Optimal. Leaf size=89 \[ 12 c d^7 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+12 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (b+2 c x)^4 \]
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Rubi [A] time = 0.06, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, 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*} 12 c d^7 \left (b^2-4 a c\right )^2 \log \left (a+b x+c x^2\right )+12 c d^7 \left (b^2-4 a c\right ) (b+2 c x)^2-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+6 c d^7 (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)^7}{\left (a+b x+c x^2\right )^2} \, dx &=-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c d^2\right ) \int \frac {(b d+2 c d x)^5}{a+b x+c x^2} \, dx\\ &=6 c d^7 (b+2 c x)^4-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx\\ &=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+\left (12 c \left (b^2-4 a c\right )^2 d^6\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx\\ &=12 c \left (b^2-4 a c\right ) d^7 (b+2 c x)^2+6 c d^7 (b+2 c x)^4-\frac {d^7 (b+2 c x)^6}{a+b x+c x^2}+12 c \left (b^2-4 a c\right )^2 d^7 \log \left (a+b x+c x^2\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 103, normalized size = 1.16 \begin {gather*} d^7 \left (-16 c^3 x^2 \left (8 a c-5 b^2\right )+16 b c^2 x \left (3 b^2-8 a c\right )-\frac {\left (b^2-4 a c\right )^3}{a+x (b+c x)}+12 c \left (b^2-4 a c\right )^2 \log (a+x (b+c x))+64 b c^4 x^3+32 c^5 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)^7}{\left (a+b x+c x^2\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.41, size = 283, normalized size = 3.18 \begin {gather*} \frac {32 \, c^{6} d^{7} x^{6} + 96 \, b c^{5} d^{7} x^{5} + 48 \, {\left (3 \, b^{2} c^{4} - 2 \, a c^{5}\right )} d^{7} x^{4} + 64 \, {\left (2 \, b^{3} c^{3} - 3 \, a b c^{4}\right )} d^{7} x^{3} + 16 \, {\left (3 \, b^{4} c^{2} - 3 \, a b^{2} c^{3} - 8 \, a^{2} c^{4}\right )} d^{7} x^{2} + 16 \, {\left (3 \, a b^{3} c^{2} - 8 \, 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} + 12 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d^{7} x^{2} + {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d^{7} x + {\left (a b^{4} c - 8 \, a^{2} b^{2} c^{2} + 16 \, a^{3} c^{3}\right )} d^{7}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 181, normalized size = 2.03 \begin {gather*} 12 \, {\left (b^{4} c d^{7} - 8 \, a b^{2} c^{2} d^{7} + 16 \, a^{2} c^{3} d^{7}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {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}}{c x^{2} + b x + a} + \frac {16 \, {\left (2 \, c^{13} d^{7} x^{4} + 4 \, b c^{12} d^{7} x^{3} + 5 \, b^{2} c^{11} d^{7} x^{2} - 8 \, a c^{12} d^{7} x^{2} + 3 \, b^{3} c^{10} d^{7} x - 8 \, a b c^{11} d^{7} x\right )}}{c^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 230, normalized size = 2.58 \begin {gather*} 32 c^{5} d^{7} x^{4}+64 b \,c^{4} d^{7} x^{3}-128 a \,c^{4} d^{7} x^{2}+80 b^{2} c^{3} d^{7} x^{2}+\frac {64 a^{3} c^{3} d^{7}}{c \,x^{2}+b x +a}-\frac {48 a^{2} b^{2} c^{2} d^{7}}{c \,x^{2}+b x +a}+192 a^{2} c^{3} d^{7} \ln \left (c \,x^{2}+b x +a \right )+\frac {12 a \,b^{4} c \,d^{7}}{c \,x^{2}+b x +a}-96 a \,b^{2} c^{2} d^{7} \ln \left (c \,x^{2}+b x +a \right )-128 a b \,c^{3} d^{7} x -\frac {b^{6} d^{7}}{c \,x^{2}+b x +a}+12 b^{4} c \,d^{7} \ln \left (c \,x^{2}+b x +a \right )+48 b^{3} c^{2} d^{7} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 155, normalized size = 1.74 \begin {gather*} 32 \, c^{5} d^{7} x^{4} + 64 \, b c^{4} d^{7} x^{3} + 16 \, {\left (5 \, b^{2} c^{3} - 8 \, a c^{4}\right )} d^{7} x^{2} + 16 \, {\left (3 \, b^{3} c^{2} - 8 \, a b c^{3}\right )} d^{7} x + 12 \, {\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d^{7} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (b^{6} - 12 \, a b^{4} c + 48 \, a^{2} b^{2} c^{2} - 64 \, a^{3} c^{3}\right )} d^{7}}{c x^{2} + b x + a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 234, normalized size = 2.63 \begin {gather*} \ln \left (c\,x^2+b\,x+a\right )\,\left (192\,a^2\,c^3\,d^7-96\,a\,b^2\,c^2\,d^7+12\,b^4\,c\,d^7\right )-x^2\,\left (64\,c^3\,d^7\,\left (b^2+2\,a\,c\right )-144\,b^2\,c^3\,d^7\right )-\frac {-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}{c\,x^2+b\,x+a}+x\,\left (560\,b^3\,c^2\,d^7+\frac {2\,b\,\left (128\,c^3\,d^7\,\left (b^2+2\,a\,c\right )-288\,b^2\,c^3\,d^7\right )}{c}-192\,b\,c^2\,d^7\,\left (b^2+2\,a\,c\right )-256\,a\,b\,c^3\,d^7\right )+32\,c^5\,d^7\,x^4+64\,b\,c^4\,d^7\,x^3 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.57, size = 160, normalized size = 1.80 \begin {gather*} 64 b c^{4} d^{7} x^{3} + 32 c^{5} d^{7} x^{4} + 12 c d^{7} \left (4 a c - b^{2}\right )^{2} \log {\left (a + b x + c x^{2} \right )} + x^{2} \left (- 128 a c^{4} d^{7} + 80 b^{2} c^{3} d^{7}\right ) + x \left (- 128 a b c^{3} d^{7} + 48 b^{3} c^{2} d^{7}\right ) + \frac {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}}{a + b x + c x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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