Optimal. Leaf size=87 \[ -\frac {35}{24 b^3 x^3}+\frac {35 c}{8 b^4 x}+\frac {1}{4 b x^3 \left (b+c x^2\right )^2}+\frac {7}{8 b^2 x^3 \left (b+c x^2\right )}+\frac {35 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {1598, 296, 331,
211} \begin {gather*} \frac {35 c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}}+\frac {35 c}{8 b^4 x}-\frac {35}{24 b^3 x^3}+\frac {7}{8 b^2 x^3 \left (b+c x^2\right )}+\frac {1}{4 b x^3 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 296
Rule 331
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^2}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{x^4 \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{4 b x^3 \left (b+c x^2\right )^2}+\frac {7 \int \frac {1}{x^4 \left (b+c x^2\right )^2} \, dx}{4 b}\\ &=\frac {1}{4 b x^3 \left (b+c x^2\right )^2}+\frac {7}{8 b^2 x^3 \left (b+c x^2\right )}+\frac {35 \int \frac {1}{x^4 \left (b+c x^2\right )} \, dx}{8 b^2}\\ &=-\frac {35}{24 b^3 x^3}+\frac {1}{4 b x^3 \left (b+c x^2\right )^2}+\frac {7}{8 b^2 x^3 \left (b+c x^2\right )}-\frac {(35 c) \int \frac {1}{x^2 \left (b+c x^2\right )} \, dx}{8 b^3}\\ &=-\frac {35}{24 b^3 x^3}+\frac {35 c}{8 b^4 x}+\frac {1}{4 b x^3 \left (b+c x^2\right )^2}+\frac {7}{8 b^2 x^3 \left (b+c x^2\right )}+\frac {\left (35 c^2\right ) \int \frac {1}{b+c x^2} \, dx}{8 b^4}\\ &=-\frac {35}{24 b^3 x^3}+\frac {35 c}{8 b^4 x}+\frac {1}{4 b x^3 \left (b+c x^2\right )^2}+\frac {7}{8 b^2 x^3 \left (b+c x^2\right )}+\frac {35 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 79, normalized size = 0.91 \begin {gather*} \frac {-8 b^3+56 b^2 c x^2+175 b c^2 x^4+105 c^3 x^6}{24 b^4 x^3 \left (b+c x^2\right )^2}+\frac {35 c^{3/2} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 64, normalized size = 0.74
method | result | size |
default | \(\frac {c^{2} \left (\frac {\frac {11}{8} c \,x^{3}+\frac {13}{8} b x}{\left (c \,x^{2}+b \right )^{2}}+\frac {35 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{b^{4}}-\frac {1}{3 b^{3} x^{3}}+\frac {3 c}{b^{4} x}\) | \(64\) |
risch | \(\frac {\frac {35 c^{3} x^{6}}{8 b^{4}}+\frac {175 c^{2} x^{4}}{24 b^{3}}+\frac {7 c \,x^{2}}{3 b^{2}}-\frac {1}{3 b}}{x^{3} \left (c \,x^{2}+b \right )^{2}}+\frac {35 \left (\munderset {\textit {\_R} =\RootOf \left (b^{9} \textit {\_Z}^{2}+c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{9}+2 c^{3}\right ) x -b^{5} c \textit {\_R} \right )\right )}{16}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 86, normalized size = 0.99 \begin {gather*} \frac {105 \, c^{3} x^{6} + 175 \, b c^{2} x^{4} + 56 \, b^{2} c x^{2} - 8 \, b^{3}}{24 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}} + \frac {35 \, c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 238, normalized size = 2.74 \begin {gather*} \left [\frac {210 \, c^{3} x^{6} + 350 \, b c^{2} x^{4} + 112 \, b^{2} c x^{2} - 16 \, b^{3} + 105 \, {\left (c^{3} x^{7} + 2 \, b c^{2} x^{5} + b^{2} c x^{3}\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right )}{48 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}}, \frac {105 \, c^{3} x^{6} + 175 \, b c^{2} x^{4} + 56 \, b^{2} c x^{2} - 8 \, b^{3} + 105 \, {\left (c^{3} x^{7} + 2 \, b c^{2} x^{5} + b^{2} c x^{3}\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right )}{24 \, {\left (b^{4} c^{2} x^{7} + 2 \, b^{5} c x^{5} + b^{6} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 138, normalized size = 1.59 \begin {gather*} - \frac {35 \sqrt {- \frac {c^{3}}{b^{9}}} \log {\left (- \frac {b^{5} \sqrt {- \frac {c^{3}}{b^{9}}}}{c^{2}} + x \right )}}{16} + \frac {35 \sqrt {- \frac {c^{3}}{b^{9}}} \log {\left (\frac {b^{5} \sqrt {- \frac {c^{3}}{b^{9}}}}{c^{2}} + x \right )}}{16} + \frac {- 8 b^{3} + 56 b^{2} c x^{2} + 175 b c^{2} x^{4} + 105 c^{3} x^{6}}{24 b^{6} x^{3} + 48 b^{5} c x^{5} + 24 b^{4} c^{2} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.20, size = 71, normalized size = 0.82 \begin {gather*} \frac {35 \, c^{2} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{4}} + \frac {11 \, c^{3} x^{3} + 13 \, b c^{2} x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{4}} + \frac {9 \, c x^{2} - b}{3 \, b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.26, size = 80, normalized size = 0.92 \begin {gather*} \frac {\frac {7\,c\,x^2}{3\,b^2}-\frac {1}{3\,b}+\frac {175\,c^2\,x^4}{24\,b^3}+\frac {35\,c^3\,x^6}{8\,b^4}}{b^2\,x^3+2\,b\,c\,x^5+c^2\,x^7}+\frac {35\,c^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,b^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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