Optimal. Leaf size=168 \[ \frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {140 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} d^4} \]
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Rubi [A]
time = 0.10, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {701, 707, 632,
212} \begin {gather*} \frac {140 c^2}{d^4 \left (b^2-4 a c\right )^4 (b+2 c x)}+\frac {140 c^2}{3 d^4 \left (b^2-4 a c\right )^3 (b+2 c x)^3}-\frac {140 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{d^4 \left (b^2-4 a c\right )^{9/2}}+\frac {7 c}{d^4 \left (b^2-4 a c\right )^2 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {1}{2 d^4 \left (b^2-4 a c\right ) (b+2 c x)^3 \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 701
Rule 707
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^3} \, dx &=-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}-\frac {(7 c) \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )^2} \, dx}{b^2-4 a c}\\ &=-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}+\frac {\left (70 c^2\right ) \int \frac {1}{(b d+2 c d x)^4 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}+\frac {\left (70 c^2\right ) \int \frac {1}{(b d+2 c d x)^2 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}+\frac {\left (70 c^2\right ) \int \frac {1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 d^4}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {\left (140 c^2\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right )^4 d^4}\\ &=\frac {140 c^2}{3 \left (b^2-4 a c\right )^3 d^4 (b+2 c x)^3}+\frac {140 c^2}{\left (b^2-4 a c\right )^4 d^4 (b+2 c x)}-\frac {1}{2 \left (b^2-4 a c\right ) d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )^2}+\frac {7 c}{\left (b^2-4 a c\right )^2 d^4 (b+2 c x)^3 \left (a+b x+c x^2\right )}-\frac {140 c^2 \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{9/2} d^4}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 140, normalized size = 0.83 \begin {gather*} \frac {\frac {64 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^3}+\frac {576 c^2}{b+2 c x}-\frac {3 \left (b^2-4 a c\right ) (b+2 c x)}{(a+x (b+c x))^2}+\frac {66 c (b+2 c x)}{a+x (b+c x)}+\frac {840 c^2 \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}}{6 \left (b^2-4 a c\right )^4 d^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.76, size = 165, normalized size = 0.98
method | result | size |
default | \(\frac {\frac {\frac {22 c^{3} x^{3}+33 b \,c^{2} x^{2}+2 c \left (13 a c +5 b^{2}\right ) x +\frac {b \left (26 a c -b^{2}\right )}{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {140 c^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{\left (4 a c -b^{2}\right )^{4}}+\frac {96 c^{2}}{\left (4 a c -b^{2}\right )^{4} \left (2 c x +b \right )}-\frac {32 c^{2}}{3 \left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )^{3}}}{d^{4}}\) | \(165\) |
risch | \(\frac {\frac {560 c^{6} x^{6}}{256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}}+\frac {1680 b \,c^{5} x^{5}}{256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}}+\frac {2800 c^{4} \left (a c +2 b^{2}\right ) x^{4}}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {2800 b \,c^{3} \left (2 a c +b^{2}\right ) x^{3}}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {7 c^{2} \left (128 a^{2} c^{2}+536 a c \,b^{2}+83 b^{4}\right ) x^{2}}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}+\frac {7 b c \left (128 a^{2} c^{2}+136 a c \,b^{2}+3 b^{4}\right ) x}{3 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}-\frac {256 a^{3} c^{3}-640 a^{2} b^{2} c^{2}-78 a \,b^{4} c +3 b^{6}}{6 \left (256 a^{4} c^{4}-256 a^{3} b^{2} c^{3}+96 a^{2} b^{4} c^{2}-16 a \,b^{6} c +b^{8}\right )}}{d^{4} \left (2 c x +b \right )^{3} \left (c \,x^{2}+b x +a \right )^{2}}+70 \left (\munderset {\textit {\_R} =\RootOf \left (\left (262144 a^{9} c^{9} d^{8}-589824 a^{8} b^{2} c^{8} d^{8}+589824 a^{7} b^{4} c^{7} d^{8}-344064 a^{6} b^{6} c^{6} d^{8}+129024 a^{5} b^{8} c^{5} d^{8}-32256 a^{4} b^{10} c^{4} d^{8}+5376 a^{3} b^{12} c^{3} d^{8}-576 a^{2} b^{14} c^{2} d^{8}+36 a \,b^{16} c \,d^{8}-b^{18} d^{8}\right ) \textit {\_Z}^{2}+c^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (1572864 a^{9} c^{10} d^{8}-3538944 a^{8} b^{2} c^{9} d^{8}+3538944 a^{7} b^{4} c^{8} d^{8}-2064384 a^{6} b^{6} c^{7} d^{8}+774144 a^{5} b^{8} c^{6} d^{8}-193536 a^{4} b^{10} c^{5} d^{8}+32256 a^{3} b^{12} c^{4} d^{8}-3456 a^{2} b^{14} c^{3} d^{8}+216 a \,b^{16} c^{2} d^{8}-6 b^{18} c \,d^{8}\right ) \textit {\_R}^{2}+4 c^{5}\right ) x +\left (786432 a^{9} b \,c^{9} d^{8}-1769472 a^{8} b^{3} c^{8} d^{8}+1769472 a^{7} b^{5} c^{7} d^{8}-1032192 a^{6} b^{7} c^{6} d^{8}+387072 a^{5} b^{9} c^{5} d^{8}-96768 a^{4} b^{11} c^{4} d^{8}+16128 a^{3} b^{13} c^{3} d^{8}-1728 a^{2} b^{15} c^{2} d^{8}+108 a \,b^{17} c \,d^{8}-3 b^{19} d^{8}\right ) \textit {\_R}^{2}+\left (-1024 a^{5} c^{7} d^{4}+1280 a^{4} b^{2} c^{6} d^{4}-640 a^{3} b^{4} c^{5} d^{4}+160 a^{2} b^{6} c^{4} d^{4}-20 a \,b^{8} c^{3} d^{4}+b^{10} c^{2} d^{4}\right ) \textit {\_R} +2 b \,c^{4}\right )\right )\) | \(974\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 1028 vs.
\(2 (160) = 320\).
time = 1.37, size = 2077, normalized size = 12.36 \begin {gather*} \text {Too large to display} \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. 1238 vs.
\(2 (165) = 330\).
time = 2.78, size = 1238, normalized size = 7.37 \begin {gather*} - \frac {70 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} \log {\left (x + \frac {- 71680 a^{5} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 89600 a^{4} b^{2} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 44800 a^{3} b^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 11200 a^{2} b^{6} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 1400 a b^{8} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b^{10} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b c^{2}}{140 c^{3}} \right )}}{d^{4}} + \frac {70 c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} \log {\left (x + \frac {71680 a^{5} c^{7} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 89600 a^{4} b^{2} c^{6} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 44800 a^{3} b^{4} c^{5} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 11200 a^{2} b^{6} c^{4} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 1400 a b^{8} c^{3} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} - 70 b^{10} c^{2} \sqrt {- \frac {1}{\left (4 a c - b^{2}\right )^{9}}} + 70 b c^{2}}{140 c^{3}} \right )}}{d^{4}} + \frac {- 256 a^{3} c^{3} + 640 a^{2} b^{2} c^{2} + 78 a b^{4} c - 3 b^{6} + 10080 b c^{5} x^{5} + 3360 c^{6} x^{6} + x^{4} \cdot \left (5600 a c^{5} + 11200 b^{2} c^{4}\right ) + x^{3} \cdot \left (11200 a b c^{4} + 5600 b^{3} c^{3}\right ) + x^{2} \cdot \left (1792 a^{2} c^{4} + 7504 a b^{2} c^{3} + 1162 b^{4} c^{2}\right ) + x \left (1792 a^{2} b c^{3} + 1904 a b^{3} c^{2} + 42 b^{5} c\right )}{1536 a^{6} b^{3} c^{4} d^{4} - 1536 a^{5} b^{5} c^{3} d^{4} + 576 a^{4} b^{7} c^{2} d^{4} - 96 a^{3} b^{9} c d^{4} + 6 a^{2} b^{11} d^{4} + x^{7} \cdot \left (12288 a^{4} c^{9} d^{4} - 12288 a^{3} b^{2} c^{8} d^{4} + 4608 a^{2} b^{4} c^{7} d^{4} - 768 a b^{6} c^{6} d^{4} + 48 b^{8} c^{5} d^{4}\right ) + x^{6} \cdot \left (43008 a^{4} b c^{8} d^{4} - 43008 a^{3} b^{3} c^{7} d^{4} + 16128 a^{2} b^{5} c^{6} d^{4} - 2688 a b^{7} c^{5} d^{4} + 168 b^{9} c^{4} d^{4}\right ) + x^{5} \cdot \left (24576 a^{5} c^{8} d^{4} + 33792 a^{4} b^{2} c^{7} d^{4} - 49152 a^{3} b^{4} c^{6} d^{4} + 20352 a^{2} b^{6} c^{5} d^{4} - 3552 a b^{8} c^{4} d^{4} + 228 b^{10} c^{3} d^{4}\right ) + x^{4} \cdot \left (61440 a^{5} b c^{7} d^{4} - 23040 a^{4} b^{3} c^{6} d^{4} - 15360 a^{3} b^{5} c^{5} d^{4} + 10560 a^{2} b^{7} c^{4} d^{4} - 2160 a b^{9} c^{3} d^{4} + 150 b^{11} c^{2} d^{4}\right ) + x^{3} \cdot \left (12288 a^{6} c^{7} d^{4} + 43008 a^{5} b^{2} c^{6} d^{4} - 38400 a^{4} b^{4} c^{5} d^{4} + 7680 a^{3} b^{6} c^{4} d^{4} + 1200 a^{2} b^{8} c^{3} d^{4} - 552 a b^{10} c^{2} d^{4} + 48 b^{12} c d^{4}\right ) + x^{2} \cdot \left (18432 a^{6} b c^{6} d^{4} + 3072 a^{5} b^{3} c^{5} d^{4} - 13056 a^{4} b^{5} c^{4} d^{4} + 5376 a^{3} b^{7} c^{3} d^{4} - 696 a^{2} b^{9} c^{2} d^{4} - 12 a b^{11} c d^{4} + 6 b^{13} d^{4}\right ) + x \left (9216 a^{6} b^{2} c^{5} d^{4} - 6144 a^{5} b^{4} c^{4} d^{4} + 384 a^{4} b^{6} c^{3} d^{4} + 576 a^{3} b^{8} c^{2} d^{4} - 156 a^{2} b^{10} c d^{4} + 12 a b^{12} d^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.25, size = 311, normalized size = 1.85 \begin {gather*} \frac {140 \, c^{2} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {44 \, c^{3} x^{3} + 66 \, b c^{2} x^{2} + 20 \, b^{2} c x + 52 \, a c^{2} x - b^{3} + 26 \, a b c}{2 \, {\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} {\left (c x^{2} + b x + a\right )}^{2}} + \frac {64 \, {\left (18 \, c^{4} x^{2} + 18 \, b c^{3} x + 5 \, b^{2} c^{2} - 2 \, a c^{3}\right )}}{3 \, {\left (b^{8} d^{4} - 16 \, a b^{6} c d^{4} + 96 \, a^{2} b^{4} c^{2} d^{4} - 256 \, a^{3} b^{2} c^{3} d^{4} + 256 \, a^{4} c^{4} d^{4}\right )} {\left (2 \, c x + b\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.36, size = 827, normalized size = 4.92 \begin {gather*} \frac {140\,c^2\,\mathrm {atan}\left (\frac {\frac {70\,c^2\,\left (256\,a^4\,b\,c^4\,d^4-256\,a^3\,b^3\,c^3\,d^4+96\,a^2\,b^5\,c^2\,d^4-16\,a\,b^7\,c\,d^4+b^9\,d^4\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{9/2}}+\frac {140\,c^3\,x\,\left (256\,a^4\,c^4\,d^4-256\,a^3\,b^2\,c^3\,d^4+96\,a^2\,b^4\,c^2\,d^4-16\,a\,b^6\,c\,d^4+b^8\,d^4\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{9/2}}}{70\,c^2}\right )}{d^4\,{\left (4\,a\,c-b^2\right )}^{9/2}}-\frac {\frac {2800\,x^3\,\left (b^3\,c^3+2\,a\,b\,c^4\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}-\frac {256\,a^3\,c^3-640\,a^2\,b^2\,c^2-78\,a\,b^4\,c+3\,b^6}{6\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {7\,x^2\,\left (128\,a^2\,c^4+536\,a\,b^2\,c^3+83\,b^4\,c^2\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {560\,c^6\,x^6}{\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {7\,b\,x\,\left (128\,a^2\,c^3+136\,a\,b^2\,c^2+3\,b^4\,c\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {1680\,b\,c^5\,x^5}{\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {2800\,c\,x^4\,\left (2\,b^2\,c^3+a\,c^4\right )}{3\,\left (4\,a\,c-b^2\right )\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}}{x^2\,\left (12\,a^2\,b\,c^2\,d^4+14\,a\,b^3\,c\,d^4+b^5\,d^4\right )+x^5\,\left (38\,b^2\,c^3\,d^4+16\,a\,c^4\,d^4\right )+x\,\left (6\,c\,a^2\,b^2\,d^4+2\,a\,b^4\,d^4\right )+x^3\,\left (8\,a^2\,c^3\,d^4+36\,a\,b^2\,c^2\,d^4+8\,b^4\,c\,d^4\right )+x^4\,\left (25\,b^3\,c^2\,d^4+40\,a\,b\,c^3\,d^4\right )+a^2\,b^3\,d^4+8\,c^5\,d^4\,x^7+28\,b\,c^4\,d^4\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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