Optimal. Leaf size=74 \[ \frac {15 x}{8 c^3}-\frac {x^5}{4 c \left (b+c x^2\right )^2}-\frac {5 x^3}{8 c^2 \left (b+c x^2\right )}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{7/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 74, 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, 294, 327,
211} \begin {gather*} -\frac {15 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{7/2}}-\frac {5 x^3}{8 c^2 \left (b+c x^2\right )}-\frac {x^5}{4 c \left (b+c x^2\right )^2}+\frac {15 x}{8 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 294
Rule 327
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^{12}}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^6}{\left (b+c x^2\right )^3} \, dx\\ &=-\frac {x^5}{4 c \left (b+c x^2\right )^2}+\frac {5 \int \frac {x^4}{\left (b+c x^2\right )^2} \, dx}{4 c}\\ &=-\frac {x^5}{4 c \left (b+c x^2\right )^2}-\frac {5 x^3}{8 c^2 \left (b+c x^2\right )}+\frac {15 \int \frac {x^2}{b+c x^2} \, dx}{8 c^2}\\ &=\frac {15 x}{8 c^3}-\frac {x^5}{4 c \left (b+c x^2\right )^2}-\frac {5 x^3}{8 c^2 \left (b+c x^2\right )}-\frac {(15 b) \int \frac {1}{b+c x^2} \, dx}{8 c^3}\\ &=\frac {15 x}{8 c^3}-\frac {x^5}{4 c \left (b+c x^2\right )^2}-\frac {5 x^3}{8 c^2 \left (b+c x^2\right )}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 66, normalized size = 0.89 \begin {gather*} \frac {15 b^2 x+25 b c x^3+8 c^2 x^5}{8 c^3 \left (b+c x^2\right )^2}-\frac {15 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 51, normalized size = 0.69
method | result | size |
default | \(\frac {x}{c^{3}}-\frac {b \left (\frac {-\frac {9}{8} c \,x^{3}-\frac {7}{8} b x}{\left (c \,x^{2}+b \right )^{2}}+\frac {15 \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \sqrt {b c}}\right )}{c^{3}}\) | \(51\) |
risch | \(\frac {x}{c^{3}}+\frac {\frac {9}{8} b c \,x^{3}+\frac {7}{8} b^{2} x}{c^{3} \left (c \,x^{2}+b \right )^{2}}+\frac {15 \sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right )}{16 c^{4}}-\frac {15 \sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right )}{16 c^{4}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 68, normalized size = 0.92 \begin {gather*} \frac {9 \, b c x^{3} + 7 \, b^{2} x}{8 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} - \frac {15 \, b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{3}} + \frac {x}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 202, normalized size = 2.73 \begin {gather*} \left [\frac {16 \, c^{2} x^{5} + 50 \, b c x^{3} + 30 \, b^{2} x + 15 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right )}{16 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}}, \frac {8 \, c^{2} x^{5} + 25 \, b c x^{3} + 15 \, b^{2} x - 15 \, {\left (c^{2} x^{4} + 2 \, b c x^{2} + b^{2}\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right )}{8 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 107, normalized size = 1.45 \begin {gather*} \frac {15 \sqrt {- \frac {b}{c^{7}}} \log {\left (- c^{3} \sqrt {- \frac {b}{c^{7}}} + x \right )}}{16} - \frac {15 \sqrt {- \frac {b}{c^{7}}} \log {\left (c^{3} \sqrt {- \frac {b}{c^{7}}} + x \right )}}{16} + \frac {7 b^{2} x + 9 b c x^{3}}{8 b^{2} c^{3} + 16 b c^{4} x^{2} + 8 c^{5} x^{4}} + \frac {x}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.61, size = 54, normalized size = 0.73 \begin {gather*} -\frac {15 \, b \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} c^{3}} + \frac {x}{c^{3}} + \frac {9 \, b c x^{3} + 7 \, b^{2} x}{8 \, {\left (c x^{2} + b\right )}^{2} c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.25, size = 64, normalized size = 0.86 \begin {gather*} \frac {\frac {7\,b^2\,x}{8}+\frac {9\,c\,b\,x^3}{8}}{b^2\,c^3+2\,b\,c^4\,x^2+c^5\,x^4}+\frac {x}{c^3}-\frac {15\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )}{8\,c^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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