Optimal. Leaf size=76 \[ -\frac {15}{8 b^3 x}+\frac {1}{4 b x \left (b+c x^2\right )^2}+\frac {5}{8 b^2 x \left (b+c x^2\right )}-\frac {15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {15 \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2}}-\frac {15}{8 b^3 x}+\frac {5}{8 b^2 x \left (b+c x^2\right )}+\frac {1}{4 b x \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^4}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {1}{x^2 \left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{4 b x \left (b+c x^2\right )^2}+\frac {5 \int \frac {1}{x^2 \left (b+c x^2\right )^2} \, dx}{4 b}\\ &=\frac {1}{4 b x \left (b+c x^2\right )^2}+\frac {5}{8 b^2 x \left (b+c x^2\right )}+\frac {15 \int \frac {1}{x^2 \left (b+c x^2\right )} \, dx}{8 b^2}\\ &=-\frac {15}{8 b^3 x}+\frac {1}{4 b x \left (b+c x^2\right )^2}+\frac {5}{8 b^2 x \left (b+c x^2\right )}-\frac {(15 c) \int \frac {1}{b+c x^2} \, dx}{8 b^3}\\ &=-\frac {15}{8 b^3 x}+\frac {1}{4 b x \left (b+c x^2\right )^2}+\frac {5}{8 b^2 x \left (b+c x^2\right )}-\frac {15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.89 \begin {gather*} -\frac {8 b^2+25 b c x^2+15 c^2 x^4}{8 b^3 x \left (b+c x^2\right )^2}-\frac {15 \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 54, normalized size = 0.71
method | result | size |
default | \(-\frac {c \left (\frac {\frac {7}{8} c \,x^{3}+\frac {9}{8} b x}{\left (c \,x^{2}+b \right )^{2}}+\frac {15 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{b^{3}}-\frac {1}{b^{3} x}\) | \(54\) |
risch | \(\frac {-\frac {15 c^{2} x^{4}}{8 b^{3}}-\frac {25 c \,x^{2}}{8 b^{2}}-\frac {1}{b}}{x \left (c \,x^{2}+b \right )^{2}}+\frac {15 \sqrt {-b c}\, \ln \left (-c x +\sqrt {-b c}\right )}{16 b^{4}}-\frac {15 \sqrt {-b c}\, \ln \left (-c x -\sqrt {-b c}\right )}{16 b^{4}}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 71, normalized size = 0.93 \begin {gather*} -\frac {15 \, c^{2} x^{4} + 25 \, b c x^{2} + 8 \, b^{2}}{8 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}} - \frac {15 \, c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 202, normalized size = 2.66 \begin {gather*} \left [-\frac {30 \, c^{2} x^{4} + 50 \, b c x^{2} - 15 \, {\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} - 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) + 16 \, b^{2}}{16 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}, -\frac {15 \, c^{2} x^{4} + 25 \, b c x^{2} + 15 \, {\left (c^{2} x^{5} + 2 \, b c x^{3} + b^{2} x\right )} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) + 8 \, b^{2}}{8 \, {\left (b^{3} c^{2} x^{5} + 2 \, b^{4} c x^{3} + b^{5} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 116, normalized size = 1.53 \begin {gather*} \frac {15 \sqrt {- \frac {c}{b^{7}}} \log {\left (- \frac {b^{4} \sqrt {- \frac {c}{b^{7}}}}{c} + x \right )}}{16} - \frac {15 \sqrt {- \frac {c}{b^{7}}} \log {\left (\frac {b^{4} \sqrt {- \frac {c}{b^{7}}}}{c} + x \right )}}{16} + \frac {- 8 b^{2} - 25 b c x^{2} - 15 c^{2} x^{4}}{8 b^{5} x + 16 b^{4} c x^{3} + 8 b^{3} c^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.03, size = 57, normalized size = 0.75 \begin {gather*} -\frac {15 \, c \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b^{3}} - \frac {7 \, c^{2} x^{3} + 9 \, b c x}{8 \, {\left (c x^{2} + b\right )}^{2} b^{3}} - \frac {1}{b^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.26, size = 66, normalized size = 0.87 \begin {gather*} -\frac {\frac {1}{b}+\frac {25\,c\,x^2}{8\,b^2}+\frac {15\,c^2\,x^4}{8\,b^3}}{b^2\,x+2\,b\,c\,x^3+c^2\,x^5}-\frac {15\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,b^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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