Integrand size = 26, antiderivative size = 39 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {696} \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 696
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 d^2 (b+2 c x)^3}{3 \left (b^2-4 a c\right ) (a+x (b+c x))^{3/2}} \]
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Time = 2.53 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.97
method | result | size |
gosper | \(\frac {2 \left (2 c x +b \right )^{3} d^{2}}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(38\) |
trager | \(\frac {2 d^{2} \left (8 c^{3} x^{3}+12 b \,c^{2} x^{2}+6 b^{2} c x +b^{3}\right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )}\) | \(58\) |
default | \(d^{2} \left (b^{2} \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+4 c^{2} \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )+4 b c \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\right )\) | \(362\) |
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Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (35) = 70\).
Time = 0.43 (sec) , antiderivative size = 146, normalized size of antiderivative = 3.74 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (8 \, c^{3} d^{2} x^{3} + 12 \, b c^{2} d^{2} x^{2} + 6 \, b^{2} c d^{2} x + b^{3} d^{2}\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left ({\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{4} + a^{2} b^{2} - 4 \, a^{3} c + 2 \, {\left (b^{3} c - 4 \, a b c^{2}\right )} x^{3} + {\left (b^{4} - 2 \, a b^{2} c - 8 \, a^{2} c^{2}\right )} x^{2} + 2 \, {\left (a b^{3} - 4 \, a^{2} b c\right )} x\right )}} \]
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Timed out. \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 195 vs. \(2 (35) = 70\).
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 5.00 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \, {\left (2 \, {\left (2 \, {\left (\frac {2 \, {\left (b^{2} c^{3} d^{2} - 4 \, a c^{4} d^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (b^{3} c^{2} d^{2} - 4 \, a b c^{3} d^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (b^{4} c d^{2} - 4 \, a b^{2} c^{2} d^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {b^{5} d^{2} - 4 \, a b^{3} c d^{2}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
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Time = 9.61 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.72 \[ \int \frac {(b d+2 c d x)^2}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2\,b^3\,d^2+12\,b^2\,c\,d^2\,x+24\,b\,c^2\,d^2\,x^2+16\,c^3\,d^2\,x^3}{\left (12\,a\,c-3\,b^2\right )\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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