Optimal. Leaf size=135 \[ \frac {6 x}{35 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}}+\frac {x \left (a+b x^2\right )}{7 a \left (a^2+2 a b x^2+b^2 x^4\right )^{9/4}}+\frac {16 x}{35 a^4 \sqrt [4]{a^2+2 a b x^2+b^2 x^4}}+\frac {8 x \left (a+b x^2\right )}{35 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/4}} \]
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Rubi [A] time = 0.04, antiderivative size = 148, normalized size of antiderivative = 1.10, 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 191
Rule 192
Rule 1089
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*}
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Mathematica [A] time = 0.02, size = 62, normalized size = 0.46 \[ \frac {x \left (35 a^3+70 a^2 b x^2+56 a b^2 x^4+16 b^3 x^6\right )}{35 a^4 \left (a+b x^2\right )^3 \sqrt [4]{\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 102, normalized size = 0.76 \[ \frac {{\left (16 \, b^{3} x^{7} + 56 \, a b^{2} x^{5} + 70 \, a^{2} b x^{3} + 35 \, a^{3} x\right )} {\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {1}{4}}}{35 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} \]
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 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 66, normalized size = 0.49 \[ \frac {\left (b \,x^{2}+a \right ) \left (16 b^{3} x^{6}+56 b^{2} x^{4} a +70 a^{2} b \,x^{2}+35 a^{3}\right ) x}{35 \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )^{\frac {9}{4}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )}^{\frac {9}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.13, size = 141, normalized size = 1.04 \[ \frac {x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/4}}{7\,a\,{\left (b\,x^2+a\right )}^5}+\frac {6\,x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/4}}{35\,a^2\,{\left (b\,x^2+a\right )}^4}+\frac {8\,x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/4}}{35\,a^3\,{\left (b\,x^2+a\right )}^3}+\frac {16\,x\,{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{3/4}}{35\,a^4\,{\left (b\,x^2+a\right )}^2} \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {9}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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