right ) \int \frac {x^2 \left (d+e x^2\right )}{a b+b^2 x^2} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {e x^3 \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (\left (-3 b^2 d+3 a b e\right ) \left (a b+b^2 x^2\right )\right ) \int \frac {x^2}{a b+b^2 x^2} \, dx}{3 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(b d-a e) x \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e x^3 \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a \left (-3 b^2 d+3 a b e\right ) \left (a b+b^2 x^2\right )\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{3 b^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(b d-a e) x \left (a+b x^2\right )}{b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {e x^3 \left (a+b x^2\right )}{3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\sqrt {a} (b d-a e) \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{5/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 80, normalized size = 0.55 \[ \frac {\left (a+b x^2\right ) \left (\sqrt {b} x \left (-3 a e+3 b d+b e x^2\right )+3 \sqrt {a} (a e-b d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right )}{3 b^{5/2} \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 129, normalized size = 0.89 \[ \left [\frac {2 \, b e x^{3} - 3 \, {\left (b d - a e\right )} \sqrt {-\frac {a}{b}} \log \left (\frac {b x^{2} + 2 \, b x \sqrt {-\frac {a}{b}} - a}{b x^{2} + a}\right ) + 6 \, {\left (b d - a e\right )} x}{6 \, b^{2}}, \frac {b e x^{3} - 3 \, {\left (b d - a e\right )} \sqrt {\frac {a}{b}} \arctan \left (\frac {b x \sqrt {\frac {a}{b}}}{a}\right ) + 3 \, {\left (b d - a e\right )} x}{3 \, b^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 101, normalized size = 0.70 \[ -\frac {{\left (a b d \mathrm {sgn}\left (b x^{2} + a\right ) - a^{2} e \mathrm {sgn}\left (b x^{2} + a\right )\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b^{2} x^{3} e \mathrm {sgn}\left (b x^{2} + a\right ) + 3 \, b^{2} d x \mathrm {sgn}\left (b x^{2} + a\right ) - 3 \, a b x e \mathrm {sgn}\left (b x^{2} + a\right )}{3 \, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 90, normalized size = 0.62 \[ \frac {\left (b \,x^{2}+a \right ) \left (\sqrt {a b}\, b e \,x^{3}+3 a^{2} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )-3 a b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )-3 \sqrt {a b}\, a e x +3 \sqrt {a b}\, b d x \right )}{3 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \sqrt {a b}\, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.52, size = 54, normalized size = 0.37 \[ -\frac {{\left (a b d - a^{2} e\right )} \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} b^{2}} + \frac {b e x^{3} + 3 \, {\left (b d - a e\right )} x}{3 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (e\,x^2+d\right )}{\sqrt {{\left (b\,x^2+a\right )}^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.36, size = 90, normalized size = 0.62 \[ x \left (- \frac {a e}{b^{2}} + \frac {d}{b}\right ) - \frac {\sqrt {- \frac {a}{b^{5}}} \left (a e - b d\right ) \log {\left (- b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{2} + \frac {\sqrt {- \frac {a}{b^{5}}} \left (a e - b d\right ) \log {\left (b^{2} \sqrt {- \frac {a}{b^{5}}} + x \right )}}{2} + \frac {e x^{3}}{3 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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