Integrand size = 22, antiderivative size = 162 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{3 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \]
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Time = 0.05 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {712} \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 (d+e x)^{3/2} \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{3 e^5}-\frac {4 \sqrt {d+e x} (2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^5}-\frac {2 \left (a e^2-b d e+c d^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 c (d+e x)^{5/2} (2 c d-b e)}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \]
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Rule 712
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2-b d e+a e^2\right )^2}{e^4 (d+e x)^{3/2}}+\frac {2 (-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^4 \sqrt {d+e x}}+\frac {\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \sqrt {d+e x}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{3/2}}{e^4}+\frac {c^2 (d+e x)^{5/2}}{e^4}\right ) \, dx \\ & = -\frac {2 \left (c d^2-b d e+a e^2\right )^2}{e^5 \sqrt {d+e x}}-\frac {4 (2 c d-b e) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x}}{e^5}+\frac {2 \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) (d+e x)^{3/2}}{3 e^5}-\frac {4 c (2 c d-b e) (d+e x)^{5/2}}{5 e^5}+\frac {2 c^2 (d+e x)^{7/2}}{7 e^5} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {-6 c^2 \left (128 d^4+64 d^3 e x-16 d^2 e^2 x^2+8 d e^3 x^3-5 e^4 x^4\right )-70 e^2 \left (3 a^2 e^2-6 a b e (2 d+e x)+b^2 \left (8 d^2+4 d e x-e^2 x^2\right )\right )+28 c e \left (5 a e \left (-8 d^2-4 d e x+e^2 x^2\right )+3 b \left (16 d^3+8 d^2 e x-2 d e^2 x^2+e^3 x^3\right )\right )}{105 e^5 \sqrt {d+e x}} \]
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.91
method | result | size |
pseudoelliptic | \(\frac {\left (30 c^{2} x^{4}+\left (84 b \,x^{3}+140 x^{2} a \right ) c +70 b^{2} x^{2}+420 a b x -210 a^{2}\right ) e^{4}+840 \left (-\frac {2 c^{2} x^{3}}{35}+\left (-\frac {1}{5} b \,x^{2}-\frac {2}{3} a x \right ) c +b \left (-\frac {b x}{3}+a \right )\right ) d \,e^{3}-1120 \left (-\frac {3 c^{2} x^{2}}{35}+\left (-\frac {3 b x}{5}+a \right ) c +\frac {b^{2}}{2}\right ) d^{2} e^{2}+1344 \left (-\frac {2 c x}{7}+b \right ) c \,d^{3} e -768 c^{2} d^{4}}{105 \sqrt {e x +d}\, e^{5}}\) | \(147\) |
risch | \(\frac {2 \left (15 c^{2} x^{3} e^{3}+42 b c \,e^{3} x^{2}-39 c^{2} d \,e^{2} x^{2}+70 a c \,e^{3} x +35 b^{2} e^{3} x -126 b c d \,e^{2} x +87 c^{2} d^{2} e x +210 a b \,e^{3}-350 a c d \,e^{2}-175 b^{2} d \,e^{2}+462 b c \,d^{2} e -279 c^{2} d^{3}\right ) \sqrt {e x +d}}{105 e^{5}}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{e^{5} \sqrt {e x +d}}\) | \(187\) |
gosper | \(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-70 a c \,e^{4} x^{2}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}-210 a b \,e^{4} x +280 a c d \,e^{3} x +140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}-420 a b d \,e^{3}+560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\) | \(194\) |
trager | \(-\frac {2 \left (-15 c^{2} x^{4} e^{4}-42 b c \,e^{4} x^{3}+24 c^{2} d \,e^{3} x^{3}-70 a c \,e^{4} x^{2}-35 b^{2} e^{4} x^{2}+84 b c d \,e^{3} x^{2}-48 c^{2} d^{2} e^{2} x^{2}-210 a b \,e^{4} x +280 a c d \,e^{3} x +140 b^{2} d \,e^{3} x -336 b c \,d^{2} e^{2} x +192 c^{2} d^{3} e x +105 a^{2} e^{4}-420 a b d \,e^{3}+560 a c \,d^{2} e^{2}+280 b^{2} d^{2} e^{2}-672 b c \,d^{3} e +384 c^{2} d^{4}\right )}{105 \sqrt {e x +d}\, e^{5}}\) | \(194\) |
derivativedivides | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a b \,e^{3} \sqrt {e x +d}-8 a c d \,e^{2} \sqrt {e x +d}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(236\) |
default | \(\frac {\frac {2 c^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {4 b c e \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {8 c^{2} d \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {4 a c \,e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {2 b^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-4 b c d e \left (e x +d \right )^{\frac {3}{2}}+4 c^{2} d^{2} \left (e x +d \right )^{\frac {3}{2}}+4 a b \,e^{3} \sqrt {e x +d}-8 a c d \,e^{2} \sqrt {e x +d}-4 b^{2} d \,e^{2} \sqrt {e x +d}+12 b c \,d^{2} e \sqrt {e x +d}-8 c^{2} d^{3} \sqrt {e x +d}-\frac {2 \left (a^{2} e^{4}-2 a b d \,e^{3}+2 a c \,d^{2} e^{2}+b^{2} d^{2} e^{2}-2 b c \,d^{3} e +c^{2} d^{4}\right )}{\sqrt {e x +d}}}{e^{5}}\) | \(236\) |
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Time = 0.26 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (15 \, c^{2} e^{4} x^{4} - 384 \, c^{2} d^{4} + 672 \, b c d^{3} e + 420 \, a b d e^{3} - 105 \, a^{2} e^{4} - 280 \, {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2} - 6 \, {\left (4 \, c^{2} d e^{3} - 7 \, b c e^{4}\right )} x^{3} + {\left (48 \, c^{2} d^{2} e^{2} - 84 \, b c d e^{3} + 35 \, {\left (b^{2} + 2 \, a c\right )} e^{4}\right )} x^{2} - 2 \, {\left (96 \, c^{2} d^{3} e - 168 \, b c d^{2} e^{2} - 105 \, a b e^{4} + 70 \, {\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} x\right )} \sqrt {e x + d}}{105 \, {\left (e^{6} x + d e^{5}\right )}} \]
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Time = 4.62 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.45 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (\frac {c^{2} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (2 b c e - 4 c^{2} d\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right )}{3 e^{4}} + \frac {\sqrt {d + e x} \left (2 a b e^{3} - 4 a c d e^{2} - 2 b^{2} d e^{2} + 6 b c d^{2} e - 4 c^{2} d^{3}\right )}{e^{4}} - \frac {\left (a e^{2} - b d e + c d^{2}\right )^{2}}{e^{4} \sqrt {d + e x}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{2} x + a b x^{2} + \frac {b c x^{4}}{2} + \frac {c^{2} x^{5}}{5} + \frac {x^{3} \cdot \left (2 a c + b^{2}\right )}{3}}{d^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} - 42 \, {\left (2 \, c^{2} d - b c e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 35 \, {\left (6 \, c^{2} d^{2} - 6 \, b c d e + {\left (b^{2} + 2 \, a c\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 210 \, {\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} + {\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} \sqrt {e x + d}}{e^{4}} - \frac {105 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} + {\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )}}{\sqrt {e x + d} e^{4}}\right )}}{105 \, e} \]
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Time = 0.28 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.59 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}}{\sqrt {e x + d} e^{5}} + \frac {2 \, {\left (15 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} e^{30} - 84 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d e^{30} + 210 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{2} e^{30} - 420 \, \sqrt {e x + d} c^{2} d^{3} e^{30} + 42 \, {\left (e x + d\right )}^{\frac {5}{2}} b c e^{31} - 210 \, {\left (e x + d\right )}^{\frac {3}{2}} b c d e^{31} + 630 \, \sqrt {e x + d} b c d^{2} e^{31} + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{2} e^{32} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a c e^{32} - 210 \, \sqrt {e x + d} b^{2} d e^{32} - 420 \, \sqrt {e x + d} a c d e^{32} + 210 \, \sqrt {e x + d} a b e^{33}\right )}}{105 \, e^{35}} \]
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Time = 0.05 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.14 \[ \int \frac {\left (a+b x+c x^2\right )^2}{(d+e x)^{3/2}} \, dx=\frac {2\,c^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^5}+\frac {{\left (d+e\,x\right )}^{3/2}\,\left (2\,b^2\,e^2-12\,b\,c\,d\,e+12\,c^2\,d^2+4\,a\,c\,e^2\right )}{3\,e^5}-\frac {2\,a^2\,e^4-4\,a\,b\,d\,e^3+4\,a\,c\,d^2\,e^2+2\,b^2\,d^2\,e^2-4\,b\,c\,d^3\,e+2\,c^2\,d^4}{e^5\,\sqrt {d+e\,x}}-\frac {\left (8\,c^2\,d-4\,b\,c\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^5}+\frac {4\,\left (b\,e-2\,c\,d\right )\,\sqrt {d+e\,x}\,\left (c\,d^2-b\,d\,e+a\,e^2\right )}{e^5} \]
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