Optimal. Leaf size=213 \[ \frac {5 x}{128 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {x}{8 b \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {x}{48 a b \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 x}{192 a^2 b \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {5 \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{7/2} b^{3/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.06, antiderivative size = 213, normalized size of antiderivative = 1.00, 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 = {1126, 294, 205,
211} \begin {gather*} \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 ) \text {ArcTan}\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}}+\frac {5 x}{128 a^3 b \sqrt {a^2+2 a b x^2+b^2 x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 211
Rule 294
Rule 1126
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*}
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Mathematica [A]
time = 0.02, size = 105, normalized size = 0.49 \begin {gather*} \frac {\sqrt {a} \sqrt {b} x \left (-15 a^3+73 a^2 b x^2+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}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 172, normalized size = 0.81
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\frac {5 b^{2} x^{7}}{128 a^{3}}+\frac {55 b \,x^{5}}{384 a^{2}}+\frac {73 x^{3}}{384 a}-\frac {5 x}{128 b}\right )}{\left (b \,x^{2}+a \right )^{5}}-\frac {5 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (b x +\sqrt {-a b}\right )}{256 \left (b \,x^{2}+a \right ) \sqrt {-a b}\, b \,a^{3}}+\frac {5 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \ln \left (-b x +\sqrt {-a b}\right )}{256 \left (b \,x^{2}+a \right ) \sqrt {-a b}\, b \,a^{3}}\) | \(149\) |
default | \(-\frac {\left (-15 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) b^{4} x^{8}-15 \sqrt {a b}\, b^{3} x^{7}-60 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a \,b^{3} x^{6}-55 \sqrt {a b}\, a \,b^{2} x^{5}-90 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{2} b^{2} x^{4}-73 \sqrt {a b}\, a^{2} b \,x^{3}-60 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{3} b \,x^{2}+15 \sqrt {a b}\, a^{3} x -15 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{4}\right ) \left (b \,x^{2}+a \right )}{384 \sqrt {a b}\, b \,a^{3} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 109, normalized size = 0.51 \begin {gather*} \frac {15 \, b^{3} x^{7} + 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} - 15 \, a^{3} x}{384 \, {\left (a^{3} b^{5} x^{8} + 4 \, a^{4} b^{4} x^{6} + 6 \, a^{5} b^{3} x^{4} + 4 \, a^{6} b^{2} x^{2} + a^{7} b\right )}} + \frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{3} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 324, normalized size = 1.52 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\left (\left (a + b x^{2}\right )^{2}\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.26, size = 93, normalized size = 0.44 \begin {gather*} \frac {5 \, \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )} + \frac {15 \, b^{3} x^{7} + 55 \, a b^{2} x^{5} + 73 \, a^{2} b x^{3} - 15 \, a^{3} x}{384 \, {\left (b x^{2} + a\right )}^{4} a^{3} b \mathrm {sgn}\left (b x^{2} + a\right )} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\left (a^2+2\,a\,b\,x^2+b^2\,x^4\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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