Optimal. Leaf size=16 \[ -\frac {1}{7 \left (a+b x+c x^2\right )^7} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {643}
\begin {gather*} -\frac {1}{7 \left (a+b x+c x^2\right )^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 643
Rubi steps
\begin {align*} \int \frac {b+2 c x}{\left (a+b x+c x^2\right )^8} \, dx &=-\frac {1}{7 \left (a+b x+c x^2\right )^7}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 15, normalized size = 0.94 \begin {gather*} -\frac {1}{7 (a+x (b+c x))^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 15, normalized size = 0.94
method | result | size |
gosper | \(-\frac {1}{7 \left (c \,x^{2}+b x +a \right )^{7}}\) | \(15\) |
derivativedivides | \(-\frac {1}{7 \left (c \,x^{2}+b x +a \right )^{7}}\) | \(15\) |
default | \(-\frac {1}{7 \left (c \,x^{2}+b x +a \right )^{7}}\) | \(15\) |
norman | \(-\frac {1}{7 \left (c \,x^{2}+b x +a \right )^{7}}\) | \(15\) |
risch | \(-\frac {1}{7 \left (c \,x^{2}+b x +a \right )^{7}}\) | \(15\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{7 \, {\left (c x^{2} + b x + a\right )}^{7}} \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. 350 vs.
\(2 (14) = 28\).
time = 0.39, size = 350, normalized size = 21.88 \begin {gather*} -\frac {1}{7 \, {\left (c^{7} x^{14} + 7 \, b c^{6} x^{13} + 7 \, {\left (3 \, b^{2} c^{5} + a c^{6}\right )} x^{12} + 7 \, {\left (5 \, b^{3} c^{4} + 6 \, a b c^{5}\right )} x^{11} + 7 \, {\left (5 \, b^{4} c^{3} + 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} x^{10} + 7 \, {\left (3 \, b^{5} c^{2} + 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} x^{9} + 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^{8} + 7 \, a^{6} b x + {\left (b^{7} + 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} + 140 \, a^{3} b c^{3}\right )} x^{7} + a^{7} + 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^{6} + 7 \, {\left (3 \, a^{2} b^{5} + 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} x^{5} + 7 \, {\left (5 \, a^{3} b^{4} + 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} x^{4} + 7 \, {\left (5 \, a^{4} b^{3} + 6 \, a^{5} b c\right )} x^{3} + 7 \, {\left (3 \, a^{5} b^{2} + a^{6} c\right )} x^{2}\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. 359 vs.
\(2 (15) = 30\).
time = 3.26, size = 359, normalized size = 22.44 \begin {gather*} - \frac {1}{7 a^{7} + 49 a^{6} b x + 49 b c^{6} x^{13} + 7 c^{7} x^{14} + x^{12} \cdot \left (49 a c^{6} + 147 b^{2} c^{5}\right ) + x^{11} \cdot \left (294 a b c^{5} + 245 b^{3} c^{4}\right ) + x^{10} \cdot \left (147 a^{2} c^{5} + 735 a b^{2} c^{4} + 245 b^{4} c^{3}\right ) + x^{9} \cdot \left (735 a^{2} b c^{4} + 980 a b^{3} c^{3} + 147 b^{5} c^{2}\right ) + x^{8} \cdot \left (245 a^{3} c^{4} + 1470 a^{2} b^{2} c^{3} + 735 a b^{4} c^{2} + 49 b^{6} c\right ) + x^{7} \cdot \left (980 a^{3} b c^{3} + 1470 a^{2} b^{3} c^{2} + 294 a b^{5} c + 7 b^{7}\right ) + x^{6} \cdot \left (245 a^{4} c^{3} + 1470 a^{3} b^{2} c^{2} + 735 a^{2} b^{4} c + 49 a b^{6}\right ) + x^{5} \cdot \left (735 a^{4} b c^{2} + 980 a^{3} b^{3} c + 147 a^{2} b^{5}\right ) + x^{4} \cdot \left (147 a^{5} c^{2} + 735 a^{4} b^{2} c + 245 a^{3} b^{4}\right ) + x^{3} \cdot \left (294 a^{5} b c + 245 a^{4} b^{3}\right ) + x^{2} \cdot \left (49 a^{6} c + 147 a^{5} b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.93, size = 14, normalized size = 0.88 \begin {gather*} -\frac {1}{7 \, {\left (c x^{2} + b x + a\right )}^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.62, size = 358, normalized size = 22.38 \begin {gather*} -\frac {1}{7\,\left (x^5\,\left (105\,a^4\,b\,c^2+140\,a^3\,b^3\,c+21\,a^2\,b^5\right )+x^9\,\left (105\,a^2\,b\,c^4+140\,a\,b^3\,c^3+21\,b^5\,c^2\right )+x^7\,\left (140\,a^3\,b\,c^3+210\,a^2\,b^3\,c^2+42\,a\,b^5\,c+b^7\right )+x^3\,\left (42\,c\,a^5\,b+35\,a^4\,b^3\right )+x^{11}\,\left (35\,b^3\,c^4+42\,a\,b\,c^5\right )+x^4\,\left (21\,a^5\,c^2+105\,a^4\,b^2\,c+35\,a^3\,b^4\right )+x^{10}\,\left (21\,a^2\,c^5+105\,a\,b^2\,c^4+35\,b^4\,c^3\right )+a^7+x^6\,\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^8\,\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^{14}+x^2\,\left (7\,c\,a^6+21\,a^5\,b^2\right )+x^{12}\,\left (21\,b^2\,c^5+7\,a\,c^6\right )+7\,b\,c^6\,x^{13}+7\,a^6\,b\,x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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