Optimal. Leaf size=18 \[ -\frac {1}{14 \left (a+b x^2+c x^4\right )^7} \]
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Rubi [A]
time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1261, 643}
\begin {gather*} -\frac {1}{14 \left (a+b x^2+c x^4\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rule 1261
Rubi steps
\begin {align*} \int \frac {x \left (b+2 c x^2\right )}{\left (a+b x^2+c x^4\right )^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx,x,x^2\right )\\ &=-\frac {1}{14 \left (a+b x^2+c x^4\right )^7}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} -\frac {1}{14 \left (a+b x^2+c x^4\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 17, normalized size = 0.94
method | result | size |
gosper | \(-\frac {1}{14 \left (c \,x^{4}+b \,x^{2}+a \right )^{7}}\) | \(17\) |
default | \(-\frac {1}{14 \left (c \,x^{4}+b \,x^{2}+a \right )^{7}}\) | \(17\) |
norman | \(-\frac {1}{14 \left (c \,x^{4}+b \,x^{2}+a \right )^{7}}\) | \(17\) |
risch | \(-\frac {1}{14 \left (c \,x^{4}+b \,x^{2}+a \right )^{7}}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (16) = 32\).
time = 0.40, size = 352, normalized size = 19.56 \begin {gather*} -\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \, {\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{24} + 7 \, {\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{22} + 7 \, {\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \, {\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 7 \, {\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} x^{16} + {\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{14} + 7 \, {\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} x^{12} + 7 \, {\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} + 7 \, {\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} + a^{7} + 7 \, {\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{6} + 7 \, {\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (16) = 32\).
time = 0.39, size = 352, normalized size = 19.56 \begin {gather*} -\frac {1}{14 \, {\left (c^{7} x^{28} + 7 \, b c^{6} x^{26} + 7 \, {\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{24} + 7 \, {\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{22} + 7 \, {\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{20} + 7 \, {\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{18} + 7 \, {\left (b^{6} c + 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} + 5 \, a^{3} c^{4}\right )} x^{16} + {\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{14} + 7 \, {\left (a b^{6} + 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} + 5 \, a^{4} c^{3}\right )} x^{12} + 7 \, {\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{10} + 7 \, a^{6} b x^{2} + 7 \, {\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{8} + a^{7} + 7 \, {\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{6} + 7 \, {\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 360 vs.
\(2 (17) = 34\).
time = 4.88, size = 360, normalized size = 20.00 \begin {gather*} - \frac {1}{14 a^{7} + 98 a^{6} b x^{2} + 98 b c^{6} x^{26} + 14 c^{7} x^{28} + x^{24} \cdot \left (98 a c^{6} + 294 b^{2} c^{5}\right ) + x^{22} \cdot \left (588 a b c^{5} + 490 b^{3} c^{4}\right ) + x^{20} \cdot \left (294 a^{2} c^{5} + 1470 a b^{2} c^{4} + 490 b^{4} c^{3}\right ) + x^{18} \cdot \left (1470 a^{2} b c^{4} + 1960 a b^{3} c^{3} + 294 b^{5} c^{2}\right ) + x^{16} \cdot \left (490 a^{3} c^{4} + 2940 a^{2} b^{2} c^{3} + 1470 a b^{4} c^{2} + 98 b^{6} c\right ) + x^{14} \cdot \left (1960 a^{3} b c^{3} + 2940 a^{2} b^{3} c^{2} + 588 a b^{5} c + 14 b^{7}\right ) + x^{12} \cdot \left (490 a^{4} c^{3} + 2940 a^{3} b^{2} c^{2} + 1470 a^{2} b^{4} c + 98 a b^{6}\right ) + x^{10} \cdot \left (1470 a^{4} b c^{2} + 1960 a^{3} b^{3} c + 294 a^{2} b^{5}\right ) + x^{8} \cdot \left (294 a^{5} c^{2} + 1470 a^{4} b^{2} c + 490 a^{3} b^{4}\right ) + x^{6} \cdot \left (588 a^{5} b c + 490 a^{4} b^{3}\right ) + x^{4} \cdot \left (98 a^{6} c + 294 a^{5} b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.94, size = 16, normalized size = 0.89 \begin {gather*} -\frac {1}{14 \, {\left (c x^{4} + b x^{2} + a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 12.16, size = 360, normalized size = 20.00 \begin {gather*} -\frac {1}{14\,\left (x^{10}\,\left (105\,a^4\,b\,c^2+140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^{18}\,\left (105\,a^2\,b\,c^4+140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^{14}\,\left (140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2+42\,a\,b^5\,c+b^7\right )+x^6\,\left (42\,c\,a^5\,b+35\,a^4\,b^3\right )+x^{22}\,\left (35\,b^3\,c^4+42\,a\,b\,c^5\right )+x^8\,\left (21\,a^5\,c^2+105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{20}\,\left (21\,a^2\,c^5+105\,a\,b^2\,c^4+35\,b^4\,c^3\right )+a^7+x^{12}\,\left (35\,a^4\,c^3+210\,a^3\,b^2\,c^2+105\,a^2\,b^4\,c+7\,a\,b^6\right )+x^{16}\,\left (35\,a^3\,c^4+210\,a^2\,b^2\,c^3+105\,a\,b^4\,c^2+7\,b^6\,c\right )+c^7\,x^{28}+x^4\,\left (7\,c\,a^6+21\,a^5\,b^2\right )+x^{24}\,\left (21\,b^2\,c^5+7\,a\,c^6\right )+7\,a^6\,b\,x^2+7\,b\,c^6\,x^{26}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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