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 = 0.08, 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 = {683} \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 683
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 = 0.04, 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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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x+c x^2\right )^3}{(b d+2 c d x)^7} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.40, 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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giac [A] time = 0.17, 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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maple [B] time = 0.06, size = 191, normalized size = 1.91 \begin {gather*} -\frac {a^{3}}{12 \left (2 c x +b \right )^{6} c \,d^{7}}+\frac {a^{2} b^{2}}{16 \left (2 c x +b \right )^{6} c^{2} d^{7}}-\frac {a \,b^{4}}{64 \left (2 c x +b \right )^{6} c^{3} d^{7}}+\frac {b^{6}}{768 \left (2 c x +b \right )^{6} c^{4} d^{7}}-\frac {3 a^{2}}{32 \left (2 c x +b \right )^{4} c^{2} d^{7}}+\frac {3 a \,b^{2}}{64 \left (2 c x +b \right )^{4} c^{3} d^{7}}-\frac {3 b^{4}}{512 \left (2 c x +b \right )^{4} c^{4} d^{7}}-\frac {3 a}{64 \left (2 c x +b \right )^{2} c^{3} d^{7}}+\frac {3 b^{2}}{256 \left (2 c x +b \right )^{2} c^{4} d^{7}}+\frac {\ln \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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maxima [B] time = 1.59, 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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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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sympy [B] time = 5.86, 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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