Optimal. Leaf size=77 \[ -\frac {b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.07, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {1247, 640, 607} \[ -\frac {b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 607
Rule 640
Rule 1247
Rubi steps
\begin {align*} \int \frac {x \left (d+e x^2\right )}{\left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+e x}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {(b d-a e) \operatorname {Subst}\left (\int \frac {1}{\left (a^2+2 a b x+b^2 x^2\right )^{3/2}} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {e}{2 b^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {b d-a e}{4 b^2 \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 45, normalized size = 0.58 \[ \frac {-a e-b \left (d+2 e x^2\right )}{4 b^2 \left (a+b x^2\right ) \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 42, normalized size = 0.55 \[ -\frac {2 \, b e x^{2} + b d + a e}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.51, size = 40, normalized size = 0.52 \[ -\frac {2 \, b x^{2} e + b d + a e}{4 \, {\left (b x^{2} + a\right )}^{2} b^{2} \mathrm {sgn}\left (b x^{2} + a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 38, normalized size = 0.49 \[ -\frac {\left (b \,x^{2}+a \right ) \left (2 b e \,x^{2}+a e +b d \right )}{4 \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {3}{2}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.64, size = 65, normalized size = 0.84 \[ -\frac {{\left (2 \, b x^{2} + a\right )} e}{4 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{2} + a^{2} b^{2}\right )}} - \frac {d}{4 \, {\left (b^{3} x^{4} + 2 \, a b^{2} x^{2} + a^{2} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 48, normalized size = 0.62 \[ -\frac {\sqrt {a^2+2\,a\,b\,x^2+b^2\,x^4}\,\left (2\,b\,e\,x^2+a\,e+b\,d\right )}{4\,b^2\,{\left (b\,x^2+a\right )}^3} \]
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 \[ \int \frac {x \left (d + e x^{2}\right )}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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