Optimal. Leaf size=156 \[ \frac {x (a e+3 b d)}{8 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x (b d-a e)}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) (a e+3 b d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.09, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1148, 385, 199, 205} \[ \frac {x (a e+3 b d)}{8 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x (b d-a e)}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (a+b x^2\right ) (a e+3 b d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 385
Rule 1148
Rubi steps
\begin {align*} \int \frac {d+e x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {d+e x^2}{\left (a b+b^2 x^2\right )^3} \, dx}{\sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(b d-a e) x}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left ((3 b d+a e) \left (a b+b^2 x^2\right )\right ) \int \frac {1}{\left (a b+b^2 x^2\right )^2} \, dx}{4 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(3 b d+a e) x}{8 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(b d-a e) x}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left ((3 b d+a e) \left (a b+b^2 x^2\right )\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{8 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {(3 b d+a e) x}{8 a^2 b \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(b d-a e) x}{4 a b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(3 b d+a e) \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 108, normalized size = 0.69 \[ \frac {\sqrt {a} \sqrt {b} x \left (a^2 (-e)+a b \left (5 d+e x^2\right )+3 b^2 d x^2\right )+\left (a+b x^2\right )^2 (a e+3 b d) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{8 a^{5/2} b^{3/2} \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 301, normalized size = 1.93 \[ \left [\frac {2 \, {\left (3 \, a b^{3} d + a^{2} b^{2} e\right )} x^{3} - {\left ({\left (3 \, b^{3} d + a b^{2} e\right )} x^{4} + 3 \, a^{2} b d + a^{3} e + 2 \, {\left (3 \, a b^{2} d + a^{2} b e\right )} x^{2}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + 2 \, {\left (5 \, a^{2} b^{2} d - a^{3} b e\right )} x}{16 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}, \frac {{\left (3 \, a b^{3} d + a^{2} b^{2} e\right )} x^{3} + {\left ({\left (3 \, b^{3} d + a b^{2} e\right )} x^{4} + 3 \, a^{2} b d + a^{3} e + 2 \, {\left (3 \, a b^{2} d + a^{2} b e\right )} x^{2}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (5 \, a^{2} b^{2} d - a^{3} b e\right )} x}{8 \, {\left (a^{3} b^{4} x^{4} + 2 \, a^{4} b^{3} x^{2} + a^{5} b^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 186, normalized size = 1.19 \[ \frac {\left (a \,b^{2} e \,x^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+3 b^{3} d \,x^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+2 a^{2} b e \,x^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+6 a \,b^{2} d \,x^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+\sqrt {a b}\, a b e \,x^{3}+3 \sqrt {a b}\, b^{2} d \,x^{3}+a^{3} e \arctan \left (\frac {b x}{\sqrt {a b}}\right )+3 a^{2} b d \arctan \left (\frac {b x}{\sqrt {a b}}\right )-\sqrt {a b}\, a^{2} e x +5 \sqrt {a b}\, a b d x \right ) \left (b \,x^{2}+a \right )}{8 \sqrt {a b}\, \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.67, size = 124, normalized size = 0.79 \[ \frac {1}{8} \, d {\left (\frac {3 \, b x^{3} + 5 \, a x}{a^{2} b^{2} x^{4} + 2 \, a^{3} b x^{2} + a^{4}} + \frac {3 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a^{2}}\right )} + \frac {1}{8} \, e {\left (\frac {b x^{3} - a x}{a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b} + \frac {\arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b} a b}\right )} \]
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 \[ \int \frac {e\,x^2+d}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/2}} \,d x \]
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 \[ \int \frac {d + e x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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