Optimal. Leaf size=154 \[ \frac {48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac {48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac {96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac {6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]
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Rubi [A] time = 0.09, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {687, 693, 681, 31, 628} \[ \frac {48 c^2 \log \left (a+b x+c x^2\right )}{d^3 \left (b^2-4 a c\right )^4}+\frac {48 c^2}{d^3 \left (b^2-4 a c\right )^3 (b+2 c x)^2}-\frac {96 c^2 \log (b+2 c x)}{d^3 \left (b^2-4 a c\right )^4}+\frac {6 c}{d^3 \left (b^2-4 a c\right )^2 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {1}{2 d^3 \left (b^2-4 a c\right ) (b+2 c x)^2 \left (a+b x+c x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 628
Rule 681
Rule 687
Rule 693
Rubi steps
\begin {align*} \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )^3} \, dx &=-\frac {1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}-\frac {(6 c) \int \frac {1}{(b d+2 c d x)^3 \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^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac {6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac {\left (48 c^2\right ) \int \frac {1}{(b d+2 c d x)^3 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^2}\\ &=\frac {48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac {1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac {6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac {\left (48 c^2\right ) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right )^3 d^2}\\ &=\frac {48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac {1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac {6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}+\frac {\left (48 c^2\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right )^4 d^4}-\frac {\left (192 c^3\right ) \int \frac {1}{b+2 c x} \, dx}{\left (b^2-4 a c\right )^4 d^3}\\ &=\frac {48 c^2}{\left (b^2-4 a c\right )^3 d^3 (b+2 c x)^2}-\frac {1}{2 \left (b^2-4 a c\right ) d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )^2}+\frac {6 c}{\left (b^2-4 a c\right )^2 d^3 (b+2 c x)^2 \left (a+b x+c x^2\right )}-\frac {96 c^2 \log (b+2 c x)}{\left (b^2-4 a c\right )^4 d^3}+\frac {48 c^2 \log \left (a+b x+c x^2\right )}{\left (b^2-4 a c\right )^4 d^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 111, normalized size = 0.72 \[ \frac {\frac {32 c^2 \left (b^2-4 a c\right )}{(b+2 c x)^2}+\frac {16 c \left (b^2-4 a c\right )}{a+x (b+c x)}-\frac {\left (b^2-4 a c\right )^2}{(a+x (b+c x))^2}+96 c^2 \log (a+x (b+c x))-192 c^2 \log (b+2 c x)}{2 d^3 \left (b^2-4 a c\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.02, size = 809, normalized size = 5.25 \[ -\frac {b^{6} - 24 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} + 128 \, a^{3} c^{3} - 96 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} x^{4} - 192 \, {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} x^{3} - 36 \, {\left (3 \, b^{4} c^{2} - 8 \, a b^{2} c^{3} - 16 \, a^{2} c^{4}\right )} x^{2} - 12 \, {\left (b^{5} c + 8 \, a b^{3} c^{2} - 48 \, a^{2} b c^{3}\right )} x - 96 \, {\left (4 \, c^{6} x^{6} + 12 \, b c^{5} x^{5} + a^{2} b^{2} c^{2} + {\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} + 2 \, a^{2} b c^{3}\right )} x\right )} \log \left (c x^{2} + b x + a\right ) + 192 \, {\left (4 \, c^{6} x^{6} + 12 \, b c^{5} x^{5} + a^{2} b^{2} c^{2} + {\left (13 \, b^{2} c^{4} + 8 \, a c^{5}\right )} x^{4} + 2 \, {\left (3 \, b^{3} c^{3} + 8 \, a b c^{4}\right )} x^{3} + {\left (b^{4} c^{2} + 10 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} x^{2} + 2 \, {\left (a b^{3} c^{2} + 2 \, a^{2} b c^{3}\right )} x\right )} \log \left (2 \, c x + b\right )}{2 \, {\left (4 \, {\left (b^{8} c^{4} - 16 \, a b^{6} c^{5} + 96 \, a^{2} b^{4} c^{6} - 256 \, a^{3} b^{2} c^{7} + 256 \, a^{4} c^{8}\right )} d^{3} x^{6} + 12 \, {\left (b^{9} c^{3} - 16 \, a b^{7} c^{4} + 96 \, a^{2} b^{5} c^{5} - 256 \, a^{3} b^{3} c^{6} + 256 \, a^{4} b c^{7}\right )} d^{3} x^{5} + {\left (13 \, b^{10} c^{2} - 200 \, a b^{8} c^{3} + 1120 \, a^{2} b^{6} c^{4} - 2560 \, a^{3} b^{4} c^{5} + 1280 \, a^{4} b^{2} c^{6} + 2048 \, a^{5} c^{7}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{11} c - 40 \, a b^{9} c^{2} + 160 \, a^{2} b^{7} c^{3} - 1280 \, a^{4} b^{3} c^{5} + 2048 \, a^{5} b c^{6}\right )} d^{3} x^{3} + {\left (b^{12} - 6 \, a b^{10} c - 60 \, a^{2} b^{8} c^{2} + 640 \, a^{3} b^{6} c^{3} - 1920 \, a^{4} b^{4} c^{4} + 1536 \, a^{5} b^{2} c^{5} + 1024 \, a^{6} c^{6}\right )} d^{3} x^{2} + 2 \, {\left (a b^{11} - 14 \, a^{2} b^{9} c + 64 \, a^{3} b^{7} c^{2} - 64 \, a^{4} b^{5} c^{3} - 256 \, a^{5} b^{3} c^{4} + 512 \, a^{6} b c^{5}\right )} d^{3} x + {\left (a^{2} b^{10} - 16 \, a^{3} b^{8} c + 96 \, a^{4} b^{6} c^{2} - 256 \, a^{5} b^{4} c^{3} + 256 \, a^{6} b^{2} c^{4}\right )} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 302, normalized size = 1.96 \[ -\frac {96 \, c^{3} \log \left ({\left | 2 \, c x + b \right |}\right )}{b^{8} c d^{3} - 16 \, a b^{6} c^{2} d^{3} + 96 \, a^{2} b^{4} c^{3} d^{3} - 256 \, a^{3} b^{2} c^{4} d^{3} + 256 \, a^{4} c^{5} d^{3}} + \frac {48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{b^{8} d^{3} - 16 \, a b^{6} c d^{3} + 96 \, a^{2} b^{4} c^{2} d^{3} - 256 \, a^{3} b^{2} c^{3} d^{3} + 256 \, a^{4} c^{4} d^{3}} + \frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} + 108 \, b^{2} c^{2} x^{2} + 144 \, a c^{3} x^{2} + 12 \, b^{3} c x + 144 \, a b c^{2} x - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2}}{2 \, {\left (b^{6} d^{3} - 12 \, a b^{4} c d^{3} + 48 \, a^{2} b^{2} c^{2} d^{3} - 64 \, a^{3} c^{3} d^{3}\right )} {\left (2 \, c^{2} x^{3} + 3 \, b c x^{2} + b^{2} x + 2 \, a c x + a b\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 332, normalized size = 2.16 \[ -\frac {32 a \,c^{3} x^{2}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}+\frac {8 b^{2} c^{2} x^{2}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}-\frac {32 a b \,c^{2} x}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}+\frac {8 b^{3} c x}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}-\frac {40 a^{2} c^{2}}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}+\frac {12 a \,b^{2} c}{\left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}-\frac {b^{4}}{2 \left (4 a c -b^{2}\right )^{4} \left (c \,x^{2}+b x +a \right )^{2} d^{3}}-\frac {96 c^{2} \ln \left (2 c x +b \right )}{\left (4 a c -b^{2}\right )^{4} d^{3}}+\frac {48 c^{2} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right )^{4} d^{3}}-\frac {16 c^{2}}{\left (4 a c -b^{2}\right )^{3} \left (2 c x +b \right )^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.64, size = 553, normalized size = 3.59 \[ \frac {96 \, c^{4} x^{4} + 192 \, b c^{3} x^{3} - b^{4} + 20 \, a b^{2} c + 32 \, a^{2} c^{2} + 36 \, {\left (3 \, b^{2} c^{2} + 4 \, a c^{3}\right )} x^{2} + 12 \, {\left (b^{3} c + 12 \, a b c^{2}\right )} x}{2 \, {\left (4 \, {\left (b^{6} c^{4} - 12 \, a b^{4} c^{5} + 48 \, a^{2} b^{2} c^{6} - 64 \, a^{3} c^{7}\right )} d^{3} x^{6} + 12 \, {\left (b^{7} c^{3} - 12 \, a b^{5} c^{4} + 48 \, a^{2} b^{3} c^{5} - 64 \, a^{3} b c^{6}\right )} d^{3} x^{5} + {\left (13 \, b^{8} c^{2} - 148 \, a b^{6} c^{3} + 528 \, a^{2} b^{4} c^{4} - 448 \, a^{3} b^{2} c^{5} - 512 \, a^{4} c^{6}\right )} d^{3} x^{4} + 2 \, {\left (3 \, b^{9} c - 28 \, a b^{7} c^{2} + 48 \, a^{2} b^{5} c^{3} + 192 \, a^{3} b^{3} c^{4} - 512 \, a^{4} b c^{5}\right )} d^{3} x^{3} + {\left (b^{10} - 2 \, a b^{8} c - 68 \, a^{2} b^{6} c^{2} + 368 \, a^{3} b^{4} c^{3} - 448 \, a^{4} b^{2} c^{4} - 256 \, a^{5} c^{5}\right )} d^{3} x^{2} + 2 \, {\left (a b^{9} - 10 \, a^{2} b^{7} c + 24 \, a^{3} b^{5} c^{2} + 32 \, a^{4} b^{3} c^{3} - 128 \, a^{5} b c^{4}\right )} d^{3} x + {\left (a^{2} b^{8} - 12 \, a^{3} b^{6} c + 48 \, a^{4} b^{4} c^{2} - 64 \, a^{5} b^{2} c^{3}\right )} d^{3}\right )}} + \frac {48 \, c^{2} \log \left (c x^{2} + b x + a\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} - \frac {96 \, c^{2} \log \left (2 \, c x + b\right )}{{\left (b^{8} - 16 \, a b^{6} c + 96 \, a^{2} b^{4} c^{2} - 256 \, a^{3} b^{2} c^{3} + 256 \, a^{4} c^{4}\right )} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 522, normalized size = 3.39 \[ \frac {\frac {32\,a^2\,c^2+20\,a\,b^2\,c-b^4}{2\,\left (-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6\right )}+\frac {48\,c^4\,x^4}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {96\,b\,c^3\,x^3}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {18\,c\,x^2\,\left (3\,b^2\,c+4\,a\,c^2\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}+\frac {6\,b\,c\,x\,\left (b^2+12\,a\,c\right )}{-64\,a^3\,c^3+48\,a^2\,b^2\,c^2-12\,a\,b^4\,c+b^6}}{x\,\left (4\,c\,a^2\,b\,d^3+2\,a\,b^3\,d^3\right )+x^4\,\left (13\,b^2\,c^2\,d^3+8\,a\,c^3\,d^3\right )+x^3\,\left (6\,b^3\,c\,d^3+16\,a\,b\,c^2\,d^3\right )+x^2\,\left (4\,a^2\,c^2\,d^3+10\,a\,b^2\,c\,d^3+b^4\,d^3\right )+a^2\,b^2\,d^3+4\,c^4\,d^3\,x^6+12\,b\,c^3\,d^3\,x^5}-\frac {96\,c^2\,\ln \left (b+2\,c\,x\right )}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3}+\frac {48\,c^2\,\ln \left (c\,x^2+b\,x+a\right )}{256\,a^4\,c^4\,d^3-256\,a^3\,b^2\,c^3\,d^3+96\,a^2\,b^4\,c^2\,d^3-16\,a\,b^6\,c\,d^3+b^8\,d^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 6.47, size = 600, normalized size = 3.90 \[ - \frac {96 c^{2} \log {\left (\frac {b}{2 c} + x \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac {48 c^{2} \log {\left (\frac {a}{c} + \frac {b x}{c} + x^{2} \right )}}{d^{3} \left (4 a c - b^{2}\right )^{4}} + \frac {- 32 a^{2} c^{2} - 20 a b^{2} c + b^{4} - 192 b c^{3} x^{3} - 96 c^{4} x^{4} + x^{2} \left (- 144 a c^{3} - 108 b^{2} c^{2}\right ) + x \left (- 144 a b c^{2} - 12 b^{3} c\right )}{128 a^{5} b^{2} c^{3} d^{3} - 96 a^{4} b^{4} c^{2} d^{3} + 24 a^{3} b^{6} c d^{3} - 2 a^{2} b^{8} d^{3} + x^{6} \left (512 a^{3} c^{7} d^{3} - 384 a^{2} b^{2} c^{6} d^{3} + 96 a b^{4} c^{5} d^{3} - 8 b^{6} c^{4} d^{3}\right ) + x^{5} \left (1536 a^{3} b c^{6} d^{3} - 1152 a^{2} b^{3} c^{5} d^{3} + 288 a b^{5} c^{4} d^{3} - 24 b^{7} c^{3} d^{3}\right ) + x^{4} \left (1024 a^{4} c^{6} d^{3} + 896 a^{3} b^{2} c^{5} d^{3} - 1056 a^{2} b^{4} c^{4} d^{3} + 296 a b^{6} c^{3} d^{3} - 26 b^{8} c^{2} d^{3}\right ) + x^{3} \left (2048 a^{4} b c^{5} d^{3} - 768 a^{3} b^{3} c^{4} d^{3} - 192 a^{2} b^{5} c^{3} d^{3} + 112 a b^{7} c^{2} d^{3} - 12 b^{9} c d^{3}\right ) + x^{2} \left (512 a^{5} c^{5} d^{3} + 896 a^{4} b^{2} c^{4} d^{3} - 736 a^{3} b^{4} c^{3} d^{3} + 136 a^{2} b^{6} c^{2} d^{3} + 4 a b^{8} c d^{3} - 2 b^{10} d^{3}\right ) + x \left (512 a^{5} b c^{4} d^{3} - 128 a^{4} b^{3} c^{3} d^{3} - 96 a^{3} b^{5} c^{2} d^{3} + 40 a^{2} b^{7} c d^{3} - 4 a b^{9} d^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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