Optimal. Leaf size=100 \[ \frac {\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac {3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac {3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac {\log (b+2 c x)}{128 c^4 d^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {697}
\begin {gather*} \frac {\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac {3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac {3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac {\log (b+2 c x)}{128 c^4 d^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 697
Rubi steps
\begin {align*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx &=\int \left (\frac {\left (-b^2+4 a c\right )^3}{64 c^3 d^7 (b+2 c x)^7}+\frac {3 \left (-b^2+4 a c\right )^2}{64 c^3 d^7 (b+2 c x)^5}+\frac {3 \left (-b^2+4 a c\right )}{64 c^3 d^7 (b+2 c x)^3}+\frac {1}{64 c^3 d^7 (b+2 c x)}\right ) \, dx\\ &=\frac {\left (b^2-4 a c\right )^3}{768 c^4 d^7 (b+2 c x)^6}-\frac {3 \left (b^2-4 a c\right )^2}{512 c^4 d^7 (b+2 c x)^4}+\frac {3 \left (b^2-4 a c\right )}{256 c^4 d^7 (b+2 c x)^2}+\frac {\log (b+2 c x)}{128 c^4 d^7}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 78, normalized size = 0.78 \begin {gather*} \frac {\frac {2 \left (b^2-4 a c\right )^3}{(b+2 c x)^6}-\frac {9 \left (b^2-4 a c\right )^2}{(b+2 c x)^4}+\frac {18 \left (b^2-4 a c\right )}{(b+2 c x)^2}+12 \log (b+2 c x)}{1536 c^4 d^7} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.75, size = 120, normalized size = 1.20
method | result | size |
default | \(\frac {-\frac {48 a^{2} c^{2}-24 a c \,b^{2}+3 b^{4}}{512 c^{4} \left (2 c x +b \right )^{4}}-\frac {64 a^{3} c^{3}-48 a^{2} b^{2} c^{2}+12 a \,b^{4} c -b^{6}}{768 c^{4} \left (2 c x +b \right )^{6}}+\frac {\ln \left (2 c x +b \right )}{128 c^{4}}-\frac {12 a c -3 b^{2}}{256 c^{4} \left (2 c x +b \right )^{2}}}{d^{7}}\) | \(120\) |
risch | \(\frac {\left (-\frac {3 a c}{4}+\frac {3 b^{2}}{16}\right ) x^{4}-\frac {3 b \left (4 a c -b^{2}\right ) x^{3}}{8 c}-\frac {3 \left (16 a^{2} c^{2}+40 a c \,b^{2}-11 b^{4}\right ) x^{2}}{128 c^{2}}-\frac {3 b \left (16 a^{2} c^{2}+8 a c \,b^{2}-3 b^{4}\right ) x}{128 c^{3}}-\frac {128 a^{3} c^{3}+48 a^{2} b^{2} c^{2}+24 a \,b^{4} c -11 b^{6}}{1536 c^{4}}}{d^{7} \left (2 c x +b \right )^{6}}+\frac {\ln \left (2 c x +b \right )}{128 c^{4} d^{7}}\) | \(157\) |
norman | \(\frac {-\frac {128 a^{3} c^{5}+48 a^{2} b^{2} c^{4}+24 a \,b^{4} c^{3}-11 c^{2} b^{6}}{1536 c^{6} d}-\frac {3 \left (4 c^{3} a -b^{2} c^{2}\right ) x^{4}}{16 c^{2} d}-\frac {3 \left (16 a^{2} c^{4}+40 a \,b^{2} c^{3}-11 b^{4} c^{2}\right ) x^{2}}{128 c^{4} d}-\frac {b \left (12 c^{3} a -3 b^{2} c^{2}\right ) x^{3}}{8 c^{3} d}-\frac {3 b \left (16 a^{2} c^{4}+8 a \,b^{2} c^{3}-3 b^{4} c^{2}\right ) x}{128 c^{5} d}}{d^{6} \left (2 c x +b \right )^{6}}+\frac {\ln \left (2 c x +b \right )}{128 c^{4} d^{7}}\) | \(201\) |
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. 238 vs.
\(2 (92) = 184\).
time = 0.29, size = 238, normalized size = 2.38 \begin {gather*} \frac {11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \, {\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\right )}} + \frac {\log \left (2 \, c x + b\right )}{128 \, c^{4} d^{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. 292 vs.
\(2 (92) = 184\).
time = 2.92, size = 292, normalized size = 2.92 \begin {gather*} \frac {11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \, {\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x + 12 \, {\left (64 \, c^{6} x^{6} + 192 \, b c^{5} x^{5} + 240 \, b^{2} c^{4} x^{4} + 160 \, b^{3} c^{3} x^{3} + 60 \, b^{4} c^{2} x^{2} + 12 \, b^{5} c x + b^{6}\right )} \log \left (2 \, c x + b\right )}{1536 \, {\left (64 \, c^{10} d^{7} x^{6} + 192 \, b c^{9} d^{7} x^{5} + 240 \, b^{2} c^{8} d^{7} x^{4} + 160 \, b^{3} c^{7} d^{7} x^{3} + 60 \, b^{4} c^{6} d^{7} x^{2} + 12 \, b^{5} c^{5} d^{7} x + b^{6} c^{4} d^{7}\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. 245 vs.
\(2 (100) = 200\).
time = 4.56, size = 245, normalized size = 2.45 \begin {gather*} \frac {- 128 a^{3} c^{3} - 48 a^{2} b^{2} c^{2} - 24 a b^{4} c + 11 b^{6} + x^{4} \left (- 1152 a c^{5} + 288 b^{2} c^{4}\right ) + x^{3} \left (- 2304 a b c^{4} + 576 b^{3} c^{3}\right ) + x^{2} \left (- 576 a^{2} c^{4} - 1440 a b^{2} c^{3} + 396 b^{4} c^{2}\right ) + x \left (- 576 a^{2} b c^{3} - 288 a b^{3} c^{2} + 108 b^{5} c\right )}{1536 b^{6} c^{4} d^{7} + 18432 b^{5} c^{5} d^{7} x + 92160 b^{4} c^{6} d^{7} x^{2} + 245760 b^{3} c^{7} d^{7} x^{3} + 368640 b^{2} c^{8} d^{7} x^{4} + 294912 b c^{9} d^{7} x^{5} + 98304 c^{10} d^{7} x^{6}} + \frac {\log {\left (b + 2 c x \right )}}{128 c^{4} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.05, size = 163, normalized size = 1.63 \begin {gather*} \frac {\log \left ({\left | 2 \, c x + b \right |}\right )}{128 \, c^{4} d^{7}} + \frac {11 \, b^{6} - 24 \, a b^{4} c - 48 \, a^{2} b^{2} c^{2} - 128 \, a^{3} c^{3} + 288 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} + 576 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} + 36 \, {\left (11 \, b^{4} c^{2} - 40 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} + 36 \, {\left (3 \, b^{5} c - 8 \, a b^{3} c^{2} - 16 \, a^{2} b c^{3}\right )} x}{1536 \, {\left (2 \, c x + b\right )}^{6} c^{4} d^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.54, size = 229, normalized size = 2.29 \begin {gather*} \frac {\ln \left (b+2\,c\,x\right )}{128\,c^4\,d^7}-\frac {\frac {128\,a^3\,c^3+48\,a^2\,b^2\,c^2+24\,a\,b^4\,c-11\,b^6}{1536\,c^4}+x^4\,\left (\frac {3\,a\,c}{4}-\frac {3\,b^2}{16}\right )+\frac {3\,x^2\,\left (16\,a^2\,c^2+40\,a\,b^2\,c-11\,b^4\right )}{128\,c^2}+\frac {3\,x\,\left (16\,a^2\,b\,c^2+8\,a\,b^3\,c-3\,b^5\right )}{128\,c^3}-\frac {3\,x^3\,\left (b^3-4\,a\,b\,c\right )}{8\,c}}{b^6\,d^7+12\,b^5\,c\,d^7\,x+60\,b^4\,c^2\,d^7\,x^2+160\,b^3\,c^3\,d^7\,x^3+240\,b^2\,c^4\,d^7\,x^4+192\,b\,c^5\,d^7\,x^5+64\,c^6\,d^7\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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