Optimal. Leaf size=135 \[ \frac{16 x}{35 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{8 x \left (a+b x^2\right )}{35 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}}+\frac{6 x}{35 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}}+\frac{x \left (a+b x^2\right )}{7 a \left (a^2+2 a b x^2+b^2 x^4\right )^{9/4}} \]
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Rubi [A] time = 0.0432201, antiderivative size = 148, normalized size of antiderivative = 1.1, number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1089, 192, 191} \[ \frac{8 x}{35 a^3 \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{7 a \left (a+b x^2\right )^3 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{16 x}{35 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}} \]
Antiderivative was successfully verified.
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Rule 1089
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{9/4}} \, dx &=\frac{\sqrt{1+\frac{b x^2}{a}} \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{9/2}} \, dx}{a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{x}{7 a \left (a+b x^2\right )^3 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (6 \sqrt{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{7/2}} \, dx}{7 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{x}{7 a \left (a+b x^2\right )^3 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (24 \sqrt{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{5/2}} \, dx}{35 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{x}{7 a \left (a+b x^2\right )^3 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{8 x}{35 a^3 \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{\left (16 \sqrt{1+\frac{b x^2}{a}}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/2}} \, dx}{35 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ &=\frac{16 x}{35 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{x}{7 a \left (a+b x^2\right )^3 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{6 x}{35 a^2 \left (a+b x^2\right )^2 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac{8 x}{35 a^3 \left (a+b x^2\right ) \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}\\ \end{align*}
Mathematica [A] time = 0.0236541, size = 62, normalized size = 0.46 \[ \frac{x \left (70 a^2 b x^2+35 a^3+56 a b^2 x^4+16 b^3 x^6\right )}{35 a^4 \left (a+b x^2\right )^3 \sqrt [4]{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 66, normalized size = 0.5 \begin{align*}{\frac{ \left ( b{x}^{2}+a \right ) x \left ( 16\,{b}^{3}{x}^{6}+56\,{b}^{2}{x}^{4}a+70\,{a}^{2}b{x}^{2}+35\,{a}^{3} \right ) }{35\,{a}^{4}} \left ({b}^{2}{x}^{4}+2\,ab{x}^{2}+{a}^{2} \right ) ^{-{\frac{9}{4}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.37772, size = 216, normalized size = 1.6 \begin{align*} \frac{{\left (16 \, b^{3} x^{7} + 56 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 35 \, a^{3} x\right )}{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{1}{4}}}{35 \,{\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\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{1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{9}{4}}}\, 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*} \int \frac{1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac{9}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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