Optimal. Leaf size=123 \[ 96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4 \]
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Rubi [A] time = 0.08, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {686, 692, 628} \begin {gather*} 96 c^2 d^9 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+96 c^2 d^9 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+48 c^2 d^9 (b+2 c x)^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 628
Rule 686
Rule 692
Rubi steps
\begin {align*} \int \frac {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}+\left (8 c d^2\right ) \int \frac {(b d+2 c d x)^7}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 d^4\right ) \int \frac {(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right ) d^6\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+\left (96 c^2 \left (b^2-4 a c\right )^2 d^8\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=96 c^2 \left (b^2-4 a c\right ) d^9 (b+2 c x)^2+48 c^2 d^9 (b+2 c x)^4-\frac {d^9 (b+2 c x)^8}{2 \left (a+b x+c x^2\right )^2}-\frac {8 c d^9 (b+2 c x)^6}{a+b x+c x^2}+96 c^2 \left (b^2-4 a c\right )^2 d^9 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.06, size = 131, normalized size = 1.07 \begin {gather*} d^9 \left (-384 c^4 x^2 \left (2 a c-b^2\right )+256 b c^3 x \left (b^2-3 a c\right )+96 c^2 \left (b^2-4 a c\right )^2 \log (a+x (b+c x))+\frac {16 c \left (4 a c-b^2\right )^3}{a+x (b+c x)}-\frac {\left (b^2-4 a c\right )^4}{2 (a+x (b+c x))^2}+256 b c^5 x^3+128 c^6 x^4\right ) \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 {(b d+2 c d x)^9}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.39, size = 490, normalized size = 3.98 \begin {gather*} \frac {256 \, c^{8} d^{9} x^{8} + 1024 \, b c^{7} d^{9} x^{7} + 1024 \, {\left (2 \, b^{2} c^{6} - a c^{7}\right )} d^{9} x^{6} + 512 \, {\left (5 \, b^{3} c^{5} - 6 \, a b c^{6}\right )} d^{9} x^{5} + 256 \, {\left (7 \, b^{4} c^{4} - 8 \, a b^{2} c^{5} - 11 \, a^{2} c^{6}\right )} d^{9} x^{4} + 512 \, {\left (b^{5} c^{3} + 2 \, a b^{3} c^{4} - 11 \, a^{2} b c^{5}\right )} d^{9} x^{3} - 32 \, {\left (b^{6} c^{2} - 44 \, a b^{4} c^{3} + 120 \, a^{2} b^{2} c^{4} - 16 \, a^{3} c^{5}\right )} d^{9} x^{2} - 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 32 \, a^{2} b^{3} c^{3} - 16 \, a^{3} b c^{4}\right )} d^{9} x - {\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9} + 192 \, {\left ({\left (b^{4} c^{4} - 8 \, a b^{2} c^{5} + 16 \, a^{2} c^{6}\right )} d^{9} x^{4} + 2 \, {\left (b^{5} c^{3} - 8 \, a b^{3} c^{4} + 16 \, a^{2} b c^{5}\right )} d^{9} x^{3} + {\left (b^{6} c^{2} - 6 \, a b^{4} c^{3} + 32 \, a^{3} c^{5}\right )} d^{9} x^{2} + 2 \, {\left (a b^{5} c^{2} - 8 \, a^{2} b^{3} c^{3} + 16 \, a^{3} b c^{4}\right )} d^{9} x + {\left (a^{2} b^{4} c^{2} - 8 \, a^{3} b^{2} c^{3} + 16 \, a^{4} c^{4}\right )} d^{9}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 299, normalized size = 2.43 \begin {gather*} 96 \, {\left (b^{4} c^{2} d^{9} - 8 \, a b^{2} c^{3} d^{9} + 16 \, a^{2} c^{4} d^{9}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{8} d^{9} + 16 \, a b^{6} c d^{9} - 288 \, a^{2} b^{4} c^{2} d^{9} + 1280 \, a^{3} b^{2} c^{3} d^{9} - 1792 \, a^{4} c^{4} d^{9} + 32 \, {\left (b^{6} c^{2} d^{9} - 12 \, a b^{4} c^{3} d^{9} + 48 \, a^{2} b^{2} c^{4} d^{9} - 64 \, a^{3} c^{5} d^{9}\right )} x^{2} + 32 \, {\left (b^{7} c d^{9} - 12 \, a b^{5} c^{2} d^{9} + 48 \, a^{2} b^{3} c^{3} d^{9} - 64 \, a^{3} b c^{4} d^{9}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2}} + \frac {128 \, {\left (c^{18} d^{9} x^{4} + 2 \, b c^{17} d^{9} x^{3} + 3 \, b^{2} c^{16} d^{9} x^{2} - 6 \, a c^{17} d^{9} x^{2} + 2 \, b^{3} c^{15} d^{9} x - 6 \, a b c^{16} d^{9} x\right )}}{c^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 465, normalized size = 3.78 \begin {gather*} \frac {1024 a^{3} c^{5} d^{9} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {768 a^{2} b^{2} c^{4} d^{9} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {192 a \,b^{4} c^{3} d^{9} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {16 b^{6} c^{2} d^{9} x^{2}}{\left (c \,x^{2}+b x +a \right )^{2}}+128 c^{6} d^{9} x^{4}+\frac {1024 a^{3} b \,c^{4} d^{9} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {768 a^{2} b^{3} c^{3} d^{9} x}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {192 a \,b^{5} c^{2} d^{9} x}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {16 b^{7} c \,d^{9} x}{\left (c \,x^{2}+b x +a \right )^{2}}+256 b \,c^{5} d^{9} x^{3}+\frac {896 a^{4} c^{4} d^{9}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {640 a^{3} b^{2} c^{3} d^{9}}{\left (c \,x^{2}+b x +a \right )^{2}}+\frac {144 a^{2} b^{4} c^{2} d^{9}}{\left (c \,x^{2}+b x +a \right )^{2}}-\frac {8 a \,b^{6} c \,d^{9}}{\left (c \,x^{2}+b x +a \right )^{2}}-768 a \,c^{5} d^{9} x^{2}-\frac {b^{8} d^{9}}{2 \left (c \,x^{2}+b x +a \right )^{2}}+384 b^{2} c^{4} d^{9} x^{2}+1536 a^{2} c^{4} d^{9} \ln \left (c \,x^{2}+b x +a \right )-768 a \,b^{2} c^{3} d^{9} \ln \left (c \,x^{2}+b x +a \right )-768 a b \,c^{4} d^{9} x +96 b^{4} c^{2} d^{9} \ln \left (c \,x^{2}+b x +a \right )+256 b^{3} c^{3} d^{9} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.53, size = 278, normalized size = 2.26 \begin {gather*} 128 \, c^{6} d^{9} x^{4} + 256 \, b c^{5} d^{9} x^{3} + 384 \, {\left (b^{2} c^{4} - 2 \, a c^{5}\right )} d^{9} x^{2} + 256 \, {\left (b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{9} x + 96 \, {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{9} \log \left (c x^{2} + b x + a\right ) - \frac {32 \, {\left (b^{6} c^{2} - 12 \, a b^{4} c^{3} + 48 \, a^{2} b^{2} c^{4} - 64 \, a^{3} c^{5}\right )} d^{9} x^{2} + 32 \, {\left (b^{7} c - 12 \, a b^{5} c^{2} + 48 \, a^{2} b^{3} c^{3} - 64 \, a^{3} b c^{4}\right )} d^{9} x + {\left (b^{8} + 16 \, a b^{6} c - 288 \, a^{2} b^{4} c^{2} + 1280 \, a^{3} b^{2} c^{3} - 1792 \, a^{4} c^{4}\right )} d^{9}}{2 \, {\left (c^{2} x^{4} + 2 \, b c x^{3} + 2 \, a b x + {\left (b^{2} + 2 \, a c\right )} x^{2} + a^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 385, normalized size = 3.13 \begin {gather*} \ln \left (c\,x^2+b\,x+a\right )\,\left (1536\,a^2\,c^4\,d^9-768\,a\,b^2\,c^3\,d^9+96\,b^4\,c^2\,d^9\right )-x^2\,\left (768\,c^4\,d^9\,\left (b^2+a\,c\right )-1152\,b^2\,c^4\,d^9\right )-\frac {\frac {b^8\,d^9}{2}-x^2\,\left (1024\,a^3\,c^5\,d^9-768\,a^2\,b^2\,c^4\,d^9+192\,a\,b^4\,c^3\,d^9-16\,b^6\,c^2\,d^9\right )-896\,a^4\,c^4\,d^9+16\,b\,x\,\left (-64\,a^3\,c^4\,d^9+48\,a^2\,b^2\,c^3\,d^9-12\,a\,b^4\,c^2\,d^9+b^6\,c\,d^9\right )-144\,a^2\,b^4\,c^2\,d^9+640\,a^3\,b^2\,c^3\,d^9+8\,a\,b^6\,c\,d^9}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3}-x\,\left (512\,c^3\,d^9\,\left (b^3+6\,a\,c\,b\right )-5376\,b^3\,c^3\,d^9-\frac {3\,b\,\left (1536\,c^4\,d^9\,\left (b^2+a\,c\right )-2304\,b^2\,c^4\,d^9\right )}{c}+2304\,b\,c^3\,d^9\,\left (b^2+a\,c\right )\right )+128\,c^6\,d^9\,x^4+256\,b\,c^5\,d^9\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 10.98, size = 320, normalized size = 2.60 \begin {gather*} 256 b c^{5} d^{9} x^{3} + 128 c^{6} d^{9} x^{4} + 96 c^{2} d^{9} \left (4 a c - b^{2}\right )^{2} \log {\left (a + b x + c x^{2} \right )} + x^{2} \left (- 768 a c^{5} d^{9} + 384 b^{2} c^{4} d^{9}\right ) + x \left (- 768 a b c^{4} d^{9} + 256 b^{3} c^{3} d^{9}\right ) + \frac {1792 a^{4} c^{4} d^{9} - 1280 a^{3} b^{2} c^{3} d^{9} + 288 a^{2} b^{4} c^{2} d^{9} - 16 a b^{6} c d^{9} - b^{8} d^{9} + x^{2} \left (2048 a^{3} c^{5} d^{9} - 1536 a^{2} b^{2} c^{4} d^{9} + 384 a b^{4} c^{3} d^{9} - 32 b^{6} c^{2} d^{9}\right ) + x \left (2048 a^{3} b c^{4} d^{9} - 1536 a^{2} b^{3} c^{3} d^{9} + 384 a b^{5} c^{2} d^{9} - 32 b^{7} c d^{9}\right )}{2 a^{2} + 4 a b x + 4 b c x^{3} + 2 c^{2} x^{4} + x^{2} \left (4 a c + 2 b^{2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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