Optimal. Leaf size=213 \[ \frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A] time = 0.0835851, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1112, 288, 199, 205} \[ \frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1112
Rule 288
Rule 199
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^2\right )\right ) \int \frac{x^2}{\left (a b+b^2 x^2\right )^5} \, dx}{\sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (b^2 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\left (a b+b^2 x^2\right )^4} \, dx}{8 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (5 b \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\left (a b+b^2 x^2\right )^3} \, dx}{48 a \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (5 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{\left (a b+b^2 x^2\right )^2} \, dx}{64 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (5 \left (a b+b^2 x^2\right )\right ) \int \frac{1}{a b+b^2 x^2} \, dx}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{5 x}{128 a^3 b \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{x}{8 b \left (a+b x^2\right )^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{48 a b \left (a+b x^2\right )^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{5 \left (a+b x^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{7/2} b^{3/2} \sqrt{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0342345, size = 105, normalized size = 0.49 \[ \frac{\sqrt{a} \sqrt{b} x \left (73 a^2 b x^2-15 a^3+55 a b^2 x^4+15 b^3 x^6\right )+15 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{384 a^{7/2} b^{3/2} \left (a+b x^2\right )^3 \sqrt{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.225, size = 172, normalized size = 0.8 \begin{align*}{\frac{b{x}^{2}+a}{384\,{a}^{3}b} \left ( 15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{8}{b}^{4}+15\,\sqrt{ab}{x}^{7}{b}^{3}+60\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{6}a{b}^{3}+55\,\sqrt{ab}{x}^{5}a{b}^{2}+90\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{4}{a}^{2}{b}^{2}+73\,\sqrt{ab}{x}^{3}{a}^{2}b+60\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){x}^{2}{a}^{3}b-15\,\sqrt{ab}x{a}^{3}+15\,\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ){a}^{4} \right ){\frac{1}{\sqrt{ab}}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57314, size = 684, normalized size = 3.21 \begin{align*} \left [\frac{30 \, a b^{4} x^{7} + 110 \, a^{2} b^{3} x^{5} + 146 \, a^{3} b^{2} x^{3} - 30 \, a^{4} b x - 15 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{768 \,{\left (a^{4} b^{6} x^{8} + 4 \, a^{5} b^{5} x^{6} + 6 \, a^{6} b^{4} x^{4} + 4 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}}, \frac{15 \, a b^{4} x^{7} + 55 \, a^{2} b^{3} x^{5} + 73 \, a^{3} b^{2} x^{3} - 15 \, a^{4} b x + 15 \,{\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{384 \,{\left (a^{4} b^{6} x^{8} + 4 \, a^{5} b^{5} x^{6} + 6 \, a^{6} b^{4} x^{4} + 4 \, a^{7} b^{3} x^{2} + a^{8} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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