Optimal. Leaf size=86 \[ d^7 \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )+\frac {1}{2} d^7 \left (b^2-4 a c\right ) (b+2 c x)^4+d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2+\frac {1}{3} d^7 (b+2 c x)^6 \]
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Rubi [A] time = 0.07, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {692, 628} \[ d^7 \left (b^2-4 a c\right )^3 \log \left (a+b x+c x^2\right )+\frac {1}{2} d^7 \left (b^2-4 a c\right ) (b+2 c x)^4+d^7 \left (b^2-4 a c\right )^2 (b+2 c x)^2+\frac {1}{3} d^7 (b+2 c x)^6 \]
Antiderivative was successfully verified.
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Rule 628
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^7}{a+b x+c x^2} \, dx &=\frac {1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right ) d^2\right ) \int \frac {(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=\frac {1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac {1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right )^2 d^4\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac {1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac {1}{3} d^7 (b+2 c x)^6+\left (\left (b^2-4 a c\right )^3 d^6\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=\left (b^2-4 a c\right )^2 d^7 (b+2 c x)^2+\frac {1}{2} \left (b^2-4 a c\right ) d^7 (b+2 c x)^4+\frac {1}{3} d^7 (b+2 c x)^6+\left (b^2-4 a c\right )^3 d^7 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 110, normalized size = 1.28 \[ d^7 \left (\frac {4}{3} c x (b+c x) \left (8 c^2 \left (6 a^2-3 a c x^2+2 c^2 x^4\right )+b^2 \left (34 c^2 x^2-36 a c\right )+8 b c^2 x \left (4 c x^2-3 a\right )+9 b^4+18 b^3 c x\right )+\left (b^2-4 a c\right )^3 \log (a+x (b+c x))\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 181, normalized size = 2.10 \[ \frac {64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \, {\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac {16}{3} \, {\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \, {\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \, {\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 219, normalized size = 2.55 \[ {\left (b^{6} d^{7} - 12 \, a b^{4} c d^{7} + 48 \, a^{2} b^{2} c^{2} d^{7} - 64 \, a^{3} c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) + \frac {4 \, {\left (16 \, c^{12} d^{7} x^{6} + 48 \, b c^{11} d^{7} x^{5} + 66 \, b^{2} c^{10} d^{7} x^{4} - 24 \, a c^{11} d^{7} x^{4} + 52 \, b^{3} c^{9} d^{7} x^{3} - 48 \, a b c^{10} d^{7} x^{3} + 27 \, b^{4} c^{8} d^{7} x^{2} - 60 \, a b^{2} c^{9} d^{7} x^{2} + 48 \, a^{2} c^{10} d^{7} x^{2} + 9 \, b^{5} c^{7} d^{7} x - 36 \, a b^{3} c^{8} d^{7} x + 48 \, a^{2} b c^{9} d^{7} x\right )}}{3 \, c^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 243, normalized size = 2.83 \[ \frac {64 c^{6} d^{7} x^{6}}{3}+64 b \,c^{5} d^{7} x^{5}-32 a \,c^{5} d^{7} x^{4}+88 b^{2} c^{4} d^{7} x^{4}-64 a b \,c^{4} d^{7} x^{3}+\frac {208 b^{3} c^{3} d^{7} x^{3}}{3}+64 a^{2} c^{4} d^{7} x^{2}-80 a \,b^{2} c^{3} d^{7} x^{2}+36 b^{4} c^{2} d^{7} x^{2}-64 a^{3} c^{3} d^{7} \ln \left (c \,x^{2}+b x +a \right )+48 a^{2} b^{2} c^{2} d^{7} \ln \left (c \,x^{2}+b x +a \right )+64 a^{2} b \,c^{3} d^{7} x -12 a \,b^{4} c \,d^{7} \ln \left (c \,x^{2}+b x +a \right )-48 a \,b^{3} c^{2} d^{7} x +b^{6} d^{7} \ln \left (c \,x^{2}+b x +a \right )+12 b^{5} c \,d^{7} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.35, size = 181, normalized size = 2.10 \[ \frac {64}{3} \, c^{6} d^{7} x^{6} + 64 \, b c^{5} d^{7} x^{5} + 8 \, {\left (11 \, b^{2} c^{4} - 4 \, a c^{5}\right )} d^{7} x^{4} + \frac {16}{3} \, {\left (13 \, b^{3} c^{3} - 12 \, a b c^{4}\right )} d^{7} x^{3} + 4 \, {\left (9 \, b^{4} c^{2} - 20 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + 4 \, {\left (3 \, b^{5} c - 12 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 418, normalized size = 4.86 \[ x^2\,\left (140\,b^4\,c^2\,d^7-\frac {b\,\left (560\,b^3\,c^3\,d^7+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}-320\,a\,b\,c^4\,d^7\right )}{2\,c}+\frac {a\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{2\,c}\right )+\ln \left (c\,x^2+b\,x+a\right )\,\left (-64\,a^3\,c^3\,d^7+48\,a^2\,b^2\,c^2\,d^7-12\,a\,b^4\,c\,d^7+b^6\,d^7\right )-x^4\,\left (32\,a\,c^5\,d^7-88\,b^2\,c^4\,d^7\right )+x^3\,\left (\frac {560\,b^3\,c^3\,d^7}{3}+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{3\,c}-\frac {320\,a\,b\,c^4\,d^7}{3}\right )-x\,\left (\frac {a\,\left (560\,b^3\,c^3\,d^7+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}-320\,a\,b\,c^4\,d^7\right )}{c}-84\,b^5\,c\,d^7+\frac {b\,\left (280\,b^4\,c^2\,d^7-\frac {b\,\left (560\,b^3\,c^3\,d^7+\frac {b\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}-320\,a\,b\,c^4\,d^7\right )}{c}+\frac {a\,\left (128\,a\,c^5\,d^7-352\,b^2\,c^4\,d^7\right )}{c}\right )}{c}\right )+\frac {64\,c^6\,d^7\,x^6}{3}+64\,b\,c^5\,d^7\,x^5 \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.93, size = 185, normalized size = 2.15 \[ 64 b c^{5} d^{7} x^{5} + \frac {64 c^{6} d^{7} x^{6}}{3} - d^{7} \left (4 a c - b^{2}\right )^{3} \log {\left (a + b x + c x^{2} \right )} + x^{4} \left (- 32 a c^{5} d^{7} + 88 b^{2} c^{4} d^{7}\right ) + x^{3} \left (- 64 a b c^{4} d^{7} + \frac {208 b^{3} c^{3} d^{7}}{3}\right ) + x^{2} \left (64 a^{2} c^{4} d^{7} - 80 a b^{2} c^{3} d^{7} + 36 b^{4} c^{2} d^{7}\right ) + x \left (64 a^{2} b c^{3} d^{7} - 48 a b^{3} c^{2} d^{7} + 12 b^{5} c d^{7}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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