Integrand size = 20, antiderivative size = 91 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {652, 627} \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {8 (b+2 c x) (2 c d-b e)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (-2 a e+x (2 c d-b e)+b d)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rule 627
Rule 652
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {(4 (2 c d-b e)) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )} \\ & = -\frac {2 (b d-2 a e+(2 c d-b e) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (2 c d-b e) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \\ \end{align*}
Time = 0.81 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (b^3 (d+3 e x)+8 c \left (a^2 e-3 a c d x-2 c^2 d x^3\right )+4 b c \left (-3 a (d-e x)+2 c x^2 (-3 d+e x)\right )+2 b^2 (a e+3 c x (-d+2 e x))\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \]
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Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.34
method | result | size |
trager | \(-\frac {2 \left (8 b \,c^{2} e \,x^{3}-16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+12 a b c e x -24 a \,c^{2} d x +3 b^{3} e x -6 b^{2} c d x +8 a^{2} c e +2 a \,b^{2} e -12 a b c d +b^{3} d \right )}{3 \left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(122\) |
gosper | \(-\frac {2 \left (8 b \,c^{2} e \,x^{3}-16 c^{3} d \,x^{3}+12 b^{2} c e \,x^{2}-24 b \,c^{2} d \,x^{2}+12 a b c e x -24 a \,c^{2} d x +3 b^{3} e x -6 b^{2} c d x +8 a^{2} c e +2 a \,b^{2} e -12 a b c d +b^{3} d \right )}{3 \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )}\) | \(131\) |
default | \(d \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )+e \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(162\) |
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Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (83) = 166\).
Time = 0.72 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.71 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (8 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x^{3} + 12 \, {\left (2 \, b c^{2} d - b^{2} c e\right )} x^{2} - {\left (b^{3} - 12 \, a b c\right )} d - 2 \, {\left (a b^{2} + 4 \, a^{2} c\right )} e + 3 \, {\left (2 \, {\left (b^{2} c + 4 \, a c^{2}\right )} d - {\left (b^{3} + 4 \, a b c\right )} e\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \]
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Timed out. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (83) = 166\).
Time = 0.29 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.16 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left (4 \, {\left (\frac {2 \, {\left (2 \, c^{3} d - b c^{2} e\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (2 \, b c^{2} d - b^{2} c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (2 \, b^{2} c d + 8 \, a c^{2} d - b^{3} e - 4 \, a b c e\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {b^{3} d - 12 \, a b c d + 2 \, a b^{2} e + 8 \, a^{2} c e}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]
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Time = 10.15 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.33 \[ \int \frac {d+e x}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2\,\left (8\,e\,a^2\,c+2\,e\,a\,b^2+12\,e\,a\,b\,c\,x-12\,d\,a\,b\,c-24\,d\,a\,c^2\,x+3\,e\,b^3\,x+d\,b^3+12\,e\,b^2\,c\,x^2-6\,d\,b^2\,c\,x+8\,e\,b\,c^2\,x^3-24\,d\,b\,c^2\,x^2-16\,d\,c^3\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \]
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