Integrand size = 24, antiderivative size = 65 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right ) \]
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Time = 0.03 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {700, 706, 642} \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=8 c d^5 \left (b^2-4 a c\right ) \log \left (a+b x+c x^2\right )-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c d^5 (b+2 c x)^2 \]
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Rule 642
Rule 700
Rule 706
Rubi steps \begin{align*} \text {integral}& = -\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c d^2\right ) \int \frac {(b d+2 c d x)^3}{a+b x+c x^2} \, dx \\ & = 8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+\left (8 c \left (b^2-4 a c\right ) d^4\right ) \int \frac {b d+2 c d x}{a+b x+c x^2} \, dx \\ & = 8 c d^5 (b+2 c x)^2-\frac {d^5 (b+2 c x)^4}{a+b x+c x^2}+8 c \left (b^2-4 a c\right ) d^5 \log \left (a+b x+c x^2\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=d^5 \left (16 b c^2 x+16 c^3 x^2-\frac {\left (b^2-4 a c\right )^2}{a+x (b+c x)}+8 c \left (b^2-4 a c\right ) \log (a+x (b+c x))\right ) \]
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Time = 2.57 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20
method | result | size |
default | \(d^{5} \left (16 c^{3} x^{2}+16 b x \,c^{2}-\frac {16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}}{c \,x^{2}+b x +a}-8 c \left (4 a c -b^{2}\right ) \ln \left (c \,x^{2}+b x +a \right )\right )\) | \(78\) |
risch | \(16 c^{3} d^{5} x^{2}+16 b \,c^{2} d^{5} x -\frac {16 d^{5} a^{2} c^{2}}{c \,x^{2}+b x +a}+\frac {8 d^{5} a \,b^{2} c}{c \,x^{2}+b x +a}-\frac {d^{5} b^{4}}{c \,x^{2}+b x +a}-32 \ln \left (c \,x^{2}+b x +a \right ) a \,c^{2} d^{5}+8 \ln \left (c \,x^{2}+b x +a \right ) b^{2} c \,d^{5}\) | \(128\) |
norman | \(\frac {\frac {b \left (32 d^{5} a^{2} c^{2}-8 a \,b^{2} c \,d^{5}+b^{4} d^{5}\right ) x}{a}+\frac {c \left (32 d^{5} a^{2} c^{2}+8 a \,b^{2} c \,d^{5}+b^{4} d^{5}\right ) x^{2}}{a}+16 c^{4} d^{5} x^{4}+32 b \,c^{3} d^{5} x^{3}}{c \,x^{2}+b x +a}+\left (-32 d^{5} c^{2} a +8 b^{2} d^{5} c \right ) \ln \left (c \,x^{2}+b x +a \right )\) | \(142\) |
parallelrisch | \(-\frac {-16 b \,c^{4} d^{5} x^{4}+32 \ln \left (c \,x^{2}+b x +a \right ) x^{2} a b \,c^{3} d^{5}-8 \ln \left (c \,x^{2}+b x +a \right ) x^{2} b^{3} c^{2} d^{5}-32 b^{2} c^{3} d^{5} x^{3}+32 \ln \left (c \,x^{2}+b x +a \right ) x a \,b^{2} c^{2} d^{5}-8 \ln \left (c \,x^{2}+b x +a \right ) x \,b^{4} c \,d^{5}-16 b^{3} c^{2} d^{5} x^{2}+32 \ln \left (c \,x^{2}+b x +a \right ) a^{2} b \,c^{2} d^{5}-8 \ln \left (c \,x^{2}+b x +a \right ) a \,b^{3} c \,d^{5}+32 a^{2} b \,c^{2} d^{5}-8 a \,b^{3} c \,d^{5}+b^{5} d^{5}}{b \left (c \,x^{2}+b x +a \right )}\) | \(226\) |
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Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (65) = 130\).
Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 2.54 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=\frac {16 \, c^{4} d^{5} x^{4} + 32 \, b c^{3} d^{5} x^{3} + 16 \, a b c^{2} d^{5} x + 16 \, {\left (b^{2} c^{2} + a c^{3}\right )} d^{5} x^{2} - {\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5} + 8 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{5} x^{2} + {\left (b^{3} c - 4 \, a b c^{2}\right )} d^{5} x + {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )} d^{5}\right )} \log \left (c x^{2} + b x + a\right )}{c x^{2} + b x + a} \]
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Time = 0.99 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.38 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=16 b c^{2} d^{5} x + 16 c^{3} d^{5} x^{2} - 8 c d^{5} \cdot \left (4 a c - b^{2}\right ) \log {\left (a + b x + c x^{2} \right )} + \frac {- 16 a^{2} c^{2} d^{5} + 8 a b^{2} c d^{5} - b^{4} d^{5}}{a + b x + c x^{2}} \]
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Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.32 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=16 \, c^{3} d^{5} x^{2} + 16 \, b c^{2} d^{5} x + 8 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d^{5} \log \left (c x^{2} + b x + a\right ) - \frac {{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{5}}{c x^{2} + b x + a} \]
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Time = 0.35 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.54 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=8 \, {\left (b^{2} c d^{5} - 4 \, a c^{2} d^{5}\right )} \log \left (c x^{2} + b x + a\right ) - \frac {b^{4} d^{5} - 8 \, a b^{2} c d^{5} + 16 \, a^{2} c^{2} d^{5}}{c x^{2} + b x + a} + \frac {16 \, {\left (c^{7} d^{5} x^{2} + b c^{6} d^{5} x\right )}}{c^{4}} \]
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Time = 9.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.49 \[ \int \frac {(b d+2 c d x)^5}{\left (a+b x+c x^2\right )^2} \, dx=16\,c^3\,d^5\,x^2-\frac {16\,a^2\,c^2\,d^5-8\,a\,b^2\,c\,d^5+b^4\,d^5}{c\,x^2+b\,x+a}-\ln \left (c\,x^2+b\,x+a\right )\,\left (32\,a\,c^2\,d^5-8\,b^2\,c\,d^5\right )+16\,b\,c^2\,d^5\,x \]
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