Optimal. Leaf size=213 \[ \frac {7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1088, 199, 205} \[ \frac {35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 199
Rule 205
Rule 1088
Rubi steps
\begin {align*} \int \frac {1}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}} \, dx &=\frac {\left (2 a b+2 b^2 x^2\right )^5 \int \frac {1}{\left (2 a b+2 b^2 x^2\right )^5} \, dx}{\left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}\\ &=\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {\left (7 \left (2 a b+2 b^2 x^2\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^2\right )^4} \, dx}{16 a b \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}\\ &=\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {\left (35 \left (2 a b+2 b^2 x^2\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^2\right )^3} \, dx}{192 a^2 b^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}\\ &=\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {\left (35 \left (2 a b+2 b^2 x^2\right )^5\right ) \int \frac {1}{\left (2 a b+2 b^2 x^2\right )^2} \, dx}{512 a^3 b^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}\\ &=\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {\left (35 \left (2 a b+2 b^2 x^2\right )^5\right ) \int \frac {1}{2 a b+2 b^2 x^2} \, dx}{2048 a^4 b^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}\\ &=\frac {x \left (a+b x^2\right )}{8 a \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {7 x \left (a+b x^2\right )^2}{48 a^2 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^3}{192 a^3 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 x \left (a+b x^2\right )^4}{128 a^4 \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}+\frac {35 \left (a+b x^2\right )^5 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{9/2} \sqrt {b} \left (a^2+2 a b x^2+b^2 x^4\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 105, normalized size = 0.49 \[ \frac {\sqrt {a} \sqrt {b} x \left (279 a^3+511 a^2 b x^2+385 a b^2 x^4+105 b^3 x^6\right )+105 \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{384 a^{9/2} \sqrt {b} \left (a+b x^2\right )^3 \sqrt {\left (a+b x^2\right )^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.12, size = 320, normalized size = 1.50 \[ \left [\frac {210 \, a b^{4} x^{7} + 770 \, a^{2} b^{3} x^{5} + 1022 \, a^{3} b^{2} x^{3} + 558 \, a^{4} b x - 105 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {-a b} \log \left (\frac {b x^{2} - 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right )}{768 \, {\left (a^{5} b^{5} x^{8} + 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} + 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}, \frac {105 \, a b^{4} x^{7} + 385 \, a^{2} b^{3} x^{5} + 511 \, a^{3} b^{2} x^{3} + 279 \, a^{4} b x + 105 \, {\left (b^{4} x^{8} + 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} + 4 \, a^{3} b x^{2} + a^{4}\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right )}{384 \, {\left (a^{5} b^{5} x^{8} + 4 \, a^{6} b^{4} x^{6} + 6 \, a^{7} b^{3} x^{4} + 4 \, a^{8} b^{2} x^{2} + a^{9} b\right )}}\right ] \]
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 = 169, normalized size = 0.79 \[ \frac {\left (105 b^{4} x^{8} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+420 a \,b^{3} x^{6} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+105 \sqrt {a b}\, b^{3} x^{7}+630 a^{2} b^{2} x^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+385 \sqrt {a b}\, a \,b^{2} x^{5}+420 a^{3} b \,x^{2} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+511 \sqrt {a b}\, a^{2} b \,x^{3}+105 a^{4} \arctan \left (\frac {b x}{\sqrt {a b}}\right )+279 \sqrt {a b}\, a^{3} x \right ) \left (b \,x^{2}+a \right )}{384 \sqrt {a b}\, \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.05, size = 102, normalized size = 0.48 \[ \frac {105 \, b^{3} x^{7} + 385 \, a b^{2} x^{5} + 511 \, a^{2} b x^{3} + 279 \, a^{3} x}{384 \, {\left (a^{4} b^{4} x^{8} + 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} + 4 \, a^{7} b x^{2} + a^{8}\right )}} + \frac {35 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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