Optimal. Leaf size=251 \[ \frac {105}{128 a^4 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3}{16 a^2 x \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21}{64 a^3 x \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {315 \left (a+b x^2\right )}{128 a^5 x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {315 \sqrt {b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}} \]
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Rubi [A]
time = 0.07, antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1126, 296, 331,
211} \begin {gather*} \frac {3}{16 a^2 x \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^2}+\frac {1}{8 a x \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )^3}-\frac {315 \sqrt {b} \left (a+b x^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {315 \left (a+b x^2\right )}{128 a^5 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {105}{128 a^4 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21}{64 a^3 x \sqrt {a^2+2 a b x^2+b^2 x^4} \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 296
Rule 331
Rule 1126
Rubi steps
\begin {align*} \int \frac {1}{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 {1}{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 {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (9 b^3 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^4} \, dx}{8 a \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3}{16 a^2 x \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (21 b^2 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^3} \, dx}{16 a^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3}{16 a^2 x \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21}{64 a^3 x \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (105 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )^2} \, dx}{64 a^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {105}{128 a^4 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3}{16 a^2 x \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21}{64 a^3 x \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {\left (315 \left (a b+b^2 x^2\right )\right ) \int \frac {1}{x^2 \left (a b+b^2 x^2\right )} \, dx}{128 a^4 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {105}{128 a^4 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3}{16 a^2 x \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21}{64 a^3 x \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {315 \left (a+b x^2\right )}{128 a^5 x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {\left (315 b \left (a b+b^2 x^2\right )\right ) \int \frac {1}{a b+b^2 x^2} \, dx}{128 a^5 \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ &=\frac {105}{128 a^4 x \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {1}{8 a x \left (a+b x^2\right )^3 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {3}{16 a^2 x \left (a+b x^2\right )^2 \sqrt {a^2+2 a b x^2+b^2 x^4}}+\frac {21}{64 a^3 x \left (a+b x^2\right ) \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {315 \left (a+b x^2\right )}{128 a^5 x \sqrt {a^2+2 a b x^2+b^2 x^4}}-\frac {315 \sqrt {b} \left (a+b x^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{11/2} \sqrt {a^2+2 a b x^2+b^2 x^4}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 115, normalized size = 0.46 \begin {gather*} \frac {-\sqrt {a} \left (128 a^4+837 a^3 b x^2+1533 a^2 b^2 x^4+1155 a b^3 x^6+315 b^4 x^8\right )-315 \sqrt {b} x \left (a+b x^2\right )^4 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{128 a^{11/2} x \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 = 191, normalized size = 0.76
method | result | size |
risch | \(\frac {\sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (-\frac {315 b^{4} x^{8}}{128 a^{5}}-\frac {1155 b^{3} x^{6}}{128 a^{4}}-\frac {1533 b^{2} x^{4}}{128 a^{3}}-\frac {837 b \,x^{2}}{128 a^{2}}-\frac {1}{a}\right )}{\left (b \,x^{2}+a \right )^{5} x}+\frac {315 \sqrt {\left (b \,x^{2}+a \right )^{2}}\, \left (\munderset {\textit {\_R} =\RootOf \left (a^{11} \textit {\_Z}^{2}+b \right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} a^{11}+2 b \right ) x +a^{6} \textit {\_R} \right )\right )}{256 \left (b \,x^{2}+a \right )}\) | \(132\) |
default | \(-\frac {\left (315 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) b^{5} x^{9}+315 \sqrt {a b}\, b^{4} x^{8}+1260 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a \,b^{4} x^{7}+1155 \sqrt {a b}\, a \,b^{3} x^{6}+1890 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{2} b^{3} x^{5}+1533 \sqrt {a b}\, a^{2} b^{2} x^{4}+1260 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{3} b^{2} x^{3}+837 \sqrt {a b}\, a^{3} b \,x^{2}+315 \arctan \left (\frac {b x}{\sqrt {a b}}\right ) a^{4} b x +128 \sqrt {a b}\, a^{4}\right ) \left (b \,x^{2}+a \right )}{128 x \sqrt {a b}\, a^{5} \left (\left (b \,x^{2}+a \right )^{2}\right )^{\frac {5}{2}}}\) | \(191\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 115, normalized size = 0.46 \begin {gather*} -\frac {315 \, b^{4} x^{8} + 1155 \, a b^{3} x^{6} + 1533 \, a^{2} b^{2} x^{4} + 837 \, a^{3} b x^{2} + 128 \, a^{4}}{128 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}} - \frac {315 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 334, normalized size = 1.33 \begin {gather*} \left [-\frac {630 \, b^{4} x^{8} + 2310 \, a b^{3} x^{6} + 3066 \, a^{2} b^{2} x^{4} + 1674 \, a^{3} b x^{2} + 256 \, a^{4} - 315 \, {\left (b^{4} x^{9} + 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right )}{256 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\right )}}, -\frac {315 \, b^{4} x^{8} + 1155 \, a b^{3} x^{6} + 1533 \, a^{2} b^{2} x^{4} + 837 \, a^{3} b x^{2} + 128 \, a^{4} + 315 \, {\left (b^{4} x^{9} + 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right )}{128 \, {\left (a^{5} b^{4} x^{9} + 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} + 4 \, a^{8} b x^{3} + a^{9} x\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 {1}{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 = 3.94, size = 109, normalized size = 0.43 \begin {gather*} -\frac {315 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{128 \, \sqrt {a b} a^{5} \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {1}{a^{5} x \mathrm {sgn}\left (b x^{2} + a\right )} - \frac {187 \, b^{4} x^{7} + 643 \, a b^{3} x^{5} + 765 \, a^{2} b^{2} x^{3} + 325 \, a^{3} b x}{128 \, {\left (b x^{2} + a\right )}^{4} a^{5} \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 {1}{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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